Applied Mathematics Seminar
Maurizio Graselli
Politecnico Milano
Phase Separation and Its Mathematics
Thursday, October 17, 2024, 11:00am
Ungar 406
Abstract: Phase separation (or segregation), namely, the creation of two (or more) distinct phases from a single homogeneous mixture (think of oil and vinegar), is a phenomenon which characterizes many important processes. In particular, it has recently become a paradigm in Cell Biology. Phase separation arises from the competition between the mixing entropy of the mixture and the demixing effects due to the reciprocal attraction of the components of the same species. This requires a temperature below some critical threshold that depends on the mixture itself. A suitable free energy functional, whose variables are the concentration of the mixture components, accounts for the competing terms. The evolution is then governed by the corresponding conserved gradient flow which is the well-known and widely studied Cahn-Hilliard equation. In its original form, the Cahn-Hilliard equation is a fourth-order nonlinear partial differential equation which was proposed in 1958 to model phase separation in binary alloys. Since then, it has been used to describe, theoretically and numerically, many different processes characterized by phase separation. An alternative to the Cahn-Hilliard equation has been proposed by J. Rubinstein and P. Sternberg in 1992. This is a second-order reaction-diffusion equation of Allen-Cahn type with a nonlocal constraint which aims to preserve the total mass. This is the reason why it is usually called the conserved Allen-Cahn equation. Moreover, a nonlocal second-order version of the Cahn-Hilliard equation has rigorously been derived by G. Giacomin and J.L. Lebowitz in 1997. I would like to give a brief introduction to the modeling and an overview of some recent results obtained for the (local and nonlocal) Cahn-Hilliard equation as well as for the conserved Allen-Cahn equation. Further issues will also be mentioned.
IMSA Seminar
Rodolfo Aguilar
University of Miami
Non-commutative Hodge Theory
Thursday, September 12, 2024, 2:00pm
Online
Abstract: I will discuss twistor structures as described by Simpson.
Applied Mathematics Seminar
Simone De Reggi
University of Udine
Numerics for the Linear Stability Analysis of Structured Population Models
Tuesday, September 10, 2024, 11:00am
Ungar 406
Abstract: In structured population models, the dynamics depend on variables, called structures, representing individual traits (e.g., age, size, immunity, spatial position). These models are often formulated as (integro-) partial differential equations (with nonlocal boundary conditions). As a result, their evolution is considered on abstract spaces and, in order to assess the local stability of equilibria, one is typically led to investigate the spectrum of linear infinite-dimensional operators, a target that can rarely be achieved analytically. In this talk, I will present a general numerical approach, based on (pseudo) spectral discretization, to approximate those spectra. Numerical results confirming the general validity of the approach and applications to epidemiology will be discussed.
IMSA Reading Seminar
Rodolfo Aguilar
University of Miami
Infinitesimal Invariants of Hodge Structures
by Joe Harris
Monday, August 26, 2024, 1:00pm
Ungar 419
Abstract: I will start with a brief overview of what an infinitesimal variation of Hodge structure is following Harris.
Topology Seminar
Jiakai Li
Harvard University
(Real) Monopole Floer Homology for Webs and Foams
Friday, May 3, 2024, 11:00am
Ungar 402
Abstract: Webs are embedded trivalent graphs in the 3-sphere and foams are singular cobordisms between webs. In this talk, I will present a construction of monopole Floer homology for webs that is functorial under foam cobordisms, based on Kronheimer and Mrowka’s monopole Floer homology. The ingredients include real and orbifold Seiberg-Witten theory, which are necessary for dealing with Klein-four symmetries. The talk will focus on the setup of the theory and compare it with the instanton counterpart J^#.
Combinatorics Seminar
Michelle Wachs
University of Miami
On the Representation of the Symmetric Group on the Free Filippov Algebra
Monday, April 29, 2024, 5:00pm
Ungar 402
Abstract: The Filippov n-algebra is a natural n-ary generalization of Lie algebra that is of interest in elementary particle physics. It is also of interest in combinatorics because it yields representations of the symmetric group that generalize the well studied Lie representation. Our ultimate aim is to determine the multiplicities of the irreducible representations in the representation of the symmetric group on the multilinear component of the free Filippov n-algebra with k brackets. This had been done for the ordinary Lie representation (n=2 case) by Kraskiewicz and Weyman. The k=2 case was handled in work with Friedmann, Hanlon, and Stanley. I will talk on continuing progress for general (n,k) obtained very recently with Friedman and Hanlon.
Probability Seminar
Shida Duan
University of Miami
Rank-Dependent Predictable Forward Performance Processes
Friday, April 26, 2024, 2:30pm
Ungar 402
Abstract: Predictable forward performance processes (PFPPs) are stochastic optimal control frameworks for an agent who controls a randomly evolving system but can only prescribe the system dynamics for a short period ahead. This is a common scenario in which a controlling agent frequently re-calibrates her model. In this talk, I will introduce a new class of PFPPs based on rank-dependent utility, generalizing existing models that are based on the more restrictive expected utility theory (EUT). Under a conditionally complete market and exogenous probability distortion functions which are updated periodically, I will show that these rank-dependent PFPPs are constructed by repeatedly solving an integral equation which generalizes the integral equation obtained under EUT in the previous studies. I will then discuss a new approach for solving the integral equation via theory of Volterra equations. Finally, I present some numerical examples in the special case of conditionally complete Black-Scholes model.
This is joint work with Bahman Angoshtari.
Topology Seminar
Fan Ye
Harvard University
Towards Isomorphisms among Floer Homologies
Friday, April 26, 2024, 11:10am
Ungar 411
Abstract: Since Floer's work in 1988, various Floer homologies have been constructed for closed 3-manifolds, knots, and sutured manifolds. In 2008, Kronheimer-Mrowka proposed a conjecture about isomorphisms among Floer homologies. In this talk, I will first introduce the history of the constructions in Floer theory and then introduce an approach to proving the isomorphisms. The idea is based on combinatorial version of Floer homology, which leads to an axiomatic construction of Floer homology. This work is joint with Baldwin, Li, Sivek, and Zemke.
IMSA Seminar
Tobias Ekholm
Department of Mathematics
Centre for Geometry and Physics
Uppsala University
Skein Valued Curve Counts and Recursion Relations
Monday, April 22, 2024, 4:00pm
Ungar 528B
Abstract: Counting open holomorphic curves with Maslow zero Lagrangian boundary condition in Calabi-Yau threefolds by the values of their boundaries in the HOMFLYPT skein module of the Lagrangian removes wall-crossing and leads to deformation invariant counts. In non-compact cases, curves at infinity play an important role. We show in several basic case how the curves at infinity determines all curves in the form of recursion relations. For example, we find three simple polynomial equations in the skein of a torus that determines all coloured HOMFLY polynomials of the Hopf link.
Probability Seminar
Christian Keller
University of Central Florida
Path-dependent PDEs:
An Introduction and Some Recent Results
Friday, April 19, 2024, 2:30pm
Ungar 402
Abstract: Path-dependent PDEs are PDEs defined on path spaces consisting of continuous or cadlag functions. Those PDEs appear naturally in non-Markovian problems in optimal control, probability, and mathematical finance, for example, optimal control of delay equations or pricing of path-dependent options.
In this talk, I will give an overview of the theory of viscosity solutions of path-dependent PDEs and I will present a new probabilistic approach for establishing uniqueness.
IMSA Seminar
Andrew Harder
Lehigh University
Hodge Numbers of Orbifold Clarke Mirrors and Mirror Symmetry
Thursday, April 18, 2024, 5:30pm
Ungar 528B
Abstract: Batyrev and Borisov gave a combinatorial mirror construction of dual pairs of toric complete intersections Calabi-Yau varieties coming from nef partitions of reflexive polytopes, and in 1996 they showed that the stringy Hodge numbers of these pairs satisfy a particular duality predicted by mirror symmetry. Later, Clarke gave a far-reaching combinatorial mirror construction which generalizes the Batyrev-Borisov construction and many other combinatorial mirror constructions.
I'll describe work in progress with Sukjoo Lee (Edinburgh) that uses new tools from tropical geometry to prove Hodge number duality for a large class of Clarke mirror pairs. This immediately recovers Batyrev and Borisov's results and leads to a proof of a conjecture of Katzarkov, Kontsevich, and Pantev for orbifold toric complete intersections. I'll describe how a construction of Doran and Harder, relating Laurent polynomials and singular toric complete intersections, fits into the Clarke mirror framework. As a consequence, my results with Sukjoo lead to predictions related to the Fanosearch Program.
IMSA/Mathematical Finance Seminar
Zachary Feinstein
Stevens Institute of Technology
Axioms for Automated Market Makers and Decentralized Finance
Tuesday, April 9, 2024, 4:00pm
Ungar 528B
Abstract: Within this talk, we introduce an axiomatic framework for Automated Market Makers (AMMs). By imposing reasonable axioms on the underlying utility function, we are able to characterize the properties of the swap size of the assets and of the resulting pricing oracle. We will analyze many existing AMMs and show that the vast majority of them satisfy our axioms. We will also consider the question of fees and divergence loss. In doing so, we will propose a new fee structure so as to make the AMM indifferent to transaction splitting. Finally, we will propose a novel AMM that has nice analytical properties and provides a large range over which there is no divergence loss.
Topology Seminar
Zedan Liu
University of Miami
A Casson-Lin Type Invariant for Links
Friday, April 5, 2024, 12:10pm
Ungar 402
Abstract: The Casson–Lin invariant of a knot is a signed count of the conjugacy classes of irreducible SU(2) representations of the knot group with a fixed trace. It is known to equal half the equivariant signature of the knot. In 2019, Bénard and Conway generalized the Casson–Lin invariant to links of any number of components. For 2-component links of linking number 1, they proved a formula expressing their invariant in terms of the Cimasoni–Florens link signature. In this talk, we will generalize the Bénard–Conway formula to 2-component links with arbitrary linking numbers.
Combinatorics Seminar
Richard Stanley
University of Miami
Some Combinatorial Aspects of Cyclotomic Polynomials
Monday, April 1, 2024, 5:00pm
Ungar 402
Abstract: A theorem of MacMahon states that the number of partitions of n for which no part appears exactly once equals the number of partitions of n into parts congruent to 1 or -1 (mod 6). The key fact behind this identity is that the numerator of the rational function 1/(1-x) - x is a product of cyclotomic polynomials Φj(x) (the monic polynomial whose zeros are the primitive j-th roots of 1), in this case just the single cyclotomic polynomial Φ6(x).
We discuss how this argument can be generalized to produce other partition identities. There is a connection with numerical semigroups, i.e., submonoids M of N={0,1,2,…} (under addition) for which N-M is finite.
We show how the proof technique carries over to some other situations, such as counting certain polynomials over the finite field Fq (such as polynomials for which every irreducible factor has multiplicity at least two) and obtaining Dirichlet series generating functions for certain subsets of integers (such as integers for which every prime factor has multiplicity at least two).
Mathematical Finance/IMSA Seminar
Stephan Sturm
Worcester Polytechnic Institute
The Distribution Builder –
A Tool for Financial Decision Making in the FinTech Era
Wednesday, March 27, 2024, 4:00pm
Ungar 528B
Abstract: The era of FinTech heralds personalized financial decision making through tools such as robo-advising. Alas, the input of personal preferences needed to personalize decision making is difficult and existing methods lack robustness. Sharpe, Goldstein, Blythe and Johnson introduced with the distribution builder a powerful tool to directly solicit user preferences on the outcomes of investments that can be used as base from decision making. In this presentation we explain how the methodology of the distribution builder can be leveraged successfully from the original setting – portfolio optimization in complete markets – to a wide array of other situations: consumption, incomplete markets and the timing of asset sales. This talk is based on joint work with Carole Bernard, Peter Carr, Mauricio Elizalde Mejia, Sixian Jin and Benjamin Rajotte.
Topology Seminar
Zhenkun Li
University of South Florida
Instanton Floer Homology and Heegaard Diagrams
Friday, March 22, 2024, 12:10pm
Ungar 402
Abstract: Floer theory is a power tool in the study of low dimensional topology, leading to many milestone results in the field. There are four major branches of Floer homologies, all of which have distinct features and applications. Among them, Heegaard Floer homology, monopole Floer homology, and embedded contact homology are known to be isomorphic, yet their relationship with Instanton Floer homology remains enigmatic. This talk will explore the connection between Instanton and Heegaard Floer homology. I will first present a result joint with Baldwin and Ye that illuminates some of the interplay between these two theories. Then, I will delve into ongoing research that further investigates these connections.
Geometric Analysis Seminar
Jesse Madnick
University of Oregon
The Morse Index of Quartic Minimal Hypersurfaces
Thursday, February 15, 2024, 4:00pm
Ungar 411
Abstract: Given a minimal hypersurface S in a round sphere, its Morse index is the number of variations that are area-decreasing to second order. In practice, computing the Morse index of a given minimal hypersurface is extremely difficult, requiring detailed information about the Laplace spectrum of S. Indeed, even for the simplest case in which S is homogeneous, the Morse index of S is not known in general.
In this talk, we compute the Morse index of two such minimal hypersurfaces. Moreover, we observe that their spectra contain (irrational) eigenvalues that are not expressible in radicals. Time permitting, we'll discuss some open problems and work-in-progress. This is joint work with Gavin Ball (Wisconsin) and Uwe Semmelmann (Stuttgart).
Combinatorics Seminar
Fabrizio Zanello
Michigan Technological University
A Broad Conjectural Framework for the Parity of Eta-quotients
Monday, Feburary 5, 2024, 5:00pm
Ungar 402
Abstract: One of the classical and most fascinating problems at the intersection between combinatorics and number theory is the study of the parity of the partition function. Even though p(n) is widely believed to be equidistributed modulo 2, progress in this area has always proven exceptionally hard. The best results available today, obtained incrementally over several decades by Serre, Soundararajan, Ono and many others, do not even guarantee that, asymptotically, p(n) is odd for √x values of n ≤ x.
In this talk, we present a new, general conjectural framework that naturally places the parity behavior of p(n) into the much broader, number-theoretic context of eta-quotients. We discuss the history of this problem as well as recent progress on our “master conjecture,” which includes novel results on multi- and regular partitions. We then show how seemingly unrelated classes of eta-quotients turn out to have surprising (and surprisingly deep) connections modulo 2. One instance is the following striking result: If any t-multipartition function, with t \≡ 0 (mod 3), is odd with positive density, then so is p(n). (Note that proving either fact unconditionally seems entirely out of reach with the existing technology.)
Throughout our talk, we will also give a feeling of the many interesting mathematical techniques that come into play in this area. They include a variety of algebraic and combinatorial methods, as well as tools from modular forms and number theory.
Much of the above work is in collaboration with my former Ph.D. student S. Judge or with W.J. Keith (see my papers in the J. Number Theory, 2015, 2018, 2021, 2022, and 2023; Annals of Comb., 2018; Int. J. Number Theory, 2021 and 2023).
Mathematical Finance Seminar
Oleksii Mostovyi
University of Connecticut
On Perturbations of Preferences and Indifference Price Invariance
Friday, December 1, 2023, 3:00pm
Ungar 411
Abstract: We investigate indifference pricing under perturbations of preferences in small and large markets. We establish stability results for small perturbations of preferences, where the latter can be stochastic. We obtain a sharp condition in terms of the associated concave and convex envelopes and provide counterexamples demonstrating that, in general, stability fails. Next, we investigate a class of models where the indifference price does not depend on the preferences or the initial wealth. Here, under the existence of an equivalent separating measure, in the settings of deterministic preferences, we show that the class of indifference price invariant models is the class of models where the dual domain is stochastically dominant of the second order. We also provide a counterexample showing that, in general, this result does not hold over stochastic preferences, where instead, we show that the indifference price invariant models are complete models (in both small and large markets). In the process, we establish a theorem of independent interest on the stability of the optimal investment problem under perturbations of preferences. Our results are new in both small and large markets, and thus, in particular, we introduce large stochastically dominant models, give examples of such models, and characterize them as the indifference price invariant ones over deterministic preferences.
CANCELLED
IMSA Reading Seminar
Rodolfo Aguilar
Sebastián Torres
University of Miami
On the Geometry of Anticanonical Pairs
by R. Friedman
Friday, November 17, 2023, 2:30pm
Ungar 528B
Combinatorics Seminar
Alex Lazar
Université Libre de Bruxelles
q-Counting Set-Valued Tableaux
Monday, November 13, 2023, 5:00pm
Ungar Room 402
Abstract: A set-valued tableau is a filling of a Ferrers diagram with nonempty sets of positive integers. These objects generalize Young tableaux and arise naturally in algebraic geometry. In this talk I will present some joint work with Sam Hopkins (Howard University) and Svante Linusson (KTH) which considers the surprisingly intricate q-enumeration of some of these tableaux in the case of the a x b rectangle, as well as some further work with Linusson in which we study a larger class of tableaux in the two-row case.
IMSA Reading Seminar
Rodolfo Aguilar
Sebastián Torres
University of Miami
On the Geometry of Anticanonical Pairs
by R. Friedman
Friday, November 10, 2023, 2:30pm
Ungar 528B
IMSA Reading Seminar
Rodolfo Aguilar
Sebastián Torres
University of Miami
On the Geometry of Anticanonical Pairs
by R. Friedman
Friday, November 3, 2023, 2:30pm
Ungar 528B
Probability Seminar
Stanislav Volkov
Centre for Mathematical Sciences
Jante's Law and Interacting Particle System
Monday, October 30, 2023, 1:30pm
Ungar 411
Abstract: Equality is a cornerstone of the Swedish culture. "We’re all the same" mentality originates from the pan-Scandinavian concept of the so-called Jante Law, the ten "commandments" derived from a 1933 Aksel Sandemose's book "A Fugitive Crosses His Tracks". For many people who live in Scandinavia, these rules are subconsciously a part of their everyday life. No one should be seen different from the "collective identity". This "law" has led to an egalitarian society with higher levels of humbleness than most other places.
In my talk, based on a series of papers, including the most recent one, I will describe some interacting stochastic particle models, which emulate the development of such societies, and the convergence to the common "mean".
IMSA Seminar
Ludmil Katzarkov
University of Miami
Milnor Spheres – 60 Years Later
Friday, October 27, 2023, 5:00pm
Ungar 528B
Abstract: Recently we were able to look at this classical differential topology result from the prospect of category theory. In this talk I will report our findings. Joint work Dr. Lino Grama and Dr. Leonardo Cavenaghi.
IMSA Reading Seminar
Rodolfo Aguilar
Sebastián Torres
University of Miami
On the Geometry of Anticanonical Pairs
by R. Friedman
Friday, October 27, 2023, 2:30pm
Ungar 528B
Geometric Analysis Seminar
Marcelo Disconzi
Vanderbilt University
The Relativistic Euler Equations with a Physical Vacuum Boundary
Friday, October 20, 2023, 4:00pm
Ungar 402
Abstract: We consider the relativistic Euler equations with a physical vacuum boundary and an equation of state $p(\rho) = \rho^\gamma$, $\gamma > 1$. We establish the following results. (i) local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and continuous dependence on the data; (ii) low regularity solutions: our uniqueness result holds at the level of Lipschitz velocity and density, while our rough solutions, obtained as unique limits of smooth solutions, have regularity near scaling; (iii) stability: our uniqueness in fact follows from a more general result, namely, we show that a certain nonlinear functional that tracks the distance between two solutions (in part by measuring the distance between their respective boundaries) is propagated by the flow; (iv) we establish sharp, essentially scale invariant energy estimates for solutions; (v) we establish a sharp continuation criterion, at the level of scaling, showing that solutions can be continued as long as the velocity is in $L^1_t Lip_x$ and a suitable weighted version of the density is at the same regularity level. This is joint work with Mihaela Ifrim and Daniel Tataru.
IMSA Reading Seminar
Rodolfo Aguilar
Sebastián Torres
University of Miami
On the Geometry of Anticanonical Pairs
by R. Friedman
Friday, October 20, 2023, 2:30pm
Ungar 528B
Geometric Analysis Seminar
Abraão Mendes
Universidade Federal de Alagoas
Rigidity of Min-max Minimal Disks in 3-balls with Non-negative Ricci Curvature
Friday, October 13, 2023, 4:00pm
Ungar 402
Abstract: In this lecture, we are going to present a rigidity statement for free boundary minimal surfaces produced via min-max methods. More precisely, for any Riemannian metric $g$ on the 3-ball $B$ with non-negative Ricci curvature and $\mbox{II}_{\partial B}\ge g_{|\partial B}$, there exists a free boundary minimal disk $\Delta$ of least area among all free boundary minimal disks in $(B, g)$. Moreover, the area of any such $\Delta$ equals to the width of $(B, g)$, $\Delta$ has index one, and the length of $\partial\Delta$ is bounded from above by $2\pi$. Furthermore, the length of $\partial\Delta$ equals to $2\pi$ if and only if $(B, g)$ is isometric to the Euclidean unit ball. This is related to a rigidity result obtained by F.C. Marques and A. Neves in the closed case. The proof uses a rigidity statement concerning half-balls with non-negative Ricci curvature which is true in any dimension.
IMSA Reading Seminar
Rodolfo Aguilar
Sebastián Torres
University of Miami
On the Geometry of Anticanonical Pairs
by R. Friedman
Friday, October 13, 2023, 2:30pm
Ungar 528B
IMSA Reading Seminar
Rodolfo Aguilar
Sebastián Torres
University of Miami
On the Geometry of Anticanonical Pairs
by R. Friedman
Friday, October 6, 2023, 2:30pm
Ungar 528B
IMSA Reading Seminar
Rodolfo Aguilar
Sebastián Torres
University of Miami
On the Geometry of Anticanonical Pairs
by R. Friedman
Friday, September 29, 2023, 2:30pm
Ungar 528B
IMSA Seminar
Ludmil Katzarkov
University of Miami
Further Invariants
Wednesday, September 27, 2023, 5:50pm
Ungar 528B
IMSA Seminar
Kaiqi Yang
University of Miami
Birational Invariants for Singular Fano over Algebraically Non-closed Fields
Wednesday, September 27, 2023, 5:00pm
Ungar 528B
IMSA Reading Seminar
Rodolfo Aguilar
Sebastián Torres
University of Miami
On the Geometry of Anticanonical Pairs
by R. Friedman
Friday, September 22, 2023, 3:00pm
Ungar 528B
IMSA Seminar
Rodolfo Aguilar
University of Miami
Recent Developments on the Shafarevich Conjecture
Wednesday, September 6, 2023, 5:00pm
Ungar 528B
Abstract: I will give a brief survey on recent works around the Shafarevich conjecture on holomorphic convexity of the universal cover of quasi-projective varieties. My main focus will be the nilpotent case and contributions of Green-Griffiths-Katzarkov and Aguilar-Campana. Some tools of the proof will be given. Time permitting, I will give some comments on the reductive case.
IMSA Seminar
René Mboro
University of Miami
Remarks on Lagrangian Submanifolds of Hyper-Kähler Manifolds
Wednesday, August 30, 2023, 5:00pm
Ungar 528B
Abstract: We will present some properties of Lagrangian subvarieties of irreducible symplectic (or hyper-Kähler) manifolds. We will focus on their interaction with a Lagrangian fibrations of the ambient hyper-Kähler and the albanese dimension of Lagrangian surfaces.
IMSA Seminar
Ludmil Katzarkov
University of Miami
A Parallel Reality Look at Generalized Geometry
Wednesday, August 23, 2023, 5:00pm
Ungar 528B
Abstract: We begin by recalling classical works of Donaldson and Simpson. Based on these works we take a new look at generalized geometries and "Hodge theory" connected with them.
Applied Math Seminar
Brian Coomes
University of Miami
Homoclinic Chaos in a Perennial Grass Model
Tuesday, May 2, 2023, 2:00pm
Ungar 406
Abstract: The second-order difference equation x n+1 = ax n + (b+cx n-1)e -x n , where 0 < a < 1, 0 < b and 0 < c < 1, has been used to model the biomass of the generations of the perennial grass Agrostis scabra. The permanence of the solutions was proved. For certain parameter values in this model, and in the data from field studies, chaotic dynamics was conjectured. In this study, with the aid of a homoclinic shadowing theorem, we prove the existence of a transversal homoclinic orbit, hence the presence of chaotic dynamics, for certain parameter values of this second-order difference equation. This is joint work with Huseyin Kocak and Ken Palmer.
IMSA/Topology Seminar
Hans Boden
McMaster University
Virtual Knots and (Algebraic) Concordance
Wednesday, April 26, 2023, 10:30am
Ungar 528B
Abstract: The motivation for this talk is a desire to better understand the algebraic structure of the concordance group of virtual knots. This group is known to contain, as a proper subgroup, the concordance group of classical knots. It is also known, by results of Chrisman, to be nonabelian. Apart from that, it is quite mysterious. We will explain how to use the Gordon-Litherland pairing to, given a virtual knot with a spanning surface, associate a square integral matrix. For classical knots, the matrix is always symmetric but for virtual knots, that is no longer true. Using these so-called mock Seifert matrices, we introduce a new set of knot invariants (signatures, LT signatures, and Alexander polynomials) and explore their behavior under virtual concordance. Using the mock Seifert matrices, we introduce a new algebraic concordance group defined in terms of non-orientable spanning surfaces. Morally speaking, this group is a linear approximation to the mystery group, with linearization given by a virtual analogue of the Levine homomorphism. The group can be seen to be abelian and infinite rank. It also contains lots of 2- and 4-torsion. We will present some results and a few open problems. Everything is joint with Homayun Karimi.
Geometric Analysis Seminar
Sven Hirsch
Duke University
On a Generalization of Geroch's Conjecture
Tuesday, April 25, 2023, 4:00pm
Ungar 402
Abstract: The theorem of Bonnet-Myers implies that manifolds with topology $M^{n-1}\times S^1$ do not admit a metric of positive Ricci curvature, while the resolution of Geroch's conjecture shows that the torus $T^n$ does not admit a metric of positive scalar curvature. In this talk I will introduce a new notion of curvature which interpolates between Ricci and scalar curvature (so-called $m$-intermediate curvature) and use stable weighted slicings to show that for $n\le7$ the manifolds $M^{n-m}\times T^m$ do not admit a metric of positive $m$-intermediate curvature. This is joint work with Simon Brendle and Florian Johne.
Geometric Analysis Seminar
Abraão Mendes
Universidade Federal de Alagoas
Some Rigidity Results for Compact Initial Data Sets
Tuesday, April 18, 2023, 4:00pm
Online
Abstract: In this lecture, we aim to present some rigidity results for compact initial data sets, in both the boundary and no boundary cases. For example, under natural energy, boundary, and topological conditions, we obtain a global version of a well-known result of H. Bray, S. Brendle, and A. Neves. We also prove an extension of a result obtained, in a previous work, by M. Eichmair, G.J. Galloway, and the author. Finally, as time permits, we are going to present an example in order to illustrate the last result presented in this lecture. This is part of a joint work with G.J. Galloway.
Combinatorics Seminar
Lisa Seccia
Max Planck Institute for Mathematics in the Sciences
Connected Domination of Graphs and v-numbers of Binomial Edge Ideals
Monday, April 17, 2023, 5:00pm
Ungar Room 402
Abstract: The v-number of a graded ideal is an algebraic invariant introduced by Cooper et al., and originally motivated by problems in coding theory. In this talk, I will provide an overview of the state-of-the-art, with a specific emphasis on the relation between v-numbers and combinatorics. In particular, I will discuss the case of binomial edge ideals and I will show how their v-numbers can be used to give an algebraic description of the connected domination number of a graph. This is a joint work with Delio Jaramillo-Velez.
Topology/IMSA Seminar
Isacco Nonino
University of Glasgow
Tight Contact Structures on Hyperbolic 3-manifolds Constructed Using Dehn Surgery on GOF-knots in Lens Spaces II
Wednesday, April 12, 2023, 10:30am
Ungar 528B
Abstract: Continuing last week, I will discuss how to obtain the desired upper bound on the number of tight contact structures on certain hyperbolic 3-manifolds constructed by Dehn surgery on some GOF-knots in lens spaces.
Topology/IMSA Seminar
Isacco Nonino
University of Glasgow
Tight Contact Structures on Hyperbolic 3-manifolds Constructed Using Dehn Surgery on GOF-knots in Lens Spaces I
Wednesday, April 5, 2023, 10:30am
Ungar 528B
Abstract: The aim of my current research project is to classify and better understand tight contact structures on a large class of hyperbolic 3-manifolds. More precisely, these manifolds are constructed using Dehn surgery on genus one fibered knots in lens spaces. In this talk, I will introduce a first classification result, which determines the number of (non-isotopic) tight contact structures on a subclass of these manifolds. Moreover, I will also show how to immediately derive Stein fillability for some of these structures. The strategy of the proof will be outlined, together with some key facts that are helpful in this specific setting and some hopes for future developments and generalizations.
FIU & IMSA Joint Seminar
Anna Fino
Florida International University
Nilmanifolds and their Cohomologies
Tuesday, April 4, 2023, 6:00pm
Ungar 528B
Abstract: Nilmanifolds constitute a well-known class of compact manifolds providing interesting explicit examples of geometric structures with special properties. A nilmanifold is a compact quotient of a connected and simply connected nilpotent Lie group G by a lattice of maximal rank in G. Hence, any left-invariant geometric structure on G descends to the nilmanifold. We will refer to such structures as invariant. In the talk we will review cohomologies of nilmanifolds endowed with an invariant (generalized) complex structure.
IMSA Seminar
Charles Doran
University of Alberta
Motivic Geometry of Two-loop Feynman Integrals
Tuesday, April 4, 2023, 5:00pm
Ungar 528B
Abstract: We study the geometry and Hodge theory of the cubic hypersurfaces attached to two-loop Feynman integrals for generic physical parameters. We show that the Hodge structure attached to planar two-loop Feynman graphs decomposes into a mixed Tate piece and a variation of Hodge structure from families of hyperelliptic curves, elliptic curves, or rational curves depending on the space-time dimension. We give more precise results for two-loop graphs with a small number of edges. In particular, we recover a result of Spencer Bloch that in the well-known double box example there is an underlying family of elliptic curves, and we give a concrete description of these elliptic curves. We show that the motive for the "non-planar" two-loop tardigrade graph is that of a family of K3 surfaces of generic Picard number 11. Lastly, we show that generic members of the multi-scoop ice cream cone family of graph hypersurfaces correspond to pairs of multi-loop sunset Calabi-Yau varieties. Our geometric realization of these motives permits us in many cases to derive in full the homogeneous differential operators for the corresponding Feynman integrals. This is joint work with Andrew Harder and Pierre Vanhove.
Geometric Analysis Seminar
Eric Ling
University of Copenhagen
Some New Results and Open Questions Related to the Cosmological Constant Appearing in Inflationary Models
Tuesday, April 4, 2023, 4:00pm
Ungar 402
Abstract: The cosmological constant appears as an initial condition for Milne-like spacetimes, a class of hyperboloidal inflationary models. In previous work it was shown that this remains true even if one removes the homogeneous and isotropic assumptions for Milne-like spacetimes. However, examples of such nonhomogeneous versions of Milne-like spacetimes are lacking. In this talk, we suggest some ways of constructing an initial value problem for Milne-like spacetimes which could provide such examples. Lastly, in joint work with Ghazal Geshnizjani and Jerome Quintin, we show that the cosmological constant also appears as an initial condition for a class flat FLRW models which are asymptotically de Sitter towards the past.
Topology/IMSA Seminar
Lev Tovstopyat-Nelip
University of Georgia
Quasipositive Surfaces and Decomposable Lagrangians
Wednesday, March 29, 2023, 10:30am
Ungar 528B
Abstract: I'll explain how an invariant of Legendrian links in knot Floer homology can be used to obstruct the existence of decomposable Lagrangian link cobordisms in a very general setting. The argument involves braiding the ends of the link cobordism about open books and appealing to an algebraic property of the Legendrian invariant called comultiplication. Much of the talk will be spent describing the topological and contact geometric ingredients.
Applied Math Seminar
Ali Hagverdiyev
University of Miami
Optimal Control of Coefficients for the Second Order Parabolic Free Boundary Problems
Tuesday, March 28, 2023, 2:00pm br />Ungar 406
Abstract: In this talk I will discuss Inverse Stefan type free boundary problem for the second order parabolic equation arising in for instance, modeling of laser ablation of biomedical tissues, where the information on the coefficients, heat flux on the fixed boundary, and density of heat sources are missing and must be found along with the temperature and free boundary. New PDE constrained optimal control framework is employed, where the missing data and the free boundary are components of the control vector, and optimality criteria are based on the final moment measurement of the temperature and position of the free boundary. Some numerical results will be shared.
IMSA/Topology Seminar
Sinem Onaran
University of Miami
Contact Surgeries on Contact 3-manifolds
Wednesday, March 8, 2023, 10:30am
Ungar 528B
Abstract: In this talk, I will discuss contact surgeries. I will discuss the behavior of contact structures under contact (+1/n) and contact (+n)-surgeries along Legendrian knots. Then, I will focus on a question: which tight contact structures on a given lens space can be obtained by a single contact (-1)-surgery along a Legendrian knot in S^3 with some contact structure? (joint with Geiges) Further, I will discuss various versions of contact surgery numbers, the minimal number of components of a surgery link L describing a contact 3-manifold under consideration. (joint with Etnyre and Kegel)
Geometric Analysis Seminar
Conghan Dong
Stony Brook University
Stability of Euclidean 3-space for the Positive Mass Theorem
Tuesday, March 7, 2023, 4:00pm
Ungar 402
Abstract:The Positive Mass Theorem of R. Schoen and S.-T. Yau in dimension 3 states that if $(M^3, g)$ is asymptotically flat and has nonnegative scalar curvature, then its ADM mass $m(g)$ satisfies $m(g) \geq 0$, and equality holds only when $(M, g)$ is the flat Euclidean 3-space $\mathbb{R}^3$. We show that $\mathbb{R}^3$ is stable in the following sense. Let $(M^3_i, g_i)$ be a sequence of asymptotically flat 3-manifolds with nonnegative scalar curvature and suppose that $m(g_i)$ converges to 0. Then for all $i$, there is a domain $Z_i$ in $M_i$ such that the area of the boundary $\partial Z_i$ converges to zero and the sequence $(M_i \setminus Z_i , \hat{d}_{g_i} , p_i )$, with induced length metric $\hat{d}_{g_i}$ and any base point $p_i \in M_i \setminus Z_i$, converges to $\mathbb{R}^3$ in the pointed measured Gromov-Hausdorff topology. This confirms a conjecture of G. Huisken and T. Ilmanen. We also find an almost optimal bound for the area of $\partial Z_i$ in terms of $m(g_i)$. This is joint work with Antoine Song.
Applied Math Seminar
Bahman Angoshtari
University of Miami
Predictable Forward Performance Processes in Complete Markets
Tuesday, March 7, 2023, 2:00pm
Ungar 406
Abstract: Predictable Forward Performance Processes (PFPPs) are stochastic optimal control frameworks for an agent who controls a dynamically evolving system but can only prescribe the system dynamics for a short period ahead. This is a common scenario in which the controlling agent must re-calibrate her model for the underlying system periodically through time. PFPPs allow the agent to form time-consistent optimal policies over time horizons that span multiple estimation periods. In this talk, I prove the existence of PFPPs in complete markets and show that the main step in their construction is solving a one-period problem involving an integral equation. I will discuss a new solution method for this integral equation using the Fourier transform. For PFPPs with completely monotonic inverse marginal functions, the integral equation has a unique solution that is obtained in closed form.
IMSA/Topology Seminar
Feride Ceren Kose
University of Georgia
Symmetric Unions and Ribbon Knots
Wednesday, March 1, 2023, 10:30am
Ungar 528B
Abstract: A symmetric union of a knot is a classical construction in knot theory introduced in the 1950s by Kinoshita and Terasaka. Because of the flexibility in their construction and the fact that they are ribbon, hence smoothly slice, symmetric unions appear quite frequently in the literature. It is still unknown whether there exists a ribbon knot which cannot be presented as a symmetric union. Thus, similar to the Slice-Ribbon conjecture, one may ask whether every ribbon knot is a symmetric union. In this talk, I will classify the simplest type of symmetric unions that are composite, two-bridge, Montesinos, or amphichiral, and in doing so, give infinite families of ribbon knots that cannot admit the simplest type of symmetric union diagrams.
Applied Math Seminar
Don Olson
University of Miami
Finding Eigenvalues:
Openings and Closures for Land and Seascapes
Tuesday, February 28, 2:00pm
Ungar 406
Abstract: The problem of posing problems involving observed features in nature is discussed in terms of identification of state variables, parameters involved and formulations of differential equations (Opening a problem). Given an opening, the estimation of scales, i.e. eigenvalues, is reviewed. The issue of closing equations such that they lead to patterns observed is then addressed. Classical problems such as the Ekman layer in the ocean and atmosphere are extended to looking at problems such as reef formations and plate tectonics.
IMSA Seminar
Gueo Grantcharov
Florida International University
Generalized Calabi-Yau Problem
Thursday, February 23, 2023, 6:00pm
Ungar 528B
Abstract: This is a continuation of the previous talk on generalized complex geometry. We'll review the definitions of holomorphically trivial generalized canonical bundle, Calabi-Yau structure and Calabi-Yau problem. Then we'll report on its current status.
IMSA/Topology Seminar
Chris Scaduto
University of Miami
Skein Exact Triangles in Equivariant Singular Instanton Theory
Wednesday, February 22, 2023, 10:30am
Ungar 528B
Abstract: Given a knot or link in the 3-sphere, its Murasugi signature is an integer-valued invariant which can easily be computed from a diagram. Work of Herald and Lin gives an alternative description of knot signatures, as signed counts of SU(2)-representations of the knot group which are traceless around meridians. There is a version of singular instanton homology for links which categorifies the Murasugi signature. We construct unoriented skein exact triangles for these Floer groups, categorifying the behavior of the Murasugi signature under unoriented skein relations. More generally, we construct skein exact triangles in the setting of equivariant singular instanton theory. This is joint work with Ali Daemi.
Applied Math Seminar
Chris Cosner
University of Miami
Reaction-diffusion-advection Models with Multiple Movement Modes
Tuesday, February 21, 2023, 2:00pm
Ungar Room 406
Abstract: Classical reaction-diffusion-advection models for population dynamics with dispersal assume that all individuals move in the same way all the time. Actually, animals may switch between faster movement when searching for resources and slower movement while exploiting them, or juveniles may move differently than adults. To describe such situations requires systems of reaction-diffusion-advection equations. The resulting systems may have features different from single equations. In models based on logistic equations, the systems may be cooperative at low densities but competitive at high densities. It is well known that for a single diffusive logistic equation in a static spatially heterogeneous bounded domain, slower diffusion is advantageous. For stage structured populations where adults and juveniles have different environmental needs this is no longer always the case. This talk will describe some recent work on the theory and applications of reaction-diffusion-advection models for populations in bounded habitats where subpopulations may have different movement rates or patterns, and individuals can switch between subpopulations by behavior or contribute to them by reproduction and aging.
Combinatorics Seminar
Hsin-Chieh Liao
University of Miami
Chow Rings and Augmented Chow Rings of Uniform Matroids and their q-analogs
Monday, February 20, 2023, 5:00pm
Ungar Room 402
Abstract: Chow rings and augmented Chow rings of matroids play important roles in the celebrated proofs of two longstanding conjectures: (1) the Adiprasito-Huh-Katz proof of the Heron-Rota-Welsh Conjecture and (2) the Braden-Huh-Matherne-Proudfoot-Wang proof of the Dowling-Wilson Top-Heavy Conjecture. These two matroid invariants have since been extensively studied. In 2021, Hameister, Rao, and Simpson gave a nice combinatorial interpretation of the Hilbert series of the Chow ring of the (q-)uniform matroid in terms of permutations and the q-Eulerian polynomials studied by Shareshian and Wachs. We present an analogous interpretation for the augmented Chow ring in terms of partial permutations and q-binomial Eulerian polynomials.
Our proof relies on a Feichtner-Yuzvinsky type basis for the augmented Chow ring of a matroid (introduced in our previous work and in independent work of Eur, Mastroeni, Mccullough). This basis is also used to obtain closed form formulas for the Hilbert series of the augmented Chow ring of the uniform matroid evaluated at -1. These are analogous to our simplification of formulas of Hameister, Rao, and Simpson for the Chow ring. We also obtain symmetric function analogs of the above results.
IMSA Seminar
Ya Deng
Université de Lorraine
A More Comprehensible Proof of the Reductive Shafarevich Conjecture II
Friday, February 17, 2023, 5:00pm
Ungar 528B
Abstract: The Shafarevich conjecture is one of the most beautiful mathematical problems in complex geometry. It connects many different subjects, especially non-abelian Hodge theories. It was first proved by Katzarkov-Ramachandran in 1998 for surfaces with reductive linear fundamental groups. In 2004 Eyssidieux proved the reductive Shafarevich conjecture for projective varieties, and this result was an important building block in the later proof of the linear case by Eyssidieux-Katzarkov-Pantev-Ramachandran. In the two lectures I will explain the recent work on the new construction of Shafarevich morphisms, and the more comprehensible proof of the reductive Shafarevich conjecture. This is a joint work with Katsutoshi Yamanoi.
IMSA Seminar
Gueo Grantcharov
Florida International University
Generalized Kaehler and Bihermitian Structures
Thursday, February 16, 2023, 6:00pm
Online
Abstract: In the talk we'll review some of the properties of generalized Kaehler structures which are similar to the Kaehloer case. It includes analog of Kodaira's result on stability under small deformations, Calabi-Yau problem and period map.
IMSA Seminar
Ya Deng
Université de Lorraine
A More Comprehensible Proof of the Reductive Shafarevich Conjecture I
Thursday, February 16, 2023, 5:00pm
Ungar 528B
Abstract: The Shafarevich conjecture is one of the most beautiful mathematical problems in complex geometry. It connects many different subjects, especially non-abelian Hodge theories. It was first proved by Katzarkov-Ramachandran in 1998 for surfaces with reductive linear fundamental groups. In 2004 Eyssidieux proved the reductive Shafarevich conjecture for projective varieties, and this result was an important building block in the later proof of the linear case by Eyssidieux-Katzarkov-Pantev-Ramachandran. In the two lectures I will explain the recent work on the new construction of Shafarevich morphisms, and the more comprehensible proof of the reductive Shafarevich conjecture. This is a joint work with Katsutoshi Yamanoi.
IMSA Seminar
Vladmir Baranovsky
University of California, Irvine
Mapping Spaces and E_2 Algebras
Thursday, February 16, 2023, 4:00pm
Online
Abstract: In a 1991 paper, Bendersky and Gitler have constructed a spectral sequence converging to the cohomology of a mapping space Maps(K, Y) where K is a simplicial set and Y is a space, such as a smooth compact manifold. The E_1 term of that spectral sequence involves chains on configuration spaces of K and cochains on Cartesian powers of Y. We assume that K is a graph (or rather its ribbon thickening) and explain a conjecture that expresses the differential of the E_1 term via standard "surjection operations" on the cochains of Y. One application is a theorem asserting that the cohomology of Maps(K, Y) may be approximated by the cohomology of Cartesian powers of Y with certain double diagonals removed. When cochains of Y are replaced by an E_2 algebra or a category with appropriate structure, one expects similar results involving factorization homology of the 2-dimensional ribbon thickening of K.
IMSA/Topology Seminar
Danny Ruberman
Brandeis University
Diffeomorphism Groups of 4-manifolds
Wednesday, February 15, 2023, 10:30am
Ungar 528B
Abstract: A phenomenon that is unique to dimension 4 is the existence of infinite families of manifolds that are homeomorphic but not diffeomorphic. This is shown via a combination of gauge theory (Seiberg-Witten theory or Yang-Mills theory) with Freedman’s topological classification results. In a joint project with Dave Auckly, we find similar `exotic’ behavior comparing the topology of the groups of diffeomorphisms and homeomorphisms of a smooth 4-manifold. Our main theorem is that the kernel of the map on homotopy groups induced by the inclusion Diff(X) -> Homeo(X) can be infinitely generated. The same techniques yield similar results about spaces of embeddings of surfaces and 3-manifolds in 4-manifolds.
Geometric Analysis Seminar
Xu Cheng
Instituto de Matematica e Estatistica
Universidade Federal Fluminense
Volume of Hypersurfaces in Rn with Bounded Weighted Mean Curvature
Tuesday, February 14, 2023, 4:00pm
Ungar 402
Abstract: In this talk, we will discuss the volume property of complete noncompact submanifolds in a gradient shrinking Ricci soliton with bounded weighted mean curvature vector. Roughly speaking, such a submanifold must have polynomial and at least linear volume growth. An example is properly immersed complete noncompact hypersurfaces in Rn with bounded Gaussian-weighted mean curvature, including self-shrinkers. This is a joint work with M. Vieira and D. Zhou.
IMSA/Topology Seminar
Kenneth L. Baker
University of Miami
Handle Numbers of Nearly Fibered Knots
Wednesday, February 1, 2023, 10:30am
Ungar Room 528B
Abstract: In the Instanton and Heegaard Floer theories, a nearly fibered knot is one for which the top grading has rank 2.
Sivek-Baldwin and Li-Ye showed that the guts (ie. the reduced sutured manifold complement) of a minimal genus Seifert surface of a nearly fibered knot has of one of three simple types.
We show that nearly fibered knots with guts of two of these types have handle number 2 while those with guts of the third type have handle number 4. Furthermore, we show that nearly fibered knots have unique incompressible Seifert surfaces rather than just unique minimal genus Siefert surfaces.
Geometric Analysis Seminar
Detang Zhou
Instituto de Matematica e Estatistica
Universidade Federal Fluminense
Rigidity of 4-dimensional Shrinking Ricci Solitons
Tuesday, January 31, 2023, 4:00pm
Ungar Room 402
Abstract: Perelman defined his W-functional and proved the entropy monotonicity formulae for Hamilton's Ricci flow. The critical points of W-functional are shrinking gradient Ricci solitons (SGRS). It is well known that gradient Ricci solitons are generalizations of Einstein manifolds and basic models for smooth metric measure spaces. In this talk I will discuss some recent progress and problems in four dimensional cases. In particular, one of the challenging problems is to classify all gradient Ricci solitons with constant scalar curvature. Recently in a joint work with X. Cheng, we prove that a 4-dimensional shrinking gradient Ricci soliton has constant scalar curvature if and only if it is either Einstein, or a finite quotient of Gaussian shrinking soliton $\mathbb{R}^4$, $\mathbb{S}^2×\mathbb{R}^2^$ or $\mathbb{S}^3×\mathbb{R}$.
Combinatorics Seminar
Michelle Wachs
University of Miami
On q-unimodality of Generalized Gaussian Coefficients and LLT Polynomials
Monday, January 30, 2023, 5:00pm
Ungar Room 402
Abstract: One of the well-known combinatorial interpretations of the Gaussian coefficients (or q-binomial coefficients) involves counting binary words by their number of inversions. Here we consider a generalization of the Gaussian coefficients obtained from a permutation statistic that interpolates between the descent number and the inversion number. We use a result of Grojnowski and Haiman on Schur-positivity of LLT polynomials to prove that the generalized Gaussian coefficients form a q-unimodal sequence. This is based on joint work with Yuval Roichman.
IMSA Seminar
Tristan Collins
Massachusetts Institute of Technology
Complete Calabi-Yau Metrics on the Complement of Two Divisors
Friday, January 20, 2023, 4:00pm
Online
Abstract: In 1990 Tian-Yau proved that if Y is a Fano manifold and D is a smooth anti-canonical divisor, the complement X=Y\D admits a complete Calabi-Yau metric. A long standing problem has been to understand the existence of Calabi-Yau metrics when D is singular. I will discuss the resolution of this problem when D=D_1+D_2 has two components and simple normal crossings. I will also explain a general picture which suggests the case of general SNC divisors should be inductive on the number of components. This is joint work with Y. Li.
IMSA Seminar
Bruno de Oliveira
University of Miami
Surface Quotient Singularities and Big Cotangent Bundle
Thursday, December 22, 2022, 11:00am
Online
Abstract: Surfaces with big cotangent bundle have hyperbolic properties, e.g. they satisfy the Green-Griffiths-Lang conjecture. The GGL-conjecture states that a variety of general type X has a proper subvariety Z such that all entire curves of X are contained in Z. We present a bigness criterion for resolutions of orbifold surfaces and obtain as a corollary the canonical model singularities criterion that can be applied to all surfaces of general type. We describe how the CMS-criterion improves upon other known criteria, such as the Rouseau-Rolleau criterion. The CMS-criterion involves an analytical based invariant of surface singularities that we calculate for A_n singularities. We then apply this criterion to the problem of finding what are the degrees $d$ (what is the minimal such d?) for which the deformation equivalence class of a smooth hypersurface of degree d in P^3 has a representative with big cotangent bundle.
Combinatorics Seminar
Yuval Roichman
Bar-Ilan University
Gallai Colorings, Transitivity and Schur-positivity
Monday, December 5, 2022, 5:00pm
Ungar Room 402
Abstract: A Gallai coloring of the complete graph is an edge-coloring with no rainbow triangle. This concept first appeared in the study of comparability graphs and anti-Ramsey theory. We introduce a transitive analogue for acyclic directed graphs and generalize these notions to Coxeter systems, loopless matroids and commutative algebras.
First, it is shown that the number of Gallai and transitive colorings in k colors is always a polynomial in k. It is further shown that for any representable matroid the maximal number of colors is equal to the rank, generalizing a result of Erdős-Simonovits-Sós for complete graphs.
We count Gallai and transitive colorings of the root system of type A with a maximal number of colors, and show that, when equipped with a natural descent set map, the resulting quasisymmetric function is Schur-positive.
The transitive commutative algebra of a Coxeter group will be presented. Open problems and conjectures regarding Hilbert series involve Stirling permutations and variants of Catalan numbers.
Based on Joint works with Ron Adin, Arkady Berenstein, Jacob Greenstein, Jianrong Li and Avichai Marmor.
Geometric Analysis Seminar
Thomas Körber
University of Vienna
The Riemannian Penrose Inequality for Asymptotically Flat Half-spaces and Rigidity
Friday, December 2, 2022, 2:00pm
Online
Abstract: Asymptotically flat half-spaces (M,g) are asymptotically flat manifolds with a non-compact boundary. They naturally arise as suitable subsets of initial data for the Einstein field equations. In this talk, I will present a proof of the Riemannian Penrose inequality for asymptotically flat half-spaces with horizon boundary (joint with M. Eichmair) that works in all dimensions up to seven. This inequality gives a sharp bound for the area of the horizon boundary in terms of the half-space mass of (M,g). To prove the inequality, we double (M,g) along its non-compact boundary and smooth the doubled manifold appropriately. To prove rigidity, we use variational methods to show that, if equality holds, the non-compact boundary of (M,g) must be totally geodesic. I will also explain how our techniques can be used to prove rigidity for the Riemannian Penrose inequality for asymptotically flat manifolds.
Geometric Analysis Seminar
Pengzi Miao
University of Miami
Positive Harmonic Functions in 3-dimension
Friday, November 18, 2022, 4:00pm
Online
Abstract: I will discuss some new properties of positive harmonic functions in dimension three. Applications include families of inequalities relating the surface capacity, Willmore functional, and the mass of asymptotically flat 3-manifolds. A by-product shows additional proofs of the 3-dimensional Riemannian positive mass theorem.
The material of this talk will be a subset of the paper arXiv:2207.03467.
Geometric Analysis Seminar
Xiaoxiang Chai
Korea Institute for Advanced Study
Band Width Estimates in CMC Initial Data Sets and Applications
Friday, November 11, 2022, 4:00pm
Ungar Room 402
Abstract: Gromov showed that a n dimensional toroical band with lower scalar curvature bound n(n-1), the distance of two boundary components of the band is bounded below by π/n. There are various generalizations of this band width estimate. We provide a generalization to the spacetime settings. In particular, we study the band width estimate torical band which is also a CMC initial data set. We give a proof using a hypersurface of prescribed null expansion and discuss other proofs. We apply this band width estimates to study the positive mass theorem for asymptotically hyperbolic manifolds with arbitrary ends. This is based joint works of Xueyuan Wan (Chongqing University of Technology).
Geometric Analysis Seminar
Abraão Mendes
Universidade Federal de Alagoas
Classification of Exterior Free Boundary Minimal Hypersurfaces
Friday, November 4, 2022, 4:00pm
Online
Abstract: In this lecture we aim to present two classification theorems for exterior free boundary minimal hypersurfaces (exterior FBMH for short) in Euclidean space. The first result states that the only exterior stable FBMH with parallel embedded regular ends are the catenoidal hypersurfaces. To achieve this we first prove a Bocher-type result for positive Jacobi functions on regular minimal ends R n+1 in which, after some calculations, implies the first theorem. The second theorem states that any exterior FBMH ∑ with one regular end is a catenoidal hypersurface. Its proof is based on a symmetrization procedure due to R. Schoen. Finally, we give a complete description of the catenoidal hypersurfaces, including the calculation of their indices. This lecture is based on a joint work with L. Mazet.
Geometric Analysis Seminar
Gaoming Wang
Cornell University
Second Order Elliptic Operators on Triple Junction Surfaces
Friday, October 28, 2022, 4:00pm
Online
Abstract: In this talk, we will consider minimal triple junction surfaces, a special class of singular minimal surfaces whose boundaries are identified in a particular manner. Hence, it is quite natural to extend the classical theory of minimal surfaces to minimal triple junction surfaces. Indeed, we can show that the classical PDE theory holds on triple junction surfaces. As a consequence, we can prove a type of Generalized Bernstein Theorem and give the definition of Morse index on minimal triple junction surfaces.
Geometric Analysis Seminar
Kwok Kun Kwong
University of Wollongong, Australia
Effect of the Average Scalar Curvature on Riemannian Manifolds
Friday, October 21, 2022, 4:00pm
Online
Abstract: The well-known Bishop-Gromov volume comparison theorem says that if the Ricci curvature is bounded below by (n-1)k, then the volume of a metric ball is at most that of the volume of the ball with the same radius in the space form with curvature k. Counterexamples show that the Ricci curvature cannot be replaced by the scalar curvature in the assumption. On the other hand, a Taylor series computation shows that the scalar curvature does tend to decrease the volume of small geodesic balls. In this talk, I will illustrate how the average scalar curvature (together with the Ricci curvature) of a closed manifold affects the average volume of its metric balls of any size. This gives an improvement of the Bishop-Gromov estimate. I will also show its effect on the average total mean curvature of geodesic spheres of radius up to the injectivity radius.
Combinatorics Seminar
Kyle Celano
University of Miami
RSK and S_n-action for P-tableaux of 3-free Natural Unit Interval Orders
Thursday, October 6, 2022, 5:00pm
Ungar Room 402
Abstract: A long-standing open problem is to find an RSK-like correspondence between permutations and pairs of tableaux coming from Gasharov's decomposition of Stanley's chromatic symmetric functions into Schur functions. In this talk we present such a correspondence for incomparability graphs of 3-free posets and use it to study an S_n action on the pairs of tableaux in the context of chromatic quasisymmetric functions and Hessenberg varieties.
IMSA Seminar
Jiachang Xu
University of Miami
Motivic Integration for Non-Archimedean Analytic Spaces I
Wednesday, October 5, 2022, 5:00pm
Ungar Room 528B
Abstract: In the first talk, I will introduce the theory of motivic integration for rigid varieties over a complete discrete valued field follows the works of François Loeser and Julien Sebag, which could be considered as a possible to develop a theory of motivic integration for Berkovich spaces over a complete discrete valued field. Also, I will also discuss some potential generalization to logarithmic geometry.
IMSA Seminar
Enrique Becerra
University of Miami
Basics on Motivic Integration
Wednesday, September 28, 2022, 5:00pm
Ungar Room 528B
Abstract: In this talk, I will expose some basic ideas on motivic integration. The goal will be to define the notion of motivic volume and show some simple computational examples.
Geometric Analysis Seminar
Da Rong Cheng
University of Miami
Existence of Free Boundary Constant Mean Curvature (CMC) Disks
Friday, September 23, 2022, 4:00pm
Ungar Room 402
Abstract: Given a surface S in R3, a classical problem is to find disk-type surfaces with prescribed constant mean curvature whose boundary meets S orthogonally. When S is diffeomorphic to a sphere, direct minimization could lead to trivial solutions and hence min-max constructions are needed. Among the earliest such constructions is the work of Struwe, who produced the desired free boundary CMC disks for almost every mean curvature value up to that of the smallest round sphere enclosing S.
In a joint work with Xin Zhou (Cornell), we combined Struwe's method with other techniques to obtain an analogous result for CMC 2-spheres in Riemannian 3-spheres and were able to remove the "almost every" restriction in the presence of positive ambient curvature. In this talk, I will report on more recent progress where the ideas in that work are applied back to the free boundary problem to refine and improve Struwe's result.
IMSA Seminar
Yixian Wu
University of Miami
Schön Varieties and Decomposition of a Semi-algebraic Set
Wednesday, September 21, 2022, 5:00pm
Ungar Room 528B
Abstract: Tropicalization provides us with a way to study varieties using its limit under degenerations. Schön varieties are those whose initial degenerations are smooth. They are the first cases whose motivic volumes are defined in Nicaise-Payne-Schroeter. In this talk, I will introduce Schön varieties, the tropical fan of them and how we decompose a semi-algebraic set into pieces where each of them can be treated as a Schön case.
Combinatorics Seminar
Bruno Benedetti
University of Miami
Random Simple Homotopy Theory
Monday, September 19, 2022, 5:00pm
Ungar Room 402
Abstract: We implement an algorithm RSHT (Random Simple-Homotopy Theory) to study the simple-homotopy types of simplicial complexes, with a particular focus on contractible spaces and on finding substructures in higher-dimensional complexes. The algorithm combines elementary simplicial collapses with pure elementary expansions, and provides an interesting alternative to discrete Morse theory. This is joint work with Crystal Lai, Davide Lofano, and Frank Lutz.
IMSA Seminar
Yixian Wu
University of Miami
Tropical Curve Counting and Correspondence Theorem
Wednesday, September 14, 2022, 5:00pm
Ungar Room 528B
Abstract: In the first talk, I will give an introduction to tropical geometry. I will set up the curve counting problems in both algebraic geometry and tropical geometry. In the case of toric surfaces, I will give a proof of the correspondence theorem that relates these two counts.
Geometry and Physics Seminar
Marco Golla
Université de Nantes
Symplectic Fillings of Divisorial Contact Structures
Thursday, May 5, 2022, 11:00am
Ungar Room 528B
Abstract: If a (possibly singular) complex curve in a Kähler surface has positive self-intersection, then it has a standard symplectically concave neighbourhood, and therefore an associated divisorial contact structure. Motivated by the study of singular symplectic curves in the complex projective plane, we will discuss the existence and classification of fillings of some of these contact structures. This is based on joint work with Laura Starkston.
Combinatorics Seminar
Michelle Wachs
University of Miami
On an n-ary Generalization of the Lie Representation
Monday, April 25, 2022, 5:00pm
Ungar Room 402
Abstract: There are two natural generalizations of the notion of Lie algebra involving an n-ary bracket; one of these was considered by Filippov in 1985 and the other was considered by the speaker and Hanlon in 1995. Both generalizations are of interest in particle physics. They are also of interest in combinatorics because they yield representations of the symmetric group that generalize the well studied Lie representation. This talk will focus on some recent work with Friedmann, Hanlon, and Stanley on the representation arising from the Filippov algebra.
Geometric Analysis Seminar
Annachiara Piubello
University of Miami
Estimates on the Bartnik Mass and their Geometric Implications
Friday, April 22, 2022, 4:00pm
Online
Abstract: In this talk, we will discuss some recent estimates on the Bartnik mass for data with non-negative Gauss curvature and positive mean curvature. In particular, if the metric is round, the estimate reduces to an estimate found by Miao and if the total mean curvature approaches 0, the estimate tends to 1/2 the area radius, which is the bound found by Mantoulidis and Schoen in the blackhole horizon case. We will then discuss some geometric implications. This is joint work with Pengzi Miao.
Applied Math Seminar
Shigui Ruan
University of Miami
Asymptotic and Transient Dynamics of SEIR Epidemic Models on Weighted Networks
Tuesday, April 19, 2022, 3:00pm
Online
Abstract: In modelling specific infectious diseases, such as COVID-19, populations tend to be inhomogeneous and there are nonlocal interactions as the disease spreads spatially via travelling. Therefore, it is very important to investigate the effects of host heterogeneity on the spatial spread of infectious diseases. In this talk we study the effect of population mobility on the transmission dynamics of infectious diseases by considering a susceptible-exposed-infectious-recovered (SEIR) epidemic model with graph Laplacian diffusion; that is, on a weighted network. First we establish the existence and uniqueness of solutions to the SEIR model defined on weighed graph. Then by the means of constructing Liapunov functions, we show that the disease-free equilibrium is globally asymptotically stable if the basic reproduction number is less than unity and the endemic equilibrium is globally asymptotically stable if the basic reproduction number is greater than unity. Finally we apply our generalized weighed graph to Watts-Strogatz network and carry out numerical simulations, which demonstrate that degrees of nodes determine the peak numbers of infectious population as well as the time to reach these peaks. It also indicates that the network has an impact on the transient dynamical behavior of the epidemic transmission. The node degrees determine the peak of infected population, where the greater the degree the higher the peak attains.
Geometric Analysis Seminar
Eric Ling
Harold H. Martin Postdoctoral Fellow
Rutgers University
Remarks on the Cosmological Constant Appearing as an Initial Condition for Milne-like Spacetimes
Friday, April 1, 2022, 5:00pm
Online
Abstract: Milne-like spacetimes are a class of k = -1 FLRW spacetimes which admit continuous spacetime extensions through the big bang. Moreover, the cosmological constant appears as an initial condition for Milne-like spacetimes under suitable assumptions on the scale factor. In this talk, we generalize this result to spacetimes which share similar geometrical properties with Milne-like spacetimes but without the strong isotropy assumption associated with them.
Applied Math Seminar
Dr. Rachidi B. Salako
University of Nevada, Las Vegas
On the Asymptotic Profiles of Endemic Equilibrium Solutions of a Diffusive Epidemic Model
Tuesday, March 29, 2022, 3:00pm
Online
Abstract: We study the asymptotic profiles of endemic equilibrium solutions of a diffusive epidemic model when the diffusion rates are suffciently low. First, we focus on the single-strain model and address the question of how the magnitude of the ratio d I / d s affects the persistence-extinction of the disease when the diffusion rates d s and d I of the susceptible and infected groups approach zero, respectively. In this case, our results indicated that when d I / d s is sufficiently small, the disease may persist and the infected group will concentrate only on the most harmful area. However, when d I / d s is significantly large, the disease may be eradicated if the total size of the population is kept below some critical number uniquely determined by the infection and recovery rates. The second part of the talk discusses how the above results extend to the two-strain model.
Combinatorics Seminar
Hsin-Chieh Liao
University of Miami
Stembridge Codes, Chow Rings of Boolean Matroids, and their Extensions
Monday, March 28, 2022, 5:00pm
Ungar Room 402
Abstract: It is well known that the Eulerian polynomial is the Poincare polynomial of the toric variety associated with the permutohedron. In 1989, Procesi and Stanley computed the Frobenius characteristic of the natural S n -action on the cohomology of this toric variety. Later Stembridge defined "codes of a permutation", on which the S n -action turns out to have the same character. We give a bijective proof of the equivalence of these representations by constructing an explicit equivariant bijection between Stembridge codes and the Feichtner-Yuzvinsky basis for the Chow ring of the Boolean matroid.
We then obtain analogous results for the binomial Eulerian polynomials by considering the augmented Chow ring introduced by Braden, Huh, Matherne, Proudfoot and Wang in 2020. We give a bijection between "extended codes" and a certain basis for the augmented Chow ring of the Boolean matroid. This basis is obtained by first showing that for any loopless matroid, the augmented Chow ring of the matroid is actually a Chow ring.
Birational Geometry Seminar
Muyuan Zhang
University of Miami
Semi Log Canonical Singularity and Its Applications - Part II
Thursday, March 24, 2022, 3:00pm
Online
Abstract: We examine further properties of Semi log canonical singularities. we will introduce the notions of stable pairs and different. After that, we will introduce important theorems regarding the moduli space of stable pairs.
Geometric Analysis Seminar
Hyun Chul Jang*
University of Miami
Mass Rigidity for Asymptotically Locally Hyperbolic Manifolds with Boundary
Tuesday, March 22, 2022, 5:00pm
Online
Abstract: Asymptotically locally hyperbolic (ALH) manifolds are a class of manifolds whose sectional curvature converges to −1 at infinity. If a given ALH manifold is asymptotic to a static reference manifold, the Wang-Chruściel-Herzlich mass integrals are well-defined for it, which is a geometric invariant that essentially measures the difference from the reference manifold. In this talk, I will present the result that an ALH manifold which minimizes the mass integrals admits a static potential. To show this, we proved the scalar curvature map is locally surjective when it is defined on (1) the space of ALH metrics that coincide exponentially toward the boundary or (2) the space of ALH metrics with arbitrarily prescribed nearby Bartnik boundary data. As an application, we establish the rigidity of the known positive mass theorems. This talk is based on joint work with L.-H. Huang.
*Hyun Chul is the department's third Fuqua Post Doc. He will begin a Harry Bateman post doc at Cal Tech in the fall.
Combinatorics Seminar
Mark Skandera
Lehigh University
Symmetric Generating Functions and Permanents of Totally Nonnegative Matrices
Monday, March 14, 2022, 5:00pm
Ungar Room 402
Abstract: For each element z of the symmetric group algebra we define a symmetric generating function Y(z)= Σ λ ε λ(z) m λ , where ε λ is the induced sign character indexed by λ. Expanding Y(z) in other symmetric function bases, we obtain other trace evaluations as coefficients. We show that all symmetric functions in span Z{m λ} are Y(z) for some z in Q[S n] . Using this fact and chromatic symmetric functions, we give new interpretations of permanents of totally nonnegative matrices. For the full paper, see arXiv:2010.00458v2.
Joint ICMS & IMSA Seminar
Kyoung-Seog Lee
University of Miami & IMSA
Higgs Bundles on Elliptic Surfaces and Logarithmic Transformations
Friday, March 11, 2022, 9:30am
Hybrid
Abstract: Logarithmic transformation is an important operation introduced by Kodaira in the 1960s. One can obtain an elliptic surface with multiple fibers by performing logarithmic transformations of an elliptic surface without multiple fibers. On the other hand, vector bundles on elliptic surfaces are important objects in many branches of mathematics, e.g., algebraic geometry, gauge theory, mathematical physics, etc. In this talk, I will discuss how certain Higgs bundles on elliptic surfaces are changed via logarithmic transformations. This talk is based on a joint work with Ludmil Katzarkov.
Applied Math Seminar
Chris Cosner
University of Miami
The Ideal Free Distribution in Temporally Varying Environments
Tuesday, March 8, 2022, 3:00pm
Online
Abstract: A population in a spatially heterogeneous environment is said to have an ideal free distribution if individuals distribute themselves so that the fitness of an individual is the same in all occupied locations. It has been shown that in various models for a single logistically growing population in a spatially varying but temporally constant environment, the dispersal strategies that are evolutionarily stable (a.k.a. evolutionarily steady) are those which produce an ideal free distribution. Recently such results have been extended to general periodic logistic reaction-advection-diffusion models and to integrodifference models with seasonal variation. For logistic reaction-advection-diffusion models and integrodifferential models in continuous time, it is possible for populations to achieve an ideal free distribution by using only local information about the environment, but in the time varying cases nonlocal information is necessary.
Combinatorics Seminar
Kyle Celano
University of Miami
Geometric Bases of Hessenberg Varieties and the Stanley-Stembridge Conjecture
Monday, March 7, 2022, 5:00pm
Ungar Room 402
Abstract: In 2012, Shareshian and Wachs described an approach to the Stanley-Stembridge e-positivity conjecture for the chromatic symmetric function: find a permutation basis of Tymozcko's S n-representation on the singular cohomology of the type A regular semisimple Hessenberg varieties. In this talk, we describe such a permutation basis arising from the variety's geometry for two special cases: one obtained by Cho, Hong, and Lee in 2020 and the other obtained more recently by the speaker.
Combinatorics Seminar
Marta Pavelka
University of Miami
2-LC Triangulated Manifolds Are Exponentially Many
Monday, February 28, 2022, 5:00pm
Ungar Room 402
Abstract: We introduce "t-LC triangulated manifolds" as those triangulations obtainable from a tree of d-simplices by recursively identifying two boundary (d−1)-faces whose intersection has dimension at least d − t − 1. The t-LC notion interpolates between the class of LC manifolds introduced by Durhuus-Jonsson (corresponding to the case t = 1), and the class of all manifolds (case t = d). Benedetti-Ziegler proved that there are at most 2^(N d^2) triangulated 1-LC d-manifolds with N facets. Here we show that there are at most 2^(N/2 d^3) triangulated 2-LC d-manifolds with N facets.
We also introduce "t-constructible complexes", interpolating between constructible complexes (the case t = 1) and all complexes (case t = d). We show that all t-constructible pseudomanifolds are t-LC, and that all t-constructible complexes have (homotopical) depth larger than d − t. This extends the famous result by Hochster that constructible complexes are (homotopy) Cohen-Macaulay.
This is joint work Bruno Benedetti. Details of the proofs and more can be found in our preprint of the same title.
Birational Geometry Seminar
Muyuan Zhang
University of Miami
Semi Log Canonical Singularity and Its Applications - Part I
Thursday, February 24, 2022, 3:00pm
Online
Abstract: We will give a general overview of what is a semi log canonical singularity. In particular, we will define semi log canonical and discuss the motivations. We will also cover some immediate properties of semi log canonical singularities.
Combinatorics Seminar
Bruno Benedetti
University of Miami
Vertex Labeling Properties for Simplicial Complexes
Monday, February 21, 2022, 5:00pm
Ungar Room 402
Abstract: Many classical graph properties (like chordality, co-comparability, having a Hamiltonian cycle, being a unit-interval graph...) can be characterized very easily in terms of vertex labelings. So there are natural, yet surprisingly unstudied, extensions of these properties to simplicial complexes. We address questions like: What is a "unit-interval simplicial complex"? What is a "Hamiltonian cycle" in higher dimensions? And do classical theorems like "all 2-connected unit-interval graphs are Hamiltonian" extend to higher dimensions? If time permits, we discuss application to commutative algebra, via Herzog et al.'s characterization of unit-interval graphs using binomial edge ideals.
This is joint work with Lisa Seccia, who is on the job market, and Matteo Varbaro, who isn't.
Joint ICMS & IMSA Seminar
Erik Paemurru
University of Miami & IMSA
Birational Geometry of Sextic Double Solids with a Compound A n Singularity
Friday, February 18, 2022, 9:30am
Hybrid
Abstract: Sextic double solids, double covers of ℙ 3 branched along a sextic surface, are the lowest degree Gorenstein Fano 3-folds, hence are expected to behave very rigidly in terms of birational geometry. Smooth sextic double solids, and those which are ℚ-factorial with ordinary double points, are known to be birationally rigid. In this talk, we discuss birational geometry of sextic double solids with an isolated compound A n singularity. I have shown that n is at most 8, and that rigidity fails for n > 3.
Combinatorics Seminar
Richard Stanley
Massachusetts Institute of Technology
University of Miami
A Fibonacci Analogue of Pascal's Triangle
Monday, February 14, 2022, 5:00pm
Ungar Room 402
Abstract: Pascal's triangle is closely associated with the expansion of the product (1+x)^n. We will discuss an analogous array of numbers that is associated with the product \prod_{i=1}^n (1+x^{F_{i+1}}), where F_{i+1} is a Fibonacci number. Both arrays are special cases of a two-parameter family that might be interesting to investigate further.
Birational Geometry Seminar
Jiachang Xu
University of Miami
Tropicalization and Skeleton of $\bar{{M}_{0,n}$ - Part II
Thursday, February 10, 2022, 3:00pm
Online
Abstract: We will first discuss the comparison between logarithmic Berkovich skeleton of $\bar{M_{0,n}}$ and a copy of the tropicalization of $M_{0,n}$ in the K-analytic variety $M_{0,n}^{an}$ in detail and the possible generalization to the Chow quotient compactification of $X(r,n)$ the moduli space of ordered $n$-tuples of hyperplanes in $\mathbb{P}^{r-1}$ whenever the compactification pair has toroidal singularities.
Joint ICMS & IMSA Seminar
Rene Mboro
University of Miami & IMSA
On the Geometry of Cubics
Wednesday, February 9, 2022, 5:00pm
Hybrid
Abstract: We will start by recalling some classical results on geometry of three and four dimensional cubics. Some new results on 5 dimensions cubics will be discussed as well.
Joint ICMS & IMSA Seminar
Sebastian Torres
University of Miami & IMSA
Windows and Geometric Invariant Theory
Friday, February 4, 2022, 9:30am
Online
Abstract: The theory of windows was introduced relatively recently by both Halpern-Leistner and Ballard, Favero and Katzarkov, and is a great tool to study derived categories of algebraic varieties that appear as GIT constructions, as well as their behavior at wall crossings as we vary the stability conditions. We will discuss this theory and explore different applications, from cohomology computations to semi-orthogonal decompositions.
Birational Geometry Seminar
Jiachang Xu
University of Miami
Tropicalization and Skeleton of $\bar{{M}_{0,n}$ - Part I
Thursday, February 3, 2022, 3:00pm
Online
Abstract: We will first discuss the comparison between logarithmic Berkovich skeleton of $\bar{M_{0,n}}$ and a copy of the tropicalization of $M_{0,n}$ in the K-analytic variety $M_{0,n}^{an}$ in detail and the possible generalization to the Chow quotient compactification of $X(r,n)$ the moduli space of ordered $n$-tuples of hyperplanes in $\mathbb{P}^{r-1}$ whenever the compactification pair has toroidal singularities.
Joint ICMS & IMSA Seminar
Rodolfo Aguilar
University of Miami & IMSA
Quantum Representations and Bogomolov-Katzarkov Surfaces
Friday, January 28, 2022, 9:30am
Online
Abstract: We will present some results of Eyssidieux-Funar as in arXiv:2112.06726, they are related to the so-called Shafarevich conjecture on holomorphic convexity.
There, they used quantum representations of the fundamental group of Riemann surfaces to show that most of the algebraic surfaces proposed by Bogomolov and Katzarkov in the late 90's are not counterexamples to this conjecture.
Joint ICMS & IMSA Seminar
Tokio Sasaki
ICMS & IMSA
Degeneration of Cubic Threefolds with Nodes
Friday, January 21, 2022, 9:30am
Online
Abstract: For a general smooth cubic threefolds $X$ in $\mathbb{P}^4$, its intermediate Jacobian and the fano variety of lines play the main role in the proof of the irrationality of $X$. When $X$ is a cubic with finitely many nodes, each node determines a sextic curve and a double covering of plane quintic curve similarly as smooth cubics. Since the Jacobian of the former curve is isomorphic to the Prym variety of the latter covering, by applying the theory of degeneration of Prym varieties and unimodular systems of vectors as matroids, one can observe the limiting behavior of the Jacobian of the sextic curve from the configuration of nodes. This is a survey talk mainly based on results by A. Collino, J. P. Murre, V. Alexeev, and T. Gwena.
CANCELLED
Joint ICMS & IMSA Seminar
August Hozie
University of Miami & IMSA
Some Examples and Asymptotics of Irregular Connections of Singularities
Friday, December 10, 2021, 9:30am
Hybrid
Abstract: A common feature of Noncommutative Hodge Structures, Frobenius Manifolds, and Irregular Hodge Structures is a connection on P1 which is Poincare rank 1 irregular at 0 and regular at infinity. In many cases we can compute local equations for these connections and investigate their asymptotics at the singular point. In this talk we'll look at some examples of the asymptotic spectra associated with these equations and some behavior under deformation.
Joint ICMS & IMSA Seminar
Dr. Ludmil Katzarkov
University of Miami
Multispectra
Monday, November 22, 2021, 5:00pm
Hybrid
Abstract: We will describe some new birational invariants.
Joint ICMS & IMSA Seminar
Dr. Shaoyun Bai
Princeton University
On the Rouquier Dimension of Wrapped Fukaya Categories
Monday, November 22, 2021, 4:00pm
Hybrid
Abstract: Given a triangulated category, its Rouquier dimension is defined to be the minimal generation time of all of its split-generators. I will explain how the Rouquier dimension of derived wrapped Fukaya categories of Weinstein manifolds/sectors is related to problems in symlectic topology of classical flavor, including quantitative intersection question of Lagrangian skeleta and estimating minimal numbers of critical points of symplectic Lefschetz fibrations. Moreover, using recent advances on symplectic flexibility (the arboreal program) and the local-to-global characterization of wrapped Fukaya categories, I will show how to resolve new cases of Orlov’s conjecture by bounding the Rouquier dimension of derived categories of algebraic varieties using homological mirror symmetry. This is joint work with Laurent Cote.
Applied Math Seminar
Bahman Angoshtari
University of Miami
Optimal Consumption under Drawdown and Habit-formation Constraints
Friday, November 19, 2021, 4:00pm
Online
Abstract: We consider an infinite horizon optimal investment and consumption problem for an agent who invests in a Black-Scholes-Samuelson market and is unwilling to consume below a fixed proportion her consumption habit. We consider two cases for the habit process. In one, it is the running maximum of past consumption rates and, in the other, it is the exponentially weighted average. In both cases, the optimal investment and consumption policies are obtained semi-explicitly and in terms of the solutions of nonlinear free-boundary problems, which we analyze in detail. This is joint work with Erhan Bayraktar and Virginia Young.
Applied Math Seminar
Chris Cosner
University of Miami & IMSA
Reaction-diffusion Models that Are Cooperative at Low Densities and Competitive at High Densities
Friday, November 5, 2021, 4:00pm
Online
Abstract: Methods based on monotone iteration or the theory of monotone dynamical systems have been widely used in the study of mathematical models in biology, and in the study of reaction-diffusion-advection systems more generally. Many models are either cooperative or competitive, or involve competing coalitions. All of those can be viewed as monotone semi-dynamical systems with respect to some ordering. However, including more effects found in nature in models related to the dispersal of organisms can lead to systems that are cooperative at low densities but competitive at high densities. This makes their analysis more challenging. In this talk I will discuss some examples of such systems and the phenomena that arise in them.
One type of system that is cooperative at low densities but competitive at high densities arises in modeling a single population where individuals can switch between different movement modes. Actual animals are often observed to switch between two or more different movement modes for large scale search to locate resources and for small scale search and exploitation once those are located.
Another place where such systems arise is in models for stage structured populations where adults and juveniles compete with each other for resources. It is well known that in bounded domains logistic reaction-diffusion models predict that slower diffusion rates are advantageous relative to faster diffusion, but in stage structured models that is not necessarily true.
A third context where systems that are cooperative at low densities but competitive at high densities arise is in models for the evolution of dispersal in a population with an Allee effects. In that modeling context, ecologically identical subpopulations with different dispersal rates or modes compete with each other. That leads to models whose dynamical terms have the forms f(x,u+v)u, f(x,x,u+v)v, where u and v are population densities. In the case of an Allee effect, f(x,u) is increasing when u is small but decreasing when u is large, which leads to a system that again is cooperative at low densities and competitive at high densities.
Joint ICMS & IMSA Seminar
Dr. Kyoung-Seog Lee
University of Miami & IMSA
Homological Mirror Symmetry for Hypersurface Singularities II
Friday, November 5, 2021, 9:30am
Hybrid
Abstract: In this talk, I will continue to introduce homological mirror symmetry for singularities. I will explain the categories of graded matrix factorizations of invertible polynomials and discuss several ways to describe them. If time permits, I will discuss stability conditions on the categories of the graded matrix factorizations of weighted homogeneous polynomials constructed by Takahashi, Kajiura-Saito-Takahashi, Toda, Otani-Takahashi.
Joint ICMS & IMSA Seminar
Dr. Kyoung-Seog Lee
University of Miami & IMSA
Homological Mirror Symmetry for Hypersurface Singularities I
Friday, October 22, 2021, 9:30am
Hybrid
Abstract: In this talk, I will briefly introduce homological mirror symmetry for certain hypersurface singularities. I will introduce basic definitions, Berglund-Hubsch duality of invertible polynomials, and some known results. Then I will discuss several examples in detail.
IMSA Seminar
Dr. Han-Bom Moon
Fordham & Stanford
Conformal Blocks in Algebraic Geometry Part III
Thursday, October 14, 2021, 2:15pm
Hybrid
Abstract: In this series of lectures, I briefly introduce mathematical aspects of the WZW model and the theory of conformal blocks. I will discuss the formal definition and basic properties of conformal blocks and how they are related to the geometric study of moduli spaces of curves and parabolic bundles. Most of the lectures will be accessible for non-experts and graduate students.
IMSA Seminar
Dr. Han-Bom Moon
Fordham & Stanford
Conformal Blocks in Algebraic Geometry Part II
Thursday, October 14, 2021, 1:00pm
Hybrid
Abstract: In this series of lectures, I briefly introduce mathematical aspects of the WZW model and the theory of conformal blocks. I will discuss the formal definition and basic properties of conformal blocks and how they are related to the geometric study of moduli spaces of curves and parabolic bundles. Most of the lectures will be accessible for non-experts and graduate students.
IMSA Seminar
Dr. Han-Bom Moon
Fordham & Stanford
Conformal Blocks in Algebraic Geometry Part I
Wednesday, October 13, 2021, 2:00pm
Hybrid
Abstract: In this series of lectures, I briefly introduce mathematical aspects of the WZW model and the theory of conformal blocks. I will discuss the formal definition and basic properties of conformal blocks and how they are related to the geometric study of moduli spaces of curves and parabolic bundles. Most of the lectures will be accessible for non-experts and graduate students.
Joint ICMS & IMSA Seminar
August Hozie
University of Miami & IMSA
More Instances of Steenbrink Spectra in Singularity Categories
Friday, October 8, 2021, 9:50am
Hybrid
Abstract: Now that we have set up the Frobenius Manifold of a singularity, we can see 2 more places where the steenbrink spectrum of a singularity appears in its singularity category, namely, the non-commutative mixed hodge structure of a singularity and the dimensional properties of the category. The former appearance is somehow natural, and the latter somewhat mysterious. In this talk we will conduct a surface level investigation of these appearances.
Joint ICMS & IMSA Seminar
Sebastian Torres
ICMS-Sofia
Windows and the BGMN Conjecture
Friday, October 8, 2021, 8:30am
Online
Abstract: Let be a smooth projective curve of genus at least 2, and let be the moduli space of semistable rank-two vector bundles of odd degree on . We construct a semi-orthogonal decomposition in the derived category of conjectured by Belmans, Galkin and Mukhopadhyay and by Narasimhan. It has blocks of the form where are -th symmetric powers of , and the semi-orthogonal complement to these blocks is conjecturally trivial.
In order to prove our result, we use the moduli spaces of stable pairs over . Such spaces are related to each other via GIT wall crossing, and the method of windows allows us to understand the relationship between the derived categories on either side of a given wall.
This is a joint work with J. Tevelev.
This event is organized by the International Center for Mathematical Sciences – Sofia (ICMS-Sofia). To view more information regarding this seminar, click here.
Joint ICMS & IMSA Seminar
August Hozie
University of Miami & IMSA
Steenbrink Spectra in Singularity Categories
Friday, October 1, 2021, 9:30am
Online
Abstract: The spectrum of a singularity as defined by Steenbrink encodes important analytic information about a singularity and describes the Hodge filtration on the vanishing cycles of the singularity. In this talk we will explore a few different ways in which the spectral numbers show up in the triangulated singularity category associated with a singularity.
Joint ICMS & IMSA Seminar
Kyoung-Seog Lee
University of Miami & IMSA
Alexander Polynomials of Algebraic Links
Friday, September 24, 2021, 9:30am
Online
Abstract: Alexander polynomial is one of the most well-known invariants in knot theory. In the first part of this talk, I will review basic definitions and examples of Alexander polynomials of algebraic links. Then I will survey several results expressing Alexander polynomials of algebraic links via tools of algebraic geometry. Then I will discuss how the spectrum of a plane curve singularity is related to the Alexander polynomial of its algebraic link.
Joint ICMS & IMSA Seminar
Kyoung-Seog Lee
University of Miami & IMSA
Plane Curve Singularities and Spectrum
Friday, September 17, 2021, 9:30am
Online
Abstract: In the first part of this talk, I will review basic notions and results about plane curve singularities, e.g. blow-ups, resolution of singularities, Newton-Puiseux series, etc. Then I will explain how to compute the spectrum of a plane curve singularity via Puiseux pairs based on Morihiko Saito's work.
IMSA Seminar
Josef Svoboda
University of Miami & IMSA
Surface Singularities and Invariants of their Links
Friday, September 10, 2021, 9:30am
Online
Abstract: To an isolated surface singularity, we can assign a natural 3-manifold - the link of the singularity. It is an old question how many of the properties of the singularity can be recovered from this purely topological information. Based on the work of Lawrence-Zagier, Hikami and others, I will show how quantum topological invariants of the link are related to the spectrum in the case of Brieskorn spheres.
IMSA Seminar
Josef Svoboda
University of Miami & IMSA
Spectrum of Singularities
Friday, September 3, 2021, 9:30am
Online
Abstract: The spectrum is a strong invariant of a hypersurface singularity, defined originally by Arnold, Varchenko and Steenbrink. I will start with examples of singularities and their spectra and sketch the definition of the spectrum. Then I will concentrate on the most important properties of the spectrum such as the Thom-Sebastiani theorem and semicontinuity, which are very powerful in applications. Finally, I will talk about computational techniques to obtain the spectrum.
Joint ICMS & IMSA Seminar
Dr. Rodolfo Aguilar
ICMS
Quantum Representations of Fundamental Groups of Curves with Infinite Image
Friday, August 20, 2021, 10:30am (5:30pm Sofia)
Online
Abstract: We will report some results due to Koberda-Santharoubane showing an element of $\pi_1(\mathbb{P}^1\setminus \{3-\text{points}\})$ having infinite order under some quantum representations of the mapping class group.
This event is organized by the International Center for Mathematical Sciences – Sofia (ICMS-Sofia). To view more information regarding this seminar, click here.
Joint ICMS & IMSA Seminar
Dr. Rene Mboro
ICMS
On Determinantal Cubic Hypersurfaces (after Iliev-Manivel, Beauville,...)
Friday, August 20, 2021, 9:30am (4:30pm Sofia)
Online
Abstract: We give an account of the problem of writting an equation of a cubic (or other degree) hypersurface $X$ as a (kind of) determinant of a matrix with homogeneous entries. Expressing the equation of $X$ as a determinant is equivalent to produce a arithmetically Cohen-Macaulay vector bundle on $X$. The talk will focus on the cases of cubic hypersurfaces of dimension at most 8.
This event is organized by the International Center for Mathematical Sciences – Sofia (ICMS-Sofia). To view more information regarding this seminar, click here.
ICMS Summer-Autumn 2021 Seminars
Dr. Ludmil Katzarkov
ICMS & IMSA
Spectra and Applications Part III
Friday, July 30, 2021, 9:00am (4:00pm Sofia)
Via Zoom
View Video
Abstract: This is a survey course on the relation of the invariants of 3 manifolds and singularity theory. Using the theory of differential equations we relate classical 3-dimensional theory with modern category theory. Applications to Birational geometry and uniformization will be discussed.
This event is organized by the International Center for Mathematical Sciences – Sofia (ICMS-Sofia). To view more information regarding this seminar, click here.
ICMS Summer-Autumn 2021 Seminars
Dr. Josef Svoboda
ICMS
Spectra and Applications Part II
Friday, July 30, 2021, 8:00am (3:00pm Sofia)
Online
View Video
Abstract: This is a survey course on the relation of the invariants of 3 manifolds and singularity theory. Using the theory of differential equations we relate classical 3-dimensional theory with modern category theory. Applications to Birational geometry and uniformization will be discussed.
This event is organized by the International Center for Mathematical Sciences – Sofia (ICMS-Sofia). To view more information regarding this seminar, click here.
ICMS Summer-Autumn 2021 Seminars
Dr. Josef Svoboda
ICMS
Spectra and Applications Part I
Thursday, July 29, 2021, 8:00am (3:00pm Sofia)
Online
View Video
Abstract This is a survey course on the relation of the invariants of 3 manifolds and singularity theory. Using the theory of differential equations we relate classical 3-dimensional theory with modern category theory. Applications to Birational geometry and uniformization will be discussed.
This event is organized by the International Center for Mathematical Sciences – Sofia (ICMS-Sofia). To view more information regarding this seminar, click here.
IMSA Seminar
Daniel Pomerleano
University of Massachusetts
Semi-Affineness of Wrapped Invariants on Affine Log Calabi-Yau Varieties
Friday, February 26, 2021, 4:00pm
Online
Abstract: A general expectation in mirror symmetry is that the mirror partner to an affine log Calabi-Yau variety is "semi-affine" (meaning it is proper over its affinization). We will discuss how the semi-affineness of the mirror can be seen directly as certain finiteness properties of Floer theoretic invariants of X (the symplectic cohomology and wrapped Fukaya category). As an application of these finiteness results, we will show that for maximally degenerate log Calabi-Yau varieties equipped with a "homological section," the wrapped Fukaya of X gives an (intrinsic) categorical crepant resolution of the affine variety Spec(SH^0(X)).
This is based on joint work with Sheel Ganatra ( https://arxiv.org/abs/1811.03609) and further work in progress.
IMSA Seminar
Ludmil Katzarkov
University of Miami
Old and New Birational Invariants
Monday, September 28, 2020, 5:00pm
Online
Abstract: In this talk we will propose new birational invariants based on combining Hodge theory and Symplectic Geometry.
IMSA Seminar
R. Paul Horja
University of Miami
A Categorical Interpretation of the GKZ D-Module
Tuesday, September 1, 2020, 5:00pm
Online
Abstract: I will explain a proposal for the B-side category in toric homological mirror symmetry along the strata of the characteristic cycle of the associated GKZ D-module. Various consistency checks will be presented. The construction builds on the string theoretical work by Aspinwall-Plesser-Wang.
IMSA Seminar
Maxime Kontsevich
University of Miami
Integral PL Actions from Birational Geometry
Monday, August 17, 2020, 9:00am
Online
Abstract: The group of birational automorphisms of a N-dimensional algebraic torus, preserving the standard logarithmic volume element, acts (by tropicalization) on N-dimensional real vector space by homogeneous piece-wise linear homeomorphisms. A similar construction exists for any compact Calabi-Yau variety over a non-archimedean field, through the notion of the "essential EEEE". I'll talk about examples coming from generalized cluster varities, and from Calabi-Yau varieties parameterizing linkages of regular graphs.
IMSA Seminar
Philip Griffiths
University of Miami
Period Mapping at Infinity
Wednesday, May 6, 2020, 10:00am
Online
Abstract: Hodge theory provides a basic invariant of complex algebraic varieties. For algebraic families of smooth varieties the global study of the Hodge structure on the cohomology of the varieties (period mapping) is a much studied and rich subject. When one completes a family to include singular varieties the local study of how the Hodge structures degenerate to limiting mixed Hodge structures is also much studied and very rich. However, the global study of the period mapping at infinity has not been similarly developed. This has now been at least partially done and will be the topic of this talk. Sample applications include
• new global invariants of limiting mixed Hodge structures
• a generic local Torelli assumption implies that moduli spaces are
• log canonical (not just log general type); and
• a proposed construction of the toroidal compactification of the image of a period mapping
The key point is that the extension data associated to a limiting mixed Hodge structure has a rich geometric structure and this provides a new tool for the study of families of singular varieties in the boundary of families of smooth varieties.
*Joint work with Mark Green and Colleen Robles
IMSA Seminar
Ernesto Lupercio
Center for Research and Advanced Studies of the
National Polytechnic Institute (Cinvestav-IPN)
Quantum Toric Geometry II:
Non-commutative Geometric Invariant Theory
Wednesday, April 29, 2020, 10:00am
Online
Abstract: In this talk I will give a bird's eye view of the field of Quantum Toric Geometry (QTG). QTG is a generalization of Classical Toric Geometry where the various classical tori appearing in the usual theory are replaced by quantum tori (also known as non-commutative tori). As a result the new theory can be though of as a deformation of the usual theory and hence, it permits the construction of a remarkable moduli space of toric varieties. I will try to convey the basic ideas required to understand this story in this first talk.
This is joint work with L. Katzarkov, L. Meersseman, and A. Verjovsky.
IMSA Seminar
Ernesto Lupercio
Center for Research and Advanced Studies of the
National Polytechnic Institute (Cinvestav-IPN)
Quantum Toric Geometry I
Wednesday, April 22, 2020, 10:00am
Online
Abstract: In this talk I will give a bird's eye view of the field of Quantum Toric Geometry (QTG). QTG is a generalization of Classical Toric Geometry where the various classical tori appearing in the usual theory are replaced by quantum tori (also known as non-commutative tori). As a result the new theory can be though of as a deformation of the usual theory and hence, it permits the construction of a remarkable moduli space of toric varieties. I will try to convey the basic ideas required to understand this story in this first talk.
This is joint work with L. Katzarkov, L. Meersseman, and A. Verjovsky.
IMSA Seminar
Kyoung-Seog Lee
University of Miami
Seiberg-Witten Gauge Theory and Complex Surfaces
Thursday, April 16, 2020, 5:00pm
Online
Abstract: Most part of this talk will be a survey about Seiberg-Witten gauge theory and how it can be understood for complex smooth projective surfaces. I will discuss several interesting examples and raise some questions.
IMSA Seminar
Benjamin Gammage
University of Miami
Mirror Symmetry and Cluster Varieties
Thursday, April 2, 2020, 5:00pm
Online
Abstract: We discuss the symplectic geometry of cluster varieties with applications to mirror symmetry, including an explanation of homological mirror symmetry for Gross-Hacking-Keel cluster varieties. This is based on work in progress with Ian Le.
IMSA Seminar
Tokio Sasaki
University of Miami
A Construction of Apéry Constants from Landau-Ginzberg Models
Thursday, March 26, 2020, 5:00pm
Online
Abstract: The irrationality of the Riemann zeta function at 3 was historically proven by R. Apéry by finding a rapidly converging sequence which is consisted of two sequences in integers and rationals satisfying certain recursive relations. Nowadays it is known that this sequence is obtained from the power series expansion of the holomorphic period function of a family of K3 surfaces, and the recurrences arise from the Picard-Fuchs differential equation.
For some Fano threefolds with Picard rank 1, V. Golyshev obtained similar special values of L-functions as Apéry limit of the quantum differential equations. If one believes the mirror symmetry also preserves these arithmetic special values, there should be a "mirror" construction in the B-model side. In this talk, as an evidence I introduce constructions of geometric higher normal functions on the mirror Landau-Ginzberg models of the above Fano threefolds. Limiting values of these normal functions toward singular fibers reconstruct the Apéry constants computed in the A-model side. With Mukais classification of the Fano threefolds, the results for V_10, V_12, V_16, V_18 are shown by M. Kerr and G. Silva Jr. A partial result for the V_14 case is given by the speaker.
Applied Math Seminar
Zhongming Wang
Florida International University
Conservative, Positivity Preserving and Free Energy Dissipative Numerical Methods for the Poisson-Nernst-Planck Equations
Wednesday, March 4, 2020, 5:00pm
Ungar Room 411
Abstract: We design and analyze some numerical methods for solving the Poisson–Nernst–Planck (PNP) equations. The numerical schemes, including finite difference method and discontinuous Galerkin method, respect three desired properties that are possessed by analytical solutions: I) conservation, II) positivity of solution, and III) free-energy dissipation. Advantages of different types of methods are discussed. Numerical experiments are performed to validate the numerical analysis. An application to an electrochemical charging system is also studied to demonstrate the effectiveness of our schemes in solving realistic problems. This is a joint work with H. Liu, D. Jie and S. Zhou.
Geometry and Physics Seminar
Siqi He
Stony Brook University
The Behavior of Sequences of Solutions to the Hitchin-Simpson Equations
Wednesday, March 4, 2020, 4:00pm
Ungar Room 402
Abstract: The Hitchin-Simpson equations defined over a Kähler manifold are first order, non-linear equations for a pair of connection on a Hermitian vector bundle and a 1-form with values in the endomorphism bundle. Under certain topological assumptions, the solutions to the Hitchin-Simpson equations are flat SL(2,C) connections. We will describe the behavior of solutions to the Hitchin-Simpson equations with norms of these 1-forms unbounded. We will discuss the relationship of this behavior with Taubes Z/2 harmonic 1-forms and SL(2,C) gauge theory.
Combinatorics Seminar
Mark Skandera
Lehigh University
Generating Functions for Induced Characters of the Hyperoctahedral Group
Monday, March 2, 2020, 5:00pm
Ungar Room 402
Abstract: Merris and Watkins interpreted results of Littlewood to give generating functions for symmetric group characters induced from one-dimensional characters of Young subgroups. Beginning with an n by n matrix X of formal variables, one obtains induced sign and trivial characters by expanding sums of products of certain determinants and permanents, respectively. We will look at a new analogous result which holds for hyperoctahedral group characters induced from four one-dimensional characters of its Young subgroups. This requires n^2 more formal variables and four combinations of determinants and permanents.
Applied Math Seminar
Jorge X. Velasco Hernández
Instituto de Matemáticas, UNAM (Mexico)
A Mathematical Model for Leptospirosis: A Neglected Infectious Disease of the Tropics
Wednesday, February 26, 2020, 5:00pm
Ungar Room 411
Abstract: As a zoonotic disease, leptospirosis is identified as one of the emerging infectious diseases of importance in the tropical regions in the world. Here, we study the propagation of leptospirosis in a cattle ranch. A mathematical model has been built with ordinary differential equations to understand the epidemiology of leptospirosis and main factors on its transmission. The model incorporates vaccination and recruitment control programs in the form of impulse actions as measures to prevent the propagation of leptospirosis in the ranch.
Geometry and Physics Seminar
Mark Green
University of California, Los Angeles
Singularities, Hodge Theory and I-Surfaces
Wednesday, February 26, 2020, 3:45pm
Ungar Room 506
Abstract: The main theme of this talk is: How are the singularities acquired on the boundary of the moduli space related to the boundary components of the space of Hodge structures? I will discuss this mainly in the case of surfaces, especially in the case of a particular family of surfaces of general type having p_g =2, q=0, K^2=1. This is joint work with Phillip Griffiths, Radu Laza and Colleen Robles.
Combinatorics Seminar
Matteo Varbaro
University of Genoa, Italy
Square-free Gröbner Degenerations
Monday, February 24, 2020, 5:00pm
Ungar Room 402
Abstract: Many homological invariants cannot go down when passing from a polynomial ideal to its initial ideal with respect to a monomial order. It turns out that in many natural situations (e.g. ideals defining Grassmannians, Schubert varieties, determinantal varieties ecc.), homological invariants like the projective dimension and the Castelnuovo-Mumford regularity stay the same. In all these cases the corresponding initial ideal is a square-free monomial ideal. So, since the 80s, it started to circulate the question whether the projective dimension and the Castelnuovo-Mumford regularity of an ideal are equal to those of its initial ideal provided the latter is square-free.
This question has later become popular as Herzog's conjecture. In this talk we discuss the attempts done to approach Herzog's conjecture, and its recent solution in positive given by Aldo Conca and myself.
Geometric Analysis Seminar
Abraão Mendes
Universidade Federal de Alagoas, Brazil
A Characterization of the Clifford Torus and the Equilateral Torus
Monday, February 24, 2020, 3:00pm
Ungar Room 506
Abstract: In this talk, we are going to prove a characterization of the Clifford torus and the equilateral torus in S^n via the second eigenvalue of the Jacobi operator. In fact, we will prove that the maximum value for the second eigenvalue of the Jacobi operator among all closed orientable immersed surfaces in S^n is -2. Furthermore, the Clifford torus in S^3 and the equilateral torus in S^5 are the only surfaces which attain the maximum.
Geometry and Physics Seminar
Enrica Mazzon
Max Planck Institute for Mathematics
The Essential Skeletons of Paris and the Geometric P=W Conjecture
Wednesday, February 19, 2020, 5:00pm
Ungar Room 402
Abstract: To a compactification of an open variety, we can associate the dual boundary complex, a topological space that encodes the combinatoric of the boundary and reflects the geometry of the open variety. The points of the dual boundary complex determine valuations on the function field of the variety, defining a so-called skeleton in the Berkovich space of the variety.
In this talk, I will explain how the Berkovich approach applies to the study of character varieties, central objects in non-abelian Hodge theory. According to the geometric P=W conjecture, it is expected that the dual boundary complex of the compactification of character varieties has the homotopy type of a sphere. In joint work with Mirko Mauri and Matthew Stevenson, we compute the first non-trivial examples of these dual boundary complexes in the compact case, providing new evidence for the conjecture.
Geometric Analysis Seminar
Pengzi Miao
University of Miami
New Interpretation of ADM Mass
Monday, February 17, 2020, 3:00pm
Ungar Room 506
Abstract: Recently Daniel Stern has discovered a formula that relates scalar curvature to the level sets of harmonic functions. Prompted by Stern's formula, we find that the mass of an asymptotically flat 3-manifold has a new geometric interpretation if evaluated along faces and edges of a large coordinate cube. In terms of the mean curvature and dihedral angle, the resulting mass formula relates to Gromov's scalar curvature comparison theory for cubic Riemannian polyhedra. In terms of the geodesic curvature and turning angle of slicing curves, the formula realizes the mass as integration of the angle defect detected by the boundary term in the Gauss-Bonnet theorem.
Combinations Seminar
Richard Stanley
University of Miami
Massachusetts Institute of Technology
Persification
Tuesday, February 11, 2020, 5:00pm
Ungar Room 402
Abstract: This talk is a variant of one given recently at a conference in honor of the 75th birthday of Persi Diaconis. "Persification" can be defined as the process of turning a mathematical result into a "story" explaining how this result applies to a concrete or real world situation, in the manner of Persi Diaconis. We will give several examples of persification related to algebraic combinatorics.
Applied Math Seminar
Hao Kang
University of Miami
A Nonlinear Age-structured Population Models with Nonlocal Diffusion and Nonlocal Boundary Conditions
Wednesday, February 5, 2020, 5:00pm
Ungar Room 411
Abstract: In this paper, we develop basic theory for age-structured population models with nonlocal diffusion and nonlocal boundary conditions. We first apply the theory of integrated semigroups and non-densely defined operators to a linear equation, study the spectrum, and analyze the asymptotic behavior via asynchronous exponential growth. Then we study a semilinear equation and a nonlinear equation with nonlocal diffusion and nonlocal boundary conditions, use the method of characteristic lines to find the resolvent of the infinitesimal generator and the variation of constant formula, apply Krasnoselskii's fixed point theorem to obtain the existence of a steady state, and establish the stability of the steady states.
Geometry and Physics Seminar
Maxim Jeffs
Harvard University
A Mirror Symmetry and Fukaya Categories of Singular Hypersurfaces
Wednesday, January 22, 2020, 5:00pm
Ungar Room 402
Abstract: I'll explain a definition of the Fukaya category of a singular hypersurface proposed by Auroux, given by localizing the Fukaya category of a nearby fiber at Seidel's natural transformation, and show that this possesses several desirable properties. Firstly, I'll explain the A-side analog of Orlov's derived Knorrer periodicity theorem, by showing that under certain hypotheses Auroux' category should be derived equivalent to the Fukaya-Seidel category of a higher-dimensional Landau-Ginzburg model. Secondly, I'll describe how this definition should imply homological mirror symmetry at various large complex structure limits, in the context of forthcoming work of Abouzaid-Auroux and Abouzaid-Gross-Siebert.
IMSA Seminar
R.Paul Horja
University of Miami
D-modules and Toric Schobers
Tuesday, January 21, 2020, 5:00pm
Ungar Room 528B
Abstract: I will present a translation of the classical mirror symmetry point of view into the more recent language of schobers. A conjecture on a categorical interpretation of the quantum toric D-module naturally appearing in mirror symmetry will be discussed.
IMSA Seminar
Ludmil Katzarkov
University of Washington
Categorical Linear Systems
Tuesday, January 14, 2020, 5:00pm
Ungar Room 528B
Geometric Analysis Seminar
Melaine Graf
University of Washington
Singularity Theorems for C 1-Lorentzian Metrics
Wednesday, December 18, 2019, 3:00pm
Ungar Room 506
Abstract: The classical singularity theorems of General Relativity show that a Lorentzian manifold with a smooth metric satisfying certain physically reasonable curvature and causality conditions cannot be causal geodesically complete. One drawback of these classical theorems is that they require smoothness of the metric while in many physical models the metric is less regular. In my talk I will present recent work concerning singularity theorems for metrics that are merely continuously differentiable – a regularity where one still has existence but not uniqueness for solutions of the geodesic equation. I will give an overview of the proof of Hawking's theorem in this regularity and, if time permits, discuss some of the estimates involved in more detail.
IMSA Seminar
Manuel Rivera
University of Miami
An Extension of a Classical Theorem of Whitehead
Thursday, December 12, 2019, 5:00pm
Ungar Room 528B
Abstract: A classical theorem of Whitehead in algebraic topology says that a continuous map between two simply connected topological spaces induces an isomorphism on homotopy groups if and only if it induces an isomorphism on integral homology groups (or equivalently if the map induces a quasi-isomorphism between singular chains with integer coefficients). This theorem is important since homology groups are in general easier to compute than homotopy groups.
In this talk I will outline a proof of the following new extension of Whitehead's classical result: a continuous map between path connected spaces induces an isomorphism on homotopy groups if and only if it induces a quasi-isomorphism after applying the singular chains functor followed by the "cobar" functor with respect to the Alexander-Whitney coalgebra structure of the singular chains. The proof of this statement uses a basic piece of the algebraic topology of spaces which was only completely understood until recently: the isomorphism class of the fundamental group of a space is completely determined by the algebraic (homological) structure of the singular chains on the space. All of this is also a step forward in answering a fundamental question posed by Loday which asks for a homological/homotopical algebra formulation of a notion of "a group up to homotopy". This is joint work with Mahmoud Zeinalian and Felix Wierstra.
IMSA Seminar
Trevor Olsen
University of South Carolina
Chow Motives and Categories
Tuesday, December 3, 2019, 5:00pm
Ungar Room 528B
Combinatorics Seminar
Trevor Olsen
University of South Carolina
Wiener Index and Remoteness in Triangulations and Quadrangulations
Monday, November 18, 2019, 5:00pm
Ungar Room 402
Abstract: Let G be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If r(v) denotes the arithmetic mean of the distances from v to all other vertices of G, then the remoteness of G is defined as the largest value of r(v) over all vertices v of G. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.
IMSA Seminar
Tokio Sasaki
University of Miami
Tyurin Degeneration and Periods
Tuesday, November 12, 2019, 5:00pm
Ungar Room 528B
IMSA Seminar
Benjamin Gammage
University of Miami
Gluing Fukaya Categories with Stops
Tuesday, November 5, 2019, 5:00pm
Ungar Room 528B
Combinatorics Seminar
Giulia Codenotti
Freie Universität Berlin
Hollow Polytopes of Large Width
Monday, November 4, 2019, 5:45pm
Ungar Room 402
Abstract: The goal of this talk is to present the construction of hollow lattice polytopes of width larger than their dimension. After defining important properties of lattice polytopes, such as width and hollowness, we introduce the flatness constant and certain specializations of it, and discuss bounds and exact values of these in low dimension. We will then show how taking a direct sum of certain polytopes yields a hollow lattice polytope (resp. a hollow lattice simplex) of dimension 14 (resp. 404) and of width 15 (resp. 408). They are the first known hollow lattice polytopes of width larger than dimension. We will also present some asymptotic results which can be obtained with this method. The talk is based on joint work with Francisco Santos.
Combinatorics Seminar
Jean-Philippe Labbé
Freie Universität Berlin
Universal Oriented Matriods for Subword Complexes of Coxeter Groups
Monday, November 4, 2019, 5:00pm
Ungar Room 402
Abstract: Steinitz's problem asks whether a triangulated sphere is realizable geometrically as the boundary of a convex polytope. The determination of the polytopality of subword complexes is a resisting instance of Steinitz's problem. Indeed, since their creation more than 15 years ago, subword complexes built up a wide portfolio of relations and applications to many other areas of research (Schubert varieties, cluster algebras, associahedra, tropical Grassmannians, to name a few) and a lot of efforts has been put into realizing them as polytopes, with relatively little success.
In this talk, I will present some reasons why this problem resisted so far, and present a glimpse of an approach to study the problem grouping together Schur functions, combinatorics of words, and oriented matroids.
IMSA Seminar
Ludmil Katzarkov
University of Miami
Bluing D Modules VI
Wednesday, October 30, 2019, 5:00pm
Ungar Room 528B
Geometry and Physics Seminar
John Etnyre
Georgia Institute of Technology
Non-isotopic Contact Submanifolds
Wednesday, October 23, 2019, 4:00pm
Ungar Room 402
Abstract: The study of transverse knots in dimension 3 has been instrumental in the development of 3 dimensional contact geometry. One natural generalization of transverse knots to higher dimensions is contact submanifolds. Embeddings of one contact manifold into another satisfies an h-principle for co-dimension greater than 2, so we will discuss the case of co-dimension 2 contact embeddings. We will give the first pair of non-isotopic contact embeddings in all dimensions (that are formally isotopic).
IMSA Seminar
Ludmil Katzarkov
University of Miami
Gluing D Modules V
Monday, October 21, 2019, 4:00pm
Ungar Room 528B
IMSA Seminar
Ludmil Katzarkov
University of Miami
Gluing D Modules IV
Friday, October 18, 2019, 4:00pm
Ungar Room 528B
IMSA Seminar
Aleksandar Petkov
University of Miami
Central Manifolds
Tuesday, October 8, 2019, 5:00pm
Ungar Room 528B
IMSA Seminar
Ludmil Katzatkov
University of Miami
Gluing D Madules III
Monday, October 7, 2019, 4:00pm
Ungar Room 528B
IMSA Seminar
Ludmil Katzatkov
University of Miami
Gluing D Madules II
Thursday, October 3, 2019, 5:00pm
Ungar Room 528B
Geometry and Physics Seminar
Langet Ma
Brandeis University
A Surgery Formual for the Casson-Seiberg-Witten Invariant of Integral Homology S 1 x S 3
Wednesday, October 2, 2019, 4:00pm
Ungar Room 402
Abstract: In this talk I will discuss a surgery formula for the Casson-Seiberg-Witten invariant $\lambda_{SW}$ introduced by Mrowka, Ruberman, and Saveliev, which is a 4-dimensional analogue of the surgery formula for the Casson invariant.
IMSA Seminar
Ludmil Katzarkov
University of Miami
Gluing D Modules I
Tuesday, October 1, 2019, 5:30pm
Ungar Room 528B
IMSA Seminar
Aleksandar Petkov
Kyoung-Seog Lee
Tokio Sasaki
Ludmil Katzarkov
University of Miami
Gluing D Modules and Chimeras
Saturday, September 28, 2019, 5:00pm
Ungar Room 528B
IMSA Seminar
Aleksandar Petkov
University of Miami
Central Manifolds
Thursday, September 26, 2019, 5:00pm
Ungar Room 528B
IMSA Seminar
Tokio Sasaki
University of Miami
More on Variations of HMS
Thursday, September 24, 2019, 5:00pm
Ungar Room 528B
IMSA Seminar
Tokio Sasaki
University of Miami
Variations of HMS
Thursday, September 19, 2019, 5:00pm
Ungar Room 528B
Geometry and Physics Seminar
Ken Baker
University of Miami
The Morse-Novikov Number is Additive
Wednesday, September 18, 2019, 4:00pm
Ungar Room 402
Abstract: M. Boileau and C. Weber asked whether the Morse-Novikov number of knots in the 3-sphere is additive under connected sum. Instead of working with homotopy classes of maps of knot exteriors to the circle, we use circular generalized Heegaard splittings introduced by Manjarrez-Gutierrez to answer the question in the affirmative.
IMSA Seminar
August Hozie
University of Miami
More on D Modules
Tuesday, September 17, 2019, 5:00pm
Ungar Room 528B
IMSA Seminar
Blagovest Sendov
Bulgarian Academy of Sciences
Smale's Mean Value Conjecture
Thursday, September 5, 2019, 3:30pm
Ungar Room 528B
Abstract: View Abstract
IMSA Seminar
Tokio Sasaki
University of Miami
The Strcture of Higher Genus Gromov-Witten Theory of Quintic 3-folds
Thursday, September 5, 2019, 2:30pm
Ungar Room 528B
Abstract: One of biggest and most difficult problems in the subject of Gromov-Witten theory is to compute the higher genus Gromov-Witten theory of a compact Calabi-Yau 3-fold. There have been a collection of remarkable conjectures from physics for so called 14 one-parameter models, the simplest compact Calabi-Yau 3-folds similar to the quintic 3-folds. These conjectures were originated from universal properties of the BCOV B-model. The backbone of this collection are four structural conjectures: (1) Yamaguchi-Yau finite generation; (2) Holomorphic anomaly equation; (3) Orbifold regularity and (4) Conifold gap condition.
In the talk, I will present background and our approach to the problem.
This is a joint work with F. Janda and S. Guo. Our proof is based on a certain localization formula from log GLSM theory developed by Q. Chen, F. Janda and myself.
IMSA Seminar
Tokio Sasaki
University of Miami
Going Down of Indecomposable Cycles to Nontrivial Elements of Griffths Groups
Tuesday, August 27, 2019, 5:00pm
Ungar Room 528B
Abstract: From a given reflexive Laurent polynomial in three variables, one can construct a degenerating family of K3 surfaces, so that the associated three dimensional Newton polytope exhibits the combinatorial geometry of the singular fiber. If a four dimensional reflexive polytope is the Minkowski sum of this polytope and another one, it defines a nef partition, and general hypersurface sections of the associated toric variety provide an example of a Tyurin degeneration. We construct some specific examples of such a degeneration together with non-trivial elements of their Griffiths groups. These elements arise from certain indecomposable cycles on the intersection K3 surface of the irreducible components on the singular fiber. This construction is based on going down by the K-theory elevator, and we expect that generally nef partitions of reflexive polytopes encode the combinatorial method to construct such a CY threefold.
Geometry and Physics Seminar
Professor Benjamin Gammage
University of Miami
Microlocal Sheaves and Mirror Symmetry
Wednesday, August 21, 2019, 5:00pm
Ungar Room 402
Abstract: Recent developments in symplectic geometry have reduced the computation of the Fukaya category of any affine variety to some local calculations involving constructible sheaves. We survey some recent developments in this theory, focusing on applications to homological mirror symmetry.
IMSA Seminar
Ludmil Katzarkov
University of Miami
New Birational Invariants
Tuesday, August 20, 2019, 4:00pm
Ungar Room 528B
Abstract: We will introduce new birational invariants coming from the interplay of A and B sides of HMS. Applications will be discussed.
Probability Seminar
Kyle Bradford
Georgia Southern University
Stable Adiabatic Times
Wednesday, May 1, 2019, 3:30pm
Ungar Room 411
Abstract: This talk will detail the stability of Markov chains. One measure of stability of a time homogeneous Markov chain is a mixing time. I will define similar measures for special types of time inhomogeneous Markov chains called the adiabatic and stable adiabatic times. I will discuss the use of these Markov chains and I will discuss how the adiabatic and stable adiabatic times relate to mixing times.
Combinatorics Seminar
Frederico Castillo
University of Kansas
Deformations of Coxeter Permutahedra and Coxeter Submodular Function
Monday, April 29, 2019, 5:00pm
Ungar Room 402
Abstract: One way to decompose a polytope is to represent it as a Minkowski of two other polytopes. These smaller pieces are naturally called summands. Starting from a polytope we want to explore the set of all summands. This set can be parametrized by a polyhedral cone, called deformation cone, in a suitable real vector space. We focus on the case where the starting polytope is a Coxeter permutahedron, which is a polytope naturally associated with a root system. This generalizes the type A case which correspond to generalized permutohedra. This is joint work with Federico Ardila, Chris Eur, and Alexander Postnikov.
Applied Math Seminar
Brian Coomes
University of Miami
Shadowing, Rigorous Numerics, and Homoclinic Orbits in Dynamical Systems (Part 2)
Friday, April 26, 2019, 4:00pm
Ungar Room 411
Abstract: We discuss the shadowing phenomenon, methods for transitioning numerical experiments into mathematical proofs, and give some examples. We show how the tools developed can be applied to the problem of proving the existence of a homoclinic orbit with applications to "Sil'nikov chaos." This work is joint with Huseyin Kocak and Ken Palmer.
Combinatorics Seminar
Morgan V. Brown
University of Miami
Chip-firing Groups of Iterated Cones
Monday, April 22, 2019, 5:00pm
Ungar Room 402
Abstract: To any finite graph $\Gamma$ we may associate the chip firing group $G(\Gamma)$, whose order is the number of spanning trees of $\Gamma$. This invariant appears in many mathematical contexts, and in particular illustrates an analogy between Riemann surfaces and graphs. I will present several results about the group $G(\Gamma)$, with a special focus on connections between algebraic geometry and graph theory.
Geometric Analysis Seminar
Marcus Khuri
SUNY Stony Brook
Stationary Vacuum Black Holes in Higher Dimensions
Monday, April 22, 2019, 4:00pm
Ungar Room 506
Abstract: A result of Galloway and Schoen asserts that horizon cross-sections must be of positive Yamabe invariant. In this talk we discuss results on a converse problem. That is, which manifolds of positive Yamabe invariant arise as horizon cross-sections in a stationary vacuum spacetime. We also discuss topological classifications of horizons and domains of outer communication.
Applied Math Seminar
Brian Coomes
University of Miami
Shadowing, Rigorous Numerics, and Homoclinic Orbits in Dynamical Systems
Friday, April 19, 2019, 4:00pm
Ungar Room 411
Abstract: We discuss the shadowing phenomenon, methods for transitioning numerical experiments into mathematical proofs, and give some examples. We show how the tools developed can be applied to the problem of proving the existence of a homoclinic orbit with applications to "Sil'nikov chaos." This work is joint with Huseyin Kocak and Ken Palmer.
Geometry and Physics Seminar
Hans Boden
McMaster University
Concordance of Virtual Knots
Wednesday, April 17, 2019, 5:00pm
Ungar Room 402
Abstract: This talk will be an overview on concordance of virtual knots, with a focus on open problems and recent results. We will examine the general problem of how to extend classical invariants such as the knot signature to the virtual setting, and discuss other invariants of virtual concordance. We show how to combine known slice obstructions with direct methods for slicing to address the question of sliceness for many of the 92,800 virtual knots with 6 or fewer crossings.
Applied Math Seminar
Hongjun Guo
University of Miami
On the Mean Speed Bistable Transition Front in Unbounded Domains
Friday, April 12, 2019, 4:00pm
Ungar Room 411
Abstract: In this talk, we will present the existence and further properties of propagation speeds of transition fronts for bistable reaction-diffusion equations in exterior domains and in some domains with multiple cylindrical branches. In exterior domains, we show that all transition fronts with complete propagation propagate with the same global mean speed, which turns out to be equal to the uniquely defined planar speed. In domains with multiple cylindrical branches, we show that the solutions emanating from some branches and propagating completely are transition fronts propagating with the unique planar speed. I will also give some geometrical and scaling conditions on the domain, either exterior or with multiple cylindrical branches, which guarantee that any transition front has a global mean speed.
Combinatorics Seminar
Rafael González D'León
Sergio Arboleda University, Colombia
On Some Conjectures and Questions Related to Whitney Labelings
Monday, April 8, 2019, 5:00pm
Ungar Room 402
Abstract: Two posets are Whitney duals to each other if the (absolute value of their) Whitney numbers of the first and second kind are switched between the two posets. This notion was introduced by González D'León-Hallam when they studied this property by means of a special family of edge labelings known as Whitney labelings. Graded posets with Whitney labelings have Whitney duals and it turns out that many families of graded posets studied in the literature have Whitney labelings. This is the case of geometric lattices, the lattice of noncrossing partitions, the poset of weighted partitions studied by González D'León-Wachs, the poset of pointed partitions studied by Chapoton-Vallette and the R*S-labelable posets studied by Simion-Stanley. In this talk we will present the main results in the theory of Whitney labelings and connect the theory with current research in other areas where Whitney labelings could be of use. I will discuss joint work with Yeison Quiceno (Universidad Nacional de Colombia) as well as joint work with Josh Hallam (Loyola Marymount University) and José Samper (University of Miami).
Applied Math Seminar
Claudio Cioffi-Revilla
George Mason University
The Nabladot Operator for an Exact and Integrated Calculus of Hybrind Functions
Friday, April 5, 2019, 4:00pm
Ungar Room 402
Abstract: Mathematical models in core areas of science contain hybrid functions with a blend of continuous and discrete variables, as opposed to functions containing either category, not both. Instances of hybrid functions are found across the natural, engineering, and social sciences, modeling a diverse variety of phenomena, such as quantum mechanics, systems reliability theory, and human and social dynamics, among others. Hybrid functions also play a significant role in interesting interdisciplinary areas, such as complex systems science, theories of resilience, sustainability, and risk. The multivariate calculus of models is mathematically exact, and supports substantive scientific theories, when classical tools from infinitesimal calculus and discrete calculus are applicable. However, in the case of hybrid functions, continuous approximations and discretization produce measurable errors, especially when the domain of a discrete variable is within Miller's range of 7 ± 2. I present a novel operator for general hybrid functions. The new operator is called "nabladot," or ∇· , after the continuous nabla operator; it provides an exact calculus for scientific models containing hybrid functions. The paper introduces nabladot and elements of nabladot calculus demonstrated through basic applications to substantive areas of science. Theoretical and methodological aspects are highlighted, including discussion of broader implications for formal scientific theories and research.
Geometry and Physics Seminar
Tye Lidman
North Carolina State University
Spineless Four-manifolds
Wednesday, April 3, 2019, 5:00pm
Ungar Room 402
Abstract: Given two homotopy equivalent manifolds with different dimensions, it is natural to ask if the smaller one embeds in the larger one. We will discuss this problem in the case of four-manifolds homotopy equivalent to surfaces. This is joint work with Adam Levine.
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
PDE and Noncommutative Motives
Wednesday, March 27, 2019, 5:00pm
Ungar Room 402
Combinatorics Seminar
Fabrizio Zanello
Michigan Technological University
On the Parity of the Partition Function
Monday, March 25, 2019, 5:00pm
Ungar Room 402
Abstract: We outline a possible new approach to one of the basic and seemingly intractable conjectures in partition theory, namely that the partition function p(n) is equidistributed modulo 2. The best results available today, obtained incrementally over the last few decades by Serre, Ono, Soundararajan and many others, don't even imply that p(n) is odd for $\sqrt{x}$ values of $n\le x$.
We present an infinite class of conjectural identities modulo 2, and show how to, in principle, prove each such identity. We describe a number of important consequences of these identities: For instance, if any t-multipartition function is odd with positive density and t is not 0 mod 3, then p(n) is also odd with positive density. All of these facts seem virtually impossible to show unconditionally today.
Our arguments employ both complex-analytic and algebraic methods, ranging from a study modulo 2 of some classical Ramanujan identities and other eta product results, to a unified approach to the Fourier coefficients of a broad class of modular forms recently introduced by Radu.
Much of this research is joint with my former PhD student S. Judge and/or with W.J. Keith (see my papers in J. Number Theory, 2015 and 2018; Annals of Comb., 2018).
Applied Math Seminar
Alexandru Hening
Tufts University
Stochastic Persistance and Extinction
Friday, March 22, 2019, 4:00pm
Ungar Room 411
Abstract: A key question in population biology is understanding the conditions under which the species from an ecosystem persist or go extinct. Theoretical and empirical studies have shown that coexistence can be facilitated or negated by both biotic interactions and environmental fluctuations. We study the dynamics of n interacting species that live in a stochastic environment. Our models are described by n dimensional piecewise deterministic Markov processes. These are processes (X(t), r(t)) where the vector X denotes the density of the n species and r(t) is a finite state space process which keeps track of the environment. In any fixed environment the process follows the flow given by a system of ordinary differential equations. The randomness comes from the changes or switches in the environment, which happen at random times. We give sharp conditions under which the the populations persist as well as conditions under which some populations go extinct exponentially fast. As an example we look at the competitive exclusion principle from ecology and show how the random switching can 'rescue' species from extinction. The talk is based on joint work with Dang H. Nguyen (University of Alabama).
Geometry and Physics Seminar
Dr. Andrei Pajitnov
Université de Nantes, France
Arnold Conjecture, Floer Homology, and Augmentation Ideals of Finite Groups
Tuesday, March 4, 2019, 5:00pm
Ungar Room 506
Abstract: Let H be a generic time-dependent 1-periodic Hamiltonian on a closed symplectic manifold M. We use a refined version of the Floer chain complex to obtain new lower bounds for the number P(H) of the 1-periodic orbits of the corresponding hamiltonian vector field. We prove in particular that if the fundamental group of M is finite and solvable or simple, then P(H) is not less than the minimal number of generators of the fundamental group. This is joint work with Kaoru Ono.
Geometric Analysis Seminar
Greg Galloway
University of Miami
On the Geometry and Topology of Initial Data Sets in General Relativity
Monday, March 4, 2019, 4:00pm
Ungar Room 506
Abstract: We present some results concerning the geometry and topology of initial data sets that model the region of space exterior to all black holes. The results to be discussed are closely connected to the Principle of Topological Censorship, which roughly asserts that the topology of the region of spacetime outside of all black holes (and white holes) should be simple. The motivation for these results was to provide support for topological censorship at the pure initial data level, thereby circumventing difficult issues of global evolution. The proofs rely on the recently developed theory of marginally outer trapped surfaces, which are natural spacetime analogues of minimal surfaces in Riemannian geometry.
The talk will begin with a brief overview of general relativity and topological censorship. The talk is based primarily on joint work with various collaborators: Lars Andersson, Mattias Dahl, Michael Eichmair and Dan Pollack.
Geometry and Physics Seminar
Angelica Cueto
Ohio State University
Anticanonical Tropical del Pezzo Cubic Surfaces Contain Exactly 27 Lines
Wednesday, February 27, 2019, 5:00pm
Ungar Room 402
Abstract: Since the beginning of tropical geometry, a persistent challenge has been to emulate tropical versions of classical results in algebraic geometry. The well-know statement "any smooth surface of degree three in P^3 contains exactly 27 lines" is known to be false tropically. Work of Vigeland from 2007 provides examples of tropical cubic surfaces with infinitely many lines and gives a classification of tropical lines on general smooth tropical surfaces in TP^3.
In this talk I will explain how to correct this pathology by viewing the surface as a del Pezzo cubic and considering its embedding in P^44 via its anticanonical bundle. The combinatorics of the root system of type E_6 and a tropical notion of convexity will play a central role in the construction. This is joint work in progress with Anand Deopurkar.
Geometric Analysis Seminar
Pengzi Miao
University of Miami
Quasi-local Mass of CMC Surfaces
Monday, February 25, 2019, 4:00pm
Ungar Room 506
Abstract: We discuss two notions of quasi-local mass associated to a surface in a 3-manifold with nonnegative scalar curvature. One is the Hawking mass and the other one is the Bartnik mass. The former has a simple expression, but lacks the positivity property; while the latter is manifestly nonnegative by defintion, but is hard to compute. In this talk, we focus on CMC (constant mean curvature) surfaces. We discuss an intrinsic condition that shows the positivity of the Hawking mass and give some estimate on the Bartnik mass. The talk is based on recent joint work with Yaohua Wang and Naqing Xie.
Combinatorics Seminar
Francesco Brenti
University of Rome Tor Vergata, Italy
Permutations, Tensor Products, and Cuntz Algebra Automorphisms
Monday, February 18, 2019, 5:00pm
Ungar Room 402
Abstract: We introduce and study a new class of permutations which arises from the automorphisms of the Cuntz algebra. I will define this class, explain its relation to the Cuntz algebra, present results about symmetries, constructions, characterizations, and enumeration of these permutations, and discuss some open problems and conjectures. This is joint work with Roberto Conti.
Applied Math Seminar
K.-Y. (Adrian) Lam
Ohio State University
Dirac Concentration in an Integro-PDE Midel from Adaptation
Monday, February 18, 2019, 10:00am
Ungar Room 411
Abstract: We consider a mutation-selection model of a population structured by the spatial variables and a trait variable which is the diffusion rate. Competition for resource is local in spatial variables, but nonlocal in the trait variable. We establish the existence and asymptotic profile of a steady state solution. Our result shows that in the limit of small mutation rate, the solution remains regular in the spatial variables and yet concentrates in the trait variable and forms a Dirac mass supported at the lowest diffusion rate. Similar result was independently obtained by B. Perthame and T. Souganidis via an elegant method. I will present a sketch of proof blending the arguments of both papers. This is joint work with Yuan Lou and Wenrui Hao (Penn State).
Geometry and Physics Seminar
Professor Carlos Simpson
Université Nice Sophia Antipolis
Spectral Networks, from WKB Theory and Harmonic Mappings to Buildings, to Stability for Fukaya Categories with Coefficients
Friday, February 15, 2019, 4:00pm
Ungar Room 402
Abstract: The spectral networks of Gaiotto-Moore-Neitzke are connected to WKB approximations for ODE's with large parameter, and they appear as singularities of harmonic mappings to buildings. They also represent special Lagrangian objects for Fukaya categories with coefficients, and hence suggest tantalizing relationships between these theories. This is based on joint work with Haiden, Katzarkov, Noll and Pandit.
Geometry and Physics Seminar
Tokio Sasaki
Washington University in St. Louis
Limits and Singularities for $K_1$ Cycles on Algebraic Surfaces
Monday, February 13, 2019, 5:00pm
Ungar Room 402
Abstract: On a projective complex variety, the rational regulator map to the Deligne cohomology gives a transcendental invariant of the motivic cohomology. By considering a family of rational regulator values on a family of such varieties, we obtain a higher normal function, which is a generalization of the usual normal functions. Its asymptotic behavior along the discriminant locus of the family is described by the "singularity invariant", or if it vanishes, "limit invariant".
In this talk, we observe how the singularity and limit invariants appear in families of real regulators, and in particular, how to detect $\mathbb{R}$-regulator indecomposable $K_1$ cycles for certain types of algebraic surfaces in $\mathbb{P}^3$. For degree 4 surfaces of this type, these indecomposable cycles give an explicit proof of Hodge-$\mathcal{D}$-conjecture. As another application, we also construct new examples of non trivial elements in the Griffiths groups on a certain Calabi-Yau threefold, which is a general fiber of a Tyurin degeneration arising from two reflexive polytopes. Since these Calabi-Yau manifolds and (higher) cycles are totally derived from the combinatorial geometry of these polytopes, we expect that their dual polytopes encodes the "mirror" objects via mirror symmetry.
Combinatorics Seminar
Eric Katz
Ohio State University
The Unipotent Torelli Theorem for Graphs
Monday, February 4, 2019, 5:00pm
Ungar Room 402
Abstract: The classical Torelli theorem says that a Riemann surface can be recovered from its Jacobian, which is a principally polarized Abelian variety. There is an analogous theorem for graphs, due to Artamkin and Caporaso-Viviani, that the 2-isomorphism class of a graph can be recovered from its cycle space, equipped with its cycle pairing. We ask what happens when one encodes mildly non-abelian data as in the Unipotent Torelli theorem for Riemann surfaces due to Hain and Pulte. This leads us to introducing the analogue of iterated integrals on graphs and encoding them in a particular structure. This structure turns out to recover bridgeless graphs up to isomorphism. We discuss some of the application of this result. This is joint work with Raymond Cheng.
Geometry and Physics Seminar
Professor Victor Przyjalkowski
National research University Higher School of Economics
On the KKP Conjecture
Friday, January 25, 2019, 4:00pm
Ungar Room 402
Geometry and Physics Seminar
Professor Daniel Pomerleano
Imperial College London
An Intrinsic Batyrev Construction via Symplectic Topology
Wednesday, January 23, 2019, 5:30pm
Ungar Room 402
Abstract: I will describe an intrinsic version of Batyrev's mirror construction associated to a general maximally degenerate log Calabi-Yau pair (M,D) using an invariant known as symplectic cohomology. The symplectic cohomology ring of log Calabi-Yau varieties comes equipped with a flat degeneration to the Stanley-Reisner ring of the dual intersection complex of a compactifying divisor. The deformation from the central fiber can be alternatively described using a symplectic version of log Gromov-Witten invariants, which modulo a certain technical conjecture enables us to relate our construction to recent mirror constructions of Gross-Hacking-Keel and Gross-Siebert.
Combinatorics Seminar
Emanuele Delucchi
Université de Fribourg, Switzerland
Stanley-Reisner Rings of Symmetric Simplical Complexes
Tuesday, January 22, 2019, 5:00pm
Ungar Room 402
Abstract: A classical theme in algebraic combinatorics is the study of face rings of finite simplicial complexes (named after Stanley and Reisner, two of the pioneers of this field). In this talk I will examine the case where the simplicial complexes at hand carry a group action and are allowed to be infinite.
I will present the foundations of this generalized theory with a special focus on simplicial complexes associated to (semi)matroids, where the associated rings enjoy especially nice algebraic properties. A main motivation for our work comes from the theory of arrangements in Abelian Lie groups (e.g., toric and elliptic arrangements), and in particular from the quest of understanding numerical properties of the coefficients of characteristic polynomials and h-polynomials of arithmetic matroids. I will describe our current results in this direction and, time permitting, I will outline some open questions that arise in this new framework. (Joint work with Alessio D'Alì.)
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
Fano Manifolds Old and New
Tuesday, January 15, 2019, 6:15pm
Ungar Room 402
Applied Math Seminar
Sergio Fernandez Rincon
Complutense University of Madrid
How Do Diffusion and Heterogeneities Affect Competition?
Friday, November 30, 2018, 12:20pm
Ungar Room 411
Abstract: It has been nearly 90 years since Alfred J. Lotka and Vito Volterra proposed the models that established the basis of the study of the interaction between species. During this time, significant improvements have been achieved in this field, with the addition of stochastic techniques and the introduction of the space.
In this talk, we will discuss the last point, analyzing fascinating results that arise when the diffusion of the species and the heterogeneity of the environment are incorporated into the basic Lotka-Volterra competition model for two species. Particularly, the Singular Perturbation Theorem and the Principle of Induced Instability will be established as a link between non-diffusive and diffusive models and, among other consequences, either uniqueness or multiplicity will be obtained depending on the configuration of the habitat.
Geometry and Physics Seminar
Tyler Foster
Florida State University
Asymptotics of Primes in Short Intervals on Curves over Finite Fields
Wednesday, November 28, 2018, 5:00pm
Ungar Room 402
Abstract: I will talk about recent joint work with E. Bank, as well as work of A. Entin, which establishes several function field analogues of conjectures concerning the distribution of primes inside "short intervals" – intervals whose widths grow at a certain rate as they move off to infinity.
Combinatorics Seminar
Alejandro Ginory
Rutgers University
Positivity Conjectures for Jack Polynomials
Monday, November 12, 2018, 5:00pm
Ungar Room 402
Abstract: In the course of investigating a statistical problem involving estimators for a parameter matrix, Donald Richards and Siddhartha Sahi have recently formulated certain positivity conjectures involving Jack polynomials. In this talk, I will present a strengthened version of the Richards-Sahi conjectures, which depends on a pair of partitions, and sketch a proof in a number of cases. This strengthened conjectures suggests new combinatorial identities involving Jack analogues of Kostka numbers and hook-length formulas.
Geometry and Physics Seminar
Dr. Raphael Zentner
Universität Regensburg
SL(2,C)-representations of Homology 3-spheres
Wednesday, November 7, 2018, 5:00pm
Ungar Room 402
Abstract: We prove that the splicing of any two non-trivial knots in the 3-sphere admits an irreducible SU(2)-representation of its fundamental group. This uses instanton gauge theory, and in particular a non-vanishing result of Kronheimer-Mrowka and some new results that we establish for holonomy perturbations of the ASD equation. Using a result of Boileau, Rubinstein and Wang (which builds on the geometrization theorem of 3-manifolds), it follows that the fundamental group of any integer homology 3-sphere different from the 3-sphere admits irreducible representations of its fundamental group in SL(2,C).
Geometric Analysis Seminar
Svetlana Roudenko
Florida International University
On Spectral Properties for Blow-up Solutions and Soliton Stability in Dispersive Equations
Monday, November 5, 2018, 4:00pm
Ungar Room 506
Abstract: When studying the blow-up dynamics and soliton stabilty in the NLS-type and KdV-type equations, we encounter spectral property questions, arising from linearization, or from the virial-type arguments. We will discuss examples of the L^2-critical blow-up dynamics, spectral properties and its consequences in the NLS and the Hartree equations, and then will describe the situation in Zakharov-Kuznetsov equation, which is a higher dimensional generalization of the KdV equation.
Geometry and Physics Seminar
Federico Buonerba
New York University
Stable Reduction of Foliated Surfaces
Friday, November 2, 2018, 4:00pm
Ungar Room 402
Abstract: In 1977 Bogomolov proved that on surfaces of general type with c_1^2>c_2, curves of a given geometric genus form a bounded family. The role played by foliations in his proof was further investigated by McQuillan, who in 1998 proved the Green-Griffiths conjecture for the same class of surfaces. In this talk I will review some basic properties of foliations on algebraic surfaces, with a focus on birational geometry as initiated by Brunella, McQuillan and others. I will then discuss the problem of their variation in families, and present the main ideas behind the proof of the stable reduction theorem in this context.
Combinatorics Seminar
Joseph Doolittle
University of Kansas
Reconstructing Spheres and Polytopes
Monday, October 29, 2018, 5:00pm
Ungar Room 402
Abstract: We review the historical progress of the problem of determining all faces of a sphere from partial information, starting in 1916 through the modern day. We culminate in a counterexample which disproves the strongest possible version of a conjecture made by Perles in 1960. This strengthened conjecture would imply that simplicial 3-spheres are reconstructible from their facet-ridge graph. While this conjecture fails, in its failure it leaves behind a new technique which may yet solve the problem of reconstructiblity of simplicial spheres.
Geometric Analysis Seminar
Rafael Montezuma
Princeton University
Extremal Metrics for the Min-max Width
Monday, October 29, 2018, 4:00pm
Ungar Room 506
Abstract: We present our study on the min-max width of Riemannian three-dimensional spheres. This is a natural geometric invariant which is closely related with critical values of the area functional acting on closed surfaces, and can be interpreted as the first eigenvalue of a non-linear spectrum of a Riemannian metric, as suggested by Gromov. We will focus first on optimal bounds for the above invariant involving their volumes in a fixed conformal classes. If time permits we will discuss some general properties of extremal metrics for the min-max width. This is all part of a joint work with Lucas Ambrozio.
Applied Math Seminar
Chiu-Ju Lin
Modeling the Trade-off between Transmissibility and Contact in Infectious Disease Dynamics
Friday, October 26, 2018, 12:20pm
Ungar Room 411
Abstract: Symptom severity affects disease transmission both by impacting contact rates, as well as by influencing the probability of transmission given contact. This involves a trade-off between these two factors, as increased symptom severity will tend to decrease contact rates, but increase the probability of transmission given contact (as pathogen shedding rates increase with symptom severity). This talk explores this trade-off between contact and transmission given contact, using a simple compartmental susceptible- infected-recovered type model. Under mild assumptions on how contact and transmission probability vary with symptom severity, we give sufficient, biologically intuitive criteria for when the basic reproduction number varies non-monotonically with symptom severity. Multiple critical points are possible. We give a complete characterization of the region in parameter space where multiple critical points are located in the special case where contact rate decreases exponentially with symptom severity. We consider a multi-strain version of the model with complete cross-immunity and no super-infection. In this model, we prove that the strain with highest basic reproduction number drives the other strains to extinction. This has both evolutionary and epidemiological implications, including the possibility of an intervention paradoxically resulting in increased infection prevalence. This is joint work with Kristen A. Deger and Joseph H. Tien.
Geometric Analysis Seminar
Sven Hirsch
Duke University
Mean Curvature Flow in a Ricci Flow Background
Monday, October 22, 2018, 4:00pm
Ungar Room 506
Abstract: Geometric flows have attracted much attention in the past years and have proven to be a powerful analytic tool leading to many groundbreaking results such as Hamilton's and Perelman's proof of the Poincare conjecture, Huisken's and Ilmanen's proof of the Penrose conjecture and Brendle's and Schoen's proof of the differentiable sphere theorem. We begin with giving a general introduction to geometric flows via the one dimensional heat equation before proceeding with the discussion of mean curvature flow in a Ricci flow backgroud. Using a variation of geodesics we derive a distance comparison theorem which was first observed by Huisken and can be considered a parabolic version of Frankel's theorem. In fact, we will obtain Frankel's theorem as a corollary of this result. Next, we discuss a long time existence result originating from our thesis, stating that suitable convex surfaces converge to a sphere after rescaling. This work is based on Huisken's approach to mean curvature flow in a fixed Riemannian manifold and employs Stampacchia iteration in order to obtain an a priori estimate pinching the eigenvalues of the second fundamental form. Finally, we present a novel rescaling technique which requires a slightly stronger gradient estimate but greatly simplifies the convergence arguments.
Geometry and Physics Seminar
Ken Baker
University of Miami
Genus Minimization of Homology Classes of Knots in Lens Spaces
Wednesday, October 17, 2018, 5:00pm
Ungar Room 402
Abstract: The genus of torsion homology class in an orientable 3-manifold is the minimal (rational) Seifert genus among knots representing that class. Knots in lens spaces with integral surgeries to S^3 are known to be minimizers of this genus. On the other hand, a classification of knots in S^3 with Dehn surgeries to lens spaces would be complete (thereby resolving the Berge Conjecture) if we knew that such minimizers were unique. We'll discuss the current state of genus minimizers and a strategy for addressing this Dehn surgery classification problem.
Geometry and Physics Seminar
Manuel Rivera
University of Miami
Singular Chains and the Fundamental Group
Tuesday, October 16, 2018, 5:00pm
Ungar Room 402
Abstract: I will explain how the singular chains on a connected space, considered as a coalgebra with extra algebraic structure, encodes the data of the fundamental group of the space. Then I will then introduce an algebraic notion of weak equivalence between differential graded coalgebras which is stronger than quasi-isomorphism to show the following version of a classical Whitehead theorem: a map between connected spaces is a weak homotopy equivalence if and only if the induced map at the level of singular chains is a weak equivalence (in the strong sense) of dg coalgebras.
Combinatorics Seminar
Amzi Jeffs
University of Washington
Convex Union Representability of Simplicial Complexes
Monday, October 15, 2018, 5:00pm
Ungar Room 402
Abstract: Given a collection of convex open sets, one can form an associated simplicial complex that records their intersection patterns. This complex retains important topological information about the sets; for example, Borsuk's Nerve Lemma states that it is homotopy equivalent to the union of the sets. I will discuss what happens when the union of the sets is convex. In this case, the associated simplicial complex has a number of rich combinatorial properties. I will also describe an application to the theory of convex codes.
Geometry and Physics Seminar
Ludmil Kaztarkov
University of Miami
Categorical Curve Complex
Wednesday, October 10, 2018, 5:00pm
Ungar Room 402
Abstract: This is an attempt to present a category theory with a human face. In the talk we will recall classical topological and geometric constructions used in the calculations of mapping class groups and lamination conjectures. We will look at them from new prospective. Applications will be discussed.
Combinatorics Seminar
Alex Lazar
University of Miami
On the Intersection Lattice of the Homogenized Linial Arrangement
Monday, October 8, 2018, 5:00pm
Ungar Room 402
Abstract: In 2017, Hetyei introduced the homogenized Linial arrangement and showed that the number of regions is equal to a median Genocchi number. In this talk, I will discuss joint work with Wachs, in which we refine Hetyie's result by computing the Möbius function of the lattice of intersections of the arrangement. We show that the Möbius invariant of the intersection lattice is a Genocchi number. Our techniques also yield a type B analog of Hetyei's result and more generally a Dowling arrangement analog involving a new q-analog of the median Genocchi numbers.
Geometric Analysis Seminar
Eric Woolgar
University of Alberta
Formal Power Series Solutions of the Bach Equation
Monday, October 8, 2018, 4:00pm
Ungar Room 506
Abstract: Conformal gravity is an alternative to Einstein gravity in 4 dimensions, obtained by replacing the Einstein equation by the Bach equation, which has many more solutions. Maldacena has proposed that the theories are equivalent, provided one imposes certain boundary and physical conditions to remove the additional solutions of the Bach equation. We test this idea. Following the method laid out by Fefferman and Graham for the Einstein equation, we expand asymptotically hyperbolic solutions of the Bach equation in power series about conformal infinity, so as to identify the free data and find those data that yield Einstein metrics. There are infinitely many free data, reflecting the conformal invariance of the 4-dimensional Bach equation, but even if we choose to break conformal invariance by imposing a constant-scalar-curvature condition, the so-called mass aspect tensor remains freely specifiable.
In dimensions greater than 4, there are many different generalizations of the Bach tensor, most of which are not well-suited to the Fefferman-Graham method. We choose a well-suited definition and find that the free data separate into two pairs of data, reflecting the separation of data for the Einstein equation into "Dirichlet" and "Neumann" data.
This talk is based on joint work with Aghil Alaee.
Applied Math Seminar
Ting-Hao Hsu
University of Miami
Number and Stability of Relaxation Oscillations for Predator-Prey Systems with Small Death Rates
Friday, October 5, 2018, 12:20pm
Ungar Room 411
Abstract: Predator-prey models that possess limit cycles can be used to explain oscillatory phenomena in real-world data, and have been studied extensively in the literature. In this talk, we will consider predator-prey systems with small predator death rate as fast-slow systems, and derive new characteristic functions that determine the location and the stability of relaxation oscillations. This criterion determines the number and the global stability of limit cycles for some planar predator-prey systems. This criterion can also been extended (joint work with Gail S. K. Wolkowicz) to be applied on some three-dimensional systems, including chemostat predator-prey systems and a class of epidemic models.
Geometry and Physics Seminar
Dr. Slawomir Kwasik
Tulane University
Decomposing Manifolds into Cartesian Products
Wednesday, October 3, 2018, 5:00pm
Ungar Room 402
Abstract: We study the decomposability of a Cartesian product of two non-decomposable manifolds into products of lower dimensional manifolds. For 3-manifolds we obtain an analog of a result due to Borsuk for surfaces, and in higher dimensions we show that similar analogs do not exist. This is a joint work with Reinhard Schultz.
Combinatorics Seminar
Bruno Benedetti
University of Miami
Random Preprocessing in Computational Topology
Monday, October 1, 2018, 5:00pm
Ungar Room 402
Abstract: Computational topology aims at understanding the 'shape' (=homotopy type, or sometimes just homology) of big data. In 2014 with Frank Lutz we introduced Random Discrete Morse theory as an experimental measure for the complicatedness of a triangulation. This measure depends both on the homotopy type of the space, and on how nicely the space is triangulated. Our approach was elementary, but sometimes successful even for huge inputs. I'll discuss some variants, drawbacks, and possible new ideas that were figured out in the meantime. At the same time, these approaches reveal that the existing libraries of examples in computational topology are all 'too easy' for testing algorithms. So let's build a new one!
Geometric Analysis Seminar
Stephen McCormick
Uppsala University
Some Recent Results Pertaining to Bartnik's Quasi-local Mass
Monday, October 1, 2018, 4:00pm
Ungar Room 506
Abstract: Bartnik's quasi-local mass is often said to be one of the most likely quantities to give a physical measure of the gravitational field mass/energy in general relativity, however it is effectively impossible to compute it general. The Bartnik mass of a domain (or its boundary data) is given by the infimum of the ADM mass over an appropriate space of asymptotically flat manifolds with nonnegative scalar curvature.
In this talk, we give a brief introduction to the Bartnik mass and give some related results that follow from a gluing technique. In particular, we give conditions that ensure two subtly different definitions of the Bartnik mass yield the same value and prove that the mass is continuous with respect to the boundary data.
Geometry/Topology Seminar
Christine Ruey Shan Lee
University of South Alabama
The Strong Slope Conjecture for Montesinos Knots
Friday, September 28, 2018, 3:00pm
Ungar Room 402
Abstract: The Strong Slope Conjecture by Garoufalidis and Kalfagianni-Tran relates the topology of essential surfaces to the colored Jones polynomial, which connects fundamental objects of 3-manifold topology to quantum topology, much like the Volume Conjecture. Much of the progress on the conjecture has been made by computing the topology of essential surfaces separately from the degree of the polynomial, and comparing the end results. This provides few insights to the reasons behind the conjecture.
To address this problem, we (joint with S. Garoufalidis and R. van der Veen) use Montesinos knots as a template and verify the conjecture for most knots in the family by establishing a close analogy between terms in the state sum defining the colored Jones polynomial of the knot, and properly embedded surfaces in the knot complement. In this talk, I will discuss these results and how they present a model for understanding the Strong Slope Conjecture for all knots, as well as the remaining difficulty.
Combinatorics Seminar
José Alejandro Samper
University of Miami
Round Polytopes
Monday, September 24, 2018, 5:00pm
Ungar Room 402
Abstract: It is known that the space of polytopes is dense in the space of closed bounded convex sets endowed with various different metrics. Given that polytopes come equipped with combinatorial structure it is reasonable to ask about the combinatorial structure of a polytope that is a good approximation to a given convex body K. We will discuss theorems about simplicial polytopes approximating convex bodies whose boundary is smooth (e.g an Euclidean ball of radius one).
Geometric Analysis Seminar
Santiago Simanca
Visiting Scholar
Mathematics Department
University of Miami
Isometric Embeddings into Spheres:
A Conformal Pascal Triangle (Cone?)
Monday, September 24, 2018, 4:00pm
Ungar Room 506
Abstract: Special properties of a Riemannian manifold (M,g) isometrically embedded into abackground manifold (N,h are reflected in special properties of extrinsic quantities, second fundamental form \alpha, mean curvature vector H. Same if (M,J,g) is almost Hermitian, but we now add the extrinsic quantity \alpha( . , J. ) into the consideration.
Any (M,J,g) admits a solution to both the usual and almost Hermitian Yamabe problems. Nash's theorem produces isometric embeddings of M with these metrics into standard spheres. The twisting by J affects the conformal level of the embedding.
We sketch a conformal Pascal triangle of (almost Hermitian) manifolds isometrically embedded into standard spheres. The description is fairly detailed if the manifold has nonnegative Ricci curvature and the level of the critical embedding is small (critical relative to the functionals associated to the extrinsic quantities).
Applied Math Seminar
Steve Cantrell
University of Miami
Resident-invader Dynamics in Infinite Dimensional Systems
Friday, September 21, 2018, 12:20pm
Ungar Room 411
Abstract: Motivated by evolutionary biology, we study general infinite-dimensional dynamical systems involving two species - the resident and the invader. Sufficient conditions for competitive exclusion phenomena are given when the two species play similar, but distinct, strategies. Those conditions are based on invasibility criteria, and allow, for example, the identification of evolutionarily stable strategies in the framework of adaptive dynamics. The results extend ideas developed by S. Geritz et al. for a class of ordinary differential equations.
Applied Math Seminar
Chris Cosner
University of Miami
Recent Work on the Ecology and Evolution of Dispersal
Friday, September 14, 2018, 12:20pm
Ungar Room 411
Abstract: This will be an informal talk describing some topics related to dispersal that I have been studying recently. Specifically, I will present some background, modeling, and and analysis of models related to optimal dispersal of organisms in time periodic environments, the use of nonlocal information in dispersal, and switching between multiple movement modes. The models are all based on reaction-diffusion-advection equations or systems.
Geometry and Physics Seminar
Morgan Brown
University of Miami
Integral Affine Structures in Non-Archimedian Geometry
Wednesday, September 12, 2018, 5:00pm
Ungar Room 402
Abstract: An integral affine structure on a manifold X consists of a flat connection and the choice of a full rank lattice inside each tangent space compatible with that connection. I will explain how integral affine structures arise in non-archimedian geometry, following the work of Kontsevich and Soibelman, with explicit examples.
Geometry and Physics Seminar
Damian Brotbek
University of Strasbourg
Jet Differentials on Complete Intersections and Applications
Thursday, June 7, 2018, 2:30pm
Ungar Room 402
Abstract: The goal of this talk is to describe a strategy allowing us to construct symmetric differential forms, and more generally jet differentials, on complete intersection varieties. We will also explain how one can deduce from this that general hypersurfaces of large degrees are hyperbolic, a result conjectured by Kobayashi in the 70's.
Combinatorics Seminar
Alex Lazar
University of Miami
The Homogenized Linial Arrangement
Monday, April 23, 2018, 5:00pm
Ungar Room 402
Abstract: The homogenized Linial arrangement was introduced by Hetyei in 2017 to prove enumerative results in graph theory. In this talk we present preliminary results of a deeper study of the combinatorics of this hyperplane arrangement.
Geometric Analysis Seminar
Mingliang Cai
Mathematics Department
University of Miami
On the Schoen-Yau Positive Mass Theorem
Monday, April 23, 2018, 4:00pm
Ungar Room 506
Abstract: The Riemannian Positive Mass Theorem states that the mass of an asymptotically flat manifold of nonnegative scalar curvature is nonnegative and zero only when the manifold is actually Euclidean. The theorem was proved by Schoen and Yau initially for dimension less than 8 in 1979 and recently for all dimensions. In this talk, we propose a new approach and, in particular, express the mass in terms of other invariants including the scalar curvature.
Hodge Theory Seminar
Morgan Brown
University of Miami
The Dual Complex of a Semi-log Canonical Surface
Friday, April 20, 2018, 4:00pm
Ungar Room 506
Abstract: Semi-log canonical surfaces arise as the limits of canonically polarized surfaces. In this sense they are the natural generalization to surfaces of nodal curves. The goal of this talk is to explore how we can associate a 2-dimensional cell complex to a semi-log canonical surface, analogous to the dual graph of a nodal curve.
Geometry and Physics Seminar
Phillip Griffiths
Institute of Advanced Study, Princeton
University of Miami
Satake-Baily-Borel Completions of Moduli Spaces
Wednesday, April 18, 2018, 5:00pm
Ungar Room 402
Abstract: In the classical cases of curves, abelian varieties, K3 surfaces, etc. the SBB compactification of quotients of Hermitian symmetric domains by arithmetic groups gives a minimal way of completing moduli moduli spaces. In the non-classical case when one doesn't have an Hermitian symmetric and when the global monodromy groups may not be arithmetic, almost nothing is known about the global structure of the boundary of KSBA moduli spaces. We will give an informal account of the general SBB completion of moduli spaces explaining what is "behind the scenes" in the construction and in its applications to moduli.
Combinatorics Seminar
Felix Gotti
University of California, Berkeley
A Connection Between Tilings and Matroids on the Lattice Points of a Regular Simplex
Monday, April 16, 2018, 5:00pm
Ungar Room 402
Abstract: The set of lattice points T(n,d) inside the regular simplex obtained by intersecting the nonnegative cone of R^d with the affine hyperplane x_1 + ... + x_d = n-1 is the ground set of a matroid M(n,d) whose independent sets are precisely those subsets S of T(n,d) satisfying that the intersection of S and T has at most k elements for each parallel translate T of the regular simplex T(k,d). We will present some matroidal properties of M(n,3) in connection to certain tilings of holey triangular regions associated to the subsets of T(n,3). In particular, we will provide characterizations for the independent sets and circuits of M(n,3) related to certain tilings of their holey triangular regions, extending a characterization of the bases of M(n,3) already given by Ardila and Billey. If time permits, we will also exhibit connections between tilings and the flats and connectivity of the matroids M(n,3).
Geometric Analysis Seminar
Lev Kapitanski
Mathematics Department
University of Miami
Hooke, Euler, Lagrange
Monday, April 16, 2018, 4:00pm
Ungar Room 506
Abstract: Mathematical theories describing the motion of an ideal fluid and the deformation of an elastic solid have many challenging problems. I will discuss some of them and explain recent progress Lars Andersson and I have made.
Combinatorics Seminar
Bruno Benedetti
University of Miami
Parity Arguments in Combinatorics and Beyond
Monday, April 9, 2018, 5:00pm
Ungar Room 402
Abstract: We survey four cute ways to apply the combinatorial concept of parity to other fields. Namely:
• in algebra, the parity distinction for permutations (dating back at least to Cauchy, 1815);
• in topology, the combinatorial proof of Brouwer's fixed point theorem (Sperner, 1928);
• in geometry, the neighborlyness of cyclic polytopes (Gale, 1963);
• and in number theory, the 'one-sentence proof' of the sum-of-squares theorem (Zagier, 1990).
This talk is intended as didactical, rather than research-oriented; it does not assume expertise in any of the four fields above.
Geometric Analysis Seminar
Lars Andersson
Max Planck Institute for Gravitational Physics
Gauge, Peeling and Linearized Gravity on the Kerr Exterior Spacetime
Monday, April 9, 2018, 4:00pm
Ungar Room 506
Abstract: The Teukolsky Master Equation governs the dynamics of linearized gravity on the Kerr rotating black hole spacetime. In this talk I will discuss some aspects of scattering and peeling for the Teukolsky equation and the issue of gauge choice in the problem of linearized stability on the Kerr background. This is based on joint work with Thomas Bäckdahl, Pieter Blue, and Siyuan Ma.
Geometry and Physics Seminar
Andrew Harder
University of Miami
Mirror Symmetry and Filtrations
Wednesday, April 4, 2018, 5:00pm
Ungar Room 402
Geometric Analysis Seminar
Matthew Haddad
Physics Department
University of Miami
Topological Defects in Anti de Sitter Space
Monday, April 2, 2018, 4:00pm
Ungar Room 506
Abstract: The solutions to the equations of motion for a topological defect in Minkowski space have been known for many years. In particular, the work of Bogomolny, Prasad, and Sommerfield allows a solution for these equations in the limit that mass of the scalar field vanishes. When the embedding space is AdS, another limit exists, which provides additional analytic solutions to appropriately-modified equations of motion.
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
P=W and Algebraic Cycles
Wednesday, March 21, 2018, 5:00pm
Ungar Room 402
Abstract: This will be a provocative report of my scintilla of progress of learning from Carlos and Phillip.
Combinatorics Seminar
José Alejandro Samper
University of Miami
Threshold Hypergraphs Revisited
Monday, March 19, 2018, 5:00pm
Ungar Room 402
Abstract: Threshold graphs were introduced by Chvatal and Hammer(1974) as tools in optimization. They coincide with the class of shifted graphs and can be described and studied in three different ways: purely combinatorial, slicing the second hypersimplex or slicing a cube. A question of Golumbic(1978) answered in the negative by Reiterman, Rodl, Sinajova and Tuma(1985), asks for higher dimensional analogues. We will give a geometric explanation for the negative answer to such question and propose a corrected version of Golumbic's question. We will then highlight the relevance of this question in the theory matroid polytopes.
Geometric Analysis Seminar
Nathan Totz
University of Miami
Rigorous Justification of Modulation Approximations to the Full Water Wave Problem
Monday, March 19, 2018, 4:00pm
Ungar Room 506
Abstract: We consider solutions to the infinite depth water wave problem in 2D and 3D which are to leading order wave packets with small $O(\epsilon)$ amplitude and slow spatial decay that are balanced. In the case of zero surface tension, multiscale calculations formally suggest that such solutions have modulations that evolve on $O(\epsilon^{-2})$ time scales according to a version of a cubic nonlinear Schrodinger (NLS) equation. Justifying this rigorously is a real problem, since standard existence results do not yield solutions to the water wave problem that exist for long enough to see the NLS dynamics. Nonetheless, given initial data suitably close to such a wave packet in $L^2$ Sobolev space, we show that there exists a unique solution to the water wave problem which remains within $o(\epsilon)$ to the formal approximation on the natural NLS time scales. The key ingredient in the proof is a formulation of the evolution equations for the water wave problem developed by Sijue Wu with either no quadratic nonlinearities (in 2D) or mild quadratic nonlinearities that can be eliminated using the method of normal forms (in 3D). If time permits, we will discuss recent work towards a justification result in the presence of the effects of both gravity and surface tension in 2D.
Combinatorics Seminar
Jessica Striker
North Dakota State University
Sign Matrix Polytopes
Monday, March 5, 2018, 5:00pm
Ungar Room 402
Abstract: Motivated by the study of polytopes formed as the convex hull of permutation matrices and alternating sign matrices, we define several new families of polytopes as convex hulls of sign matrices, which are certain {0,1,-1}-matrices in bijection with semistandard Young tableaux. We investigate various properties of these polytopes, including their inequality descriptions, vertices, facets, and face lattices, as well as connections to alternating sign matrix polytopes and transportation polytopes.
Geometry and Physics Seminar
Pranav Pandit
University of Vienna
From Homotopical Mathematics to Emergent Geometry
Wednesday, February 28, 2018, 5:00pm
Ungar Room 402
Abstract: At the root of the fundamental mathematical notion of symmetry is the idea that it is useful to keep track of the multitude of ways in which two objects can be identified, rather than to simply ask if they are the same. Taking this idea to its logical conclusion leads to a mathematical universe where shapes (homotopy types) are the fundamental building blocks of mathematical structures instead of sets. Derived geometry is geometry in this homotopy-theoretic context. It provides an intuitive language for quantum field theory, and a powerful framework in which "classical geometry" can be seen to emerge from the structure of quantum field theory.
After introducing this paradigm, I will touch upon joint work with Fabian Haiden, Ludmil Katzarkov, and Maxim Kontsevich, in which we attempt to formalize and understand the mathematical structures underlying the physical notion of stability for D-branes in string theory using the language of derived noncommutative geometry. Our work builds upon Bridgeland's notion of stability conditions on triangulated categories, and is inspired by ideas from symplectic geometry, non-Archimedean geometry, dynamical systems, geometric invariant theory, and the Donaldson-Uhlenbeck-Yau correspondence.
Geometry and Physics Seminar
Ben Davison
University of Edinburgh
BPS Cohomology and Character Varieties
Tuesday, February 27, 2018, 5:00pm
Ungar Room 506
Abstract: In this talk I will review a general definition of BPS invariants counting stable sheaves on a Calabi-Yau 3-fold. It turns out that the definition is really a theorem, which in turn can be categorified to give a notion of the cohomology of the space of BPS states. Although this cohomology can be explicitly defined, the explicit definition is not really the cohomology of a space at all, but the vanishing cycle cohomology of the intersection complex of a coarse moduli space.
Despite being more closely analogous to the category of sheaves on a K3 surface than a CY3 variety, the category of representations of the fundamental group of a Riemann surface fits naturally into this theory, and in contrast with the general case, the BPS cohomology has a (conjecturally) much more down-to-earth description: it is the cohomology of the twisted character variety, a central and mysterious object in the study of nonabelian Hodge theory.
Geometric Analysis Seminar
Nikolai Saveliev
University of Miami
End-periodic Index Theory and Metrics of Positive Scalar Curvature
Monday, February 26, 2018, 4:00pm
Ungar Room 506
Abstract: We study metrics of positive scalar curvature on certain closed manifolds of even dimension. We provide a new obstruction to the existence of such metrics, and give examples of manifolds with infinitely many path components in the moduli space of metrics of positive scalar curvature. The methods include the index theorem for end-periodic Dirac operators (due to Mrowka, Ruberman, and the speaker) and some Seiberg-Witten theory. This is a joint project with Jianfeng Lin, Tom Mrowka, and Danny Ruberman.
Geometry and Physics Seminar
Dmitry Kaledin
Russian Academy of Sciences
Simplicial Sets and Partially Ordered Sets
Wednesday, February 21, 2018, 5:00pm
Ungar Room 402
Abstract: I am going to discuss how to describe families of groupoids over simplicial sets in terms of families over partially ordered sets, and why this seems a good thing to do.
Combinatorics Seminar
Michelle Wachs
University of Miami
On Enumerators of Smirnov Words by Descents and by Cyclic Descents
Monday, February 19, 2018, 5:00pm
Ungar Room 402
Abstract: Smirnov words are words over the alphabet of positive integers with no adjacent equal letters. The enumerator of these words by descent number is a symmetric function which arose in work with Shareshian on q-Eulerian polynomials, on Rees products of posets, and on chromatic quasisymmetric functions. In this talk I will discuss this work withShareshian and recent work with Ellzey on the enumerators of Smirnov words by cyclic descents.
Geometric Analysis Seminar
Pengzi Miao
University of Miami
Some Recent Developments on Manifolds with Nonnegative Scalar Curvature
Monday, February 19, 2018, 4:00pm
Ungar Room 506
Abstract: Scalar curvature is a basic scalar quantity of curvature. It is tied to local energy density in relativity. For noncompact manifolds, the Riemannian positive mass theorem and Riemannian Penrose inequality are fundamental results formulated on asymptotically flat manifolds that model isolated systems. For compact manifolds with boundary, understanding the impact of scalar curvature on the manifold boundary is tied to the quasi-local mass problem in relativity. In this talk, I will discuss recent developments on compact Riemannian manifolds with nonnegative scalar curvature, with boundary. The talk will be based on my join work with Christos Mantoulidis, and with Siyuan Lu, respectively.
Hodge Theory Seminar
Dmitry Kaledin
Russian Academy of Sciences
Brown Representability for Unpointed Spaces
Friday, February 16, 2018, 4:00pm
Ungar Room 411
Geometry and Physics Seminar
R. Paul Horja
University of Miami
Toric Schobers and D-modules
Wednesday, February 14, 2018, 5:00pm
Ungar Room 402
Abstract: Many classical mirror symmetry results can be recast using the more recent language of perverse sheaves of categories and schobers. In this context, I will explain a Riemann-Hilbert type conjectural connection with the D-modules naturally appearing in mirror symmetry.
Geometric Analysis Seminar
Eric Ling
University of Miami
Milne-like Spacetimes
Monday, February 12, 2018, 4:00pm
Ungar Room 506
Abstract: The study of the (in)-extendibility of spacetimes is motivated by the strong cosmic censorship conjecture in general relativity. Recently there has been an interest in extendibility results with regularity less than C^2. This began with Sbierski's work where he demonstrated that the maximally analytic extension of the Schwarzschild spacetime is C^0-inextendible. In his paper Sbierski posed the question of whether or not the FLRW cosmological models admit C^0 extensions. In this talk we present a class of cosmological models, dubbed Milne-like spacetimes, which admit C^0 extensions through the big bang. We discuss their properties and how they fit in the modern view of cosmology.
Geometry and Physics Seminar
Chris Scaduto
Stony Brook University
Yang-Mills Theory and Definite Intersection Forms Bounding Homology 3-spheres
Friday, February 9, 2018, 4:00pm
Ungar Room 402
Abstract: Using Yang-Mills instanton Floer theory, we find new constraints on the possible definite intersection forms of smooth 4-manifolds that bound integer homology 3-spheres. We will give examples of 3-manifolds such that the set of all bounding negative definite lattices consists of essentially two distinct non-standard lattices. The methods used follow the work of Froyshov.
Geometry and Physics Seminar
John Francis
Northwestern University
Factorization Homology
Wednesday, February 7, 2018, 5:00pm
Ungar Room 402
Abstract: The Ran space Ran(X) is the space of finite subsets of X, topologized so that points can collide. Ran spaces have been studied in diverse works from Borsuk-Ulam and Bott, to Beilinson-Drinfeld, Gaitsgory-Lurie and others. The alpha form of factorization homology takes as input a manifold or variety X together with a suitable algebraic coefficient system A, and it outputs the sheaf homology of Ran(X) with coefficients defined by A. Factorization homology simultaneously generalizes singular homology, Hochschild homology, and conformal blocks or observables in conformal field theory. I'll discuss applications of this alpha form of factorization homology in the study of mapping spaces in algebraic topology, bundles on algebraic curves, and perturbative quantum field theory. I'll also describe a beta form of factorization homology, where one replaces Ran(X) with a moduli space of stratifications of X, designed to overcome certain strict limitations of the alpha form. One such application is to proving the Cobordism Hypothesis, after Baez-Dolan, Costello, Hopkins-Lurie, and Lurie. This is joint work with David Ayala.
Geometry and Physics Seminar
Dmitry Kaledin
Higher School of Economics
On the Notion of an Enhanced Category
Monday, February 5, 2018, 6:00pm
Ungar Room 411
Abstract: By now, it is a well-established general principle that when you localize a category with respect to a class of morphisms, you get a category "enriched in homotopy types". Several precise definition of this notion exist in the literature (complete Segal spaces of Rezk, infinity-categories of Lurie), and they are all equivalent in some sense, but they all depend on some auxiliary choices, and the precise sense in which they are all equivalent also depends on choices. In these lectures, I am going to sketch a somewhat more invariant approach to the subject that seems to be much more model-independent. This is based on Grothendieck's idea of a derivator, also well-established in the literature; however, and this seems to be new, there is also a theorem that states that when you pass to the derivator, you lose no information, so that the approach is equivalent to the earlier ones. In the first lecture, I am going to give a general overview, and then discuss a version of Brown representability theorem for unpointed topological spaces.
Combinatorics Seminar
Bruno Benedetti
University of Miami
Some Contractible 2-complexes Do Not Embed in R^4
Monday, February 5, 2018, 5:00pm
Ungar Room 402
Abstract: We discuss the problem of whether all contractible d-complexes can be drawn in R^{2d}. This is clear only for d=1 (in which case the answer is: "yes, all trees are planar graphs".) We also look at combinatorial strengthenings of contractibility, like collapsibility and non-evasiveness. This is work in progress with Karim Adiprasito.
Geometric Analysis Seminar
Melanie Graf
Faculty of Mathematics
University of Vienna
Causality and Geodesics in Low Regularity
Monday, February 5, 2018, 4:00pm
Ungar Room 506
Abstract: As a concrete example by Chruściel and Grant demonstrates, many classical results from causality theory fail for metrics that are not at least Lipschitz continuous; for example lightcones need no longer be hypersurfaces and one may have maximizing causal curves that are neither timelike nor null. In this talk I will try to give a bit of an overview over what is currently known for which regularity classes of metrics and some of the remaining open problems. We are also going to take a look at the geodesic equation for Lipschitz metrics: To deal with the right-hand-side of the geodesic equation being only locally bounded one can make use of a solution concept of Filippov allowing for a general existence result.
Combinatorics Seminar
Jai Aslam
Northeastern University
Intersection Patterns of Sets
Monday, January 29, 2018, 5:00pm
Ungar Room 402
Abstract: We present Kneser's conjecture and its reformulation into a graph coloring problem. We then introduce the generalized Erdos-Kneser conjecture partially proven by Sarkaria in 1990 and its associated hypergraph coloring problem. We prove this conjecture for r-uniform hypergraphs with the size of intersection s, not too close to r. We discuss what's still open related to this conjecture and possible methods for further proofs.
Hodge Theory Seminar
D. Yang
Institute for Advanced Study
Integral Virtual Fundamental Chains
Friday, January 26, 2018, 4:00pm
Ungar Room 402
Abstract: To define invariants using moduli spaces of holomorphic curves in general symplectic manifolds, a virtual technique is typically required, such as Kuranishi theory or polyfolds. All the methods in full generality use perturbation or duality, involve locally breaking the symmetry then taking the weighted averages, and thus yield virtual fundamental chains over rationals. We carry out a program of Fukaya-Ono outlined in their 2001 paper. The key notions are a group-normal structure that one can always construct for a good coordinate system, and a group-normal complex structure that is always present on the moduli space of holomorphic curves, and their combined group-normal complex good coordinate system. Using this, one can perform a single-valued group-normally polynomial perturbation to yield integral virtual fundamental chains/pseudocycles for Floer/GW moduli spaces on general symplectic manifolds. This method is expected to be applicable to all moduli spaces based on holomorphic curves. This is a joint work with Guangbo Xu.
Geometry and Physics Seminar
Victor Przyjalkowski
Steklov Institute of Mathematics
Higher School of Economics
Smooth Fano Weighted Complete Intersections and Landau-Ginzburg Models
Wednesday, January 24, 2018, 5:00pm
Ungar Room 402
Abstract: Smooth Fano varieties are classified in dimensions up to three, while in higher dimensions only some examples are known. The typical examples in the most interesting Picard rank one case are smooth complete intersections in weighted projective spaces and Grassmannians. It turns out that smoothness is a strong restriction for weighted complete intersections that lets to get bounds on their numerical invariants and, thus, lets classify all of them for any given dimension. We observe what is known in this direction, as well as we discuss their numerical invariants like Hodge numbers. We also outline, by an analogy with three-dimensional case, the way to construct their Landau-Ginzburg models and to prove Hodge numbers mirror symmetry in the spirit of Katzarkov-Kontsevich-Pantev conjecture.
Geometry and Physics Seminar
Alexander Efimov
Higher School of Economics
Russian Academy of Sciences
On the (Non-)L-equivalence of Algebraic Varieties
Tuesday, January 23, 2018, 5:00pm
Ungar Room 506
Abstract: Recall that two complex algebraic varieties are called L-equivalent if they have the same classes in the localization of the Grothendieck ring of varieties with respect to L (the class of affine line).
In this talk I will disprove (the original versions of) two conjectures on L-equivalence, due to Huybrechts and to Kuznetsov-Shinder. The first one states that isogenous K3 surfaces are L-equivalent, and the second one states that derived equivalence of smooth projective varieties implies L-equivalence (the second conjecture fails already for abelian varieties). Moreover, it will be shown that both for K3 surfaces and for abelian varieties each L-equivalence class contains only finitely many isomorphism classes.
Our results on non-L-equivalence are deduced (via integral Hodge realization) from the very general (and quite surprisingly, new) results on the Grothendieck group of an additive category whose morphisms are finitely generated abelian groups.
Combinatorics Seminar
Richard Stanley
University of Miami
Massachusetts Institute of Technology
The Sperner Property
Monday, January 22, 2018, 5:00pm
Ungar Room 402
Abstract: A finite graded partially ordered set $P$ has the \emph{Sperner property} if the largest level of $P$ is an antichain of maximum size. Most of the talk will be a survey of the Sperner property, beginning with Sperner's result that the boolean algebra of all subsets of a finite set has the Sperner property. (Of course Sperner did not use this terminology.) We will focus our attention on the use of linear algebra. We conclude with a discussion of the weak Bruhat order of the symmetric group. It is an open problem whether this poset has the Sperner property. We will discuss a determinantal conjecture which would imply the Sperner property.
Geometry and Physics Seminar
Carlos Simpson
University of Miami
Université Nice Sophia Antipolis
Structures on the Boundary of the Character Variety of a Compact Riemann Surface
Wednesday, January 17, 2018, 5:00pm
Ungar Room 402
Abstract: Consider a compact Riemann surface or an orbicurve, and let M be the moduli space of representations of the fundamental group into a fixed complex algebraic group such as SL_r(C). In this talk, we'll describe a conjecture on the structure of the boundary of M. This conjecture, a relative of the "P=W conjecture", relates natural spherical structures that are visible, one on the "Dolbeault" side of Hitchin's description of M and the other on the "Betti" side of the most naive expression of M as an affine variety. Here we'll give an overview of what is known in this direction, and some of the techniques that can be used. Understanding in a detailed way the main examples, and theorems, will be the subject of my course.
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
Categorical Brill Noether Invariants
Tuesday, January 16, 2018, 5:00pm
Ungar Room 402
Combinatorics Seminar
José Alejandro Samper Casas
University of Miami
Hopf Algebras in Combinatorics
(continuation)
Monday, December 18, 2017, 5:00pm
Ungar Room 402
Combinatorics Seminar
José Alejandro Samper Casas
University of Miami
Hopf Algebras in Combinatorics
(continuation)
Monday, December 11, 2017, 5:00pm
Ungar Room 402
Combinatorics Seminar
José Alejandro Samper Casas
University of Miami
Hopf Algebras in Combinatorics
(continuation)
Monday, December 4, 2017, 5:00pm
Ungar Room 402
Geometry and Physics Seminar
Dr. Dmitri Vassiliev
University College London
Classification of First Order Sesquilinear Forms
Wednesday, November 29, 2017, 5:00pm
Ungar Room 402
Abstract: We work with n complex-valued scalar fields over an m-dimensional real manifold M without boundary. Our object of study is a first order Hermitian sesquilinear form, i.e. an integral over the manifold whose integrand is a linear combination of terms "product of gradient of scalar field and scalar field" and "product of two scalar fields".
We call two sesquilinear forms equivalent if one is obtained from the other by some x-dependent GL(n,C) transformation, i.e. by a change of basis in the vector space of n-tuples of complex-valued scalar fields. Our aim is to provide a description of equivalence classes of sesquilinear forms.
The main result of the talk is that in the special case m=4, n=2 we provide explicit necessary and sufficient conditions for two sesquilinear forms to be equivalent. In the process of formulating these necessary and sufficient conditions we show that a first order Hermitian sesquilinear form implicitly contains geometric constructs such as Lorentzian metric, spin structure, connection coefficients and electromagnetic covector potential.
The talk is based on the paper Z. Avetisyan, Y.-L. Fang, N. Saveliev and D. Vassiliev, "Analytic definition of spin structure", Journal of Mathematical Physics 58 (2017), 082301.
Combinatorics Seminar
James McKeown
University of Miami
Tilings of Space and the Dedekind-MacNeille Completion of Bruhat Order
Monday, November 13, 2017, 5:00pm
Ungar Room 402
Abstract: It is quite ordinary to consider how a group acts on an object. What if instead, one fixes a representation and lets the set of linear transformations (id-g) act on the object? In 2005, Waldspurger showed that, for the regular representation of a finite reflection group, the action of (id-g) on the cone over the fundamental weights gives a tiling of the cone over the positive roots. Shortly thereafter, Meinrenken considered the case of affine Weyl groups, and showed that the action of (id-g) on a fundamental alcove gives a tiling of the whole vector space. Bibikov and Zhgoon then proved analogous results for all cocompact hyperbolic reflection groups. We will look at some combinatorial consequences of these theorems for finite and affine types A and B. In particular, we will investigate the Dedekind-MacNeille completion or Bruhat order—the smallest lattice containing Bruhat order as a subposet.
Geometry and Physics Seminar
Jianfeng Lin
Massachusetts Institute of Technology
Froyshov Invariants of Branched Double Covers
Wednesday, November 8, 2017, 5:00pm
Ungar Room 402
Abstract: Let Y be a double cover of the 3-sphere branched over a knot K. The Froyshov invariant of Y (for the unique spin structure) is a useful concordance invariant of the knot. When the knot is quasi-alternating, this invariant equals the signature of the knot divided by 8 (as proved by Manolescu-Owens and Lisca-Owens). While this relation does not hold in general, I will give a generalization that holds for all knots (even links) in the 3-sphere and sketch the proof. Various applications will be discussed, including an interesting relation between Seiberg-Witten type invariants and Donaldson type invariants of homology S1 cross S3. This is a joint work with Daniel Ruberman and Nikolai Saveliev.
Applied Math Seminar
Xi Huo
University of Miami
Modelling Antimicrobial De-escalation:
Implications for Stewardship Programs
Tuesday, November 7, 2017, 2:15pm
Ungar Room 402
Abstract: Antimicrobial de-escalation aims to minimize resistance to high-value broad-spectrum empiric antimicrobials by switching to alternative drugs when testing confirms susceptibility. Though widely practiced in ICUs, the effects of de-escalation are not well understood. We develop a high-dimensional ODE model to assess the effect of de-escalation on a broad range of outcomes, and clarify expectations. In this talk, I will present the medical background and conclusions of this study, and show how we numerically analyze the model output with a broad range of undetermined parameters and limited data. Ongoing work on the model simplification and relevant mathematical analysis will be discussed at the end.
Combinatorics Seminar
Michelle Wachs
University of Miami
Chromatic Quasisymmetric Functions and Hessenberg Varieties
Monday, November 6, 2017, 5:00pm
Ungar Room 402
Abstract: I will discuss an algebro-geometric approach to proving the longstanding Stanley-Stembridge e-positivity conjecture for chromatic symmetric functions that was proposed by Shareshian and myself several years ago. Our approach to this conjecture involves a refinement of Stanley's chromatic symmetric functions. We conjectured a certain relationship between our refinement and Hessenberg varieties. Our conjecture was recently proved by Brosnan and Chow using techniques from algebraic geometry, and more recently by Guay-Paquet using Hopf algebras. I will describe this result, some of its consequences, and what still needs to be done to prove the Stanley-Stembridge conjecture.
Geometry and Physics Seminar
Enrica Mazzon
Imperial College London
Berkovich Spaces and Dual Complexes of Degenerations
Wednesday, November 1, 2017, 5:00pm
Ungar Room 402
Abstract: In the late nineteen-nineties Berkovich developed a new approach to non-archimedean analytic geometry. This theory has quickly found many applications in algebraic and arithmetic geometry. In particular it turned out that there are strong connections between Berkovich spaces of degenerations of varieties and the birational geometry of dual complexes. In this talk, I will explain how the dual complex of a degeneration can be interpreted as a simplicial subset of a Berkovich space. I will introduce the central objects of this theory: the weight function and the essential skeleton of the degeneration. Finally, I will use them to study the dual complex of products and of some examples of hyperkahler varieties.
This is a joint work with Morgan Brown and a complementary talk to the seminar he gave on the dual complex of a product of degenerations.
Applied Math Seminar
Chris Cosner
University of Miami
Dynamics of Populations with Individual Variation in Dispersal on Bounded Domains
(Joint work with Steve Cantrell and Xiao Yu)
Tuesday, October 31, 2017, 2:15pm
Ungar Room 406
Abstract: Most classical models for the movement of organisms assume that all individuals have the same patterns and rates of movement, but there is empirical evidence that movement rates and patterns may vary among individuals. One way to capture variation in dispersal is to allow individuals to switch between two distinct dispersal modes. We consider models for populations with logistic-type local population dynamics whose members can switch between two different nonzero rates of diffusion. The resulting reaction-diffusion systems can be cooperative at some population densities and competitive at others. We analyze the dynamics of such systems on bounded regions. (Traveling waves and spread rates have been studied by others for similar models in the context of biological invasions.) The analytic methods include ideas and results from reaction-diffusion theory, semi-dynamical systems, and bifurcation/continuation theory.
Combinatorics Seminar
Brittney Ellzey
University of Miami
Chromatic Quasisymmetric Functions of Directed Graphs
Monday, October 30, 2017, 5:00pm
Ungar Room 402
Abstract: I will be presenting my work on expansions (in various bases for the ring of symmetric and quasisymmetric functions) of chromatic quasisymmetric functions for digraphs. This is a version of the talk I will be giving at the Combinatorics Seminar at Brandeis.
Combinatorics Seminar
Manuel Rivera
University of Miami
A Combinatorial Model for the Based Loop Space
Monday, October 23, 2017, 5:00pm
Ungar Room 402
Abstract: To any topological space we may associate a topological monoid called the based loop space: as a set it consists of all loops in the space based at a fixed point and the multiplication is given by concatenation of loops. The homology of the based loop space has the structure of a Hopf algebra: the product is induced by concatenation of loops, the coproduct by the Alexander-Whitney diagonal, and the antipode by the map sending a loop to its inverse. From a classical result of homotopy theory we know that sufficiently nice topological spaces may be modeled by combinatorial objects called simplicial sets. I will explain how to model the above construction in purely combinatorial terms, namely, to any connected simplicial set S I will construct a natural differential graded Hopf algebra, based on the combinatorics of S, having the property that its homology is isomorphic to the homology Hopf algebra of the the based loop space of the geometric realization of S. This is joint work with Samson Saneblidze and generalizes classical results of Adams and Baues.
Geometry and Physics Seminar
Andrew Harder
University of Miami
Perverse Sheaves of Categories
Wednesday, October 18, 2017, 5:00pm
Ungar Room 402
Abstract: A perverse sheaf of categories is a graph on a punctured Riemann surface with categorical data associated to each edge and vertex. In this talk, I will explain how these thing can be used to encode the derived category of coherent sheaves on certain algebraic varieties and what this means for homological mirror symmetry.
Combinatorics Seminar
José Alejandro Samper Casas
University of Miami
Hopf Algebras in Combinatorics
(continuation)
Monday, October 16, 2017, 5:00pm
Ungar Room 402
Geometry and Physics Seminar
Manuel Rivera
University of Miami
Categorical and Algebraic Constructions Related to Path Spaces
Wednesday, October 11, 2017, 5:00pm
Ungar Room 402
Abstract: In this talk I will describe explicitly how the following three functors are related:
1) the path space functor and its relatives (based path space, based loop space, free loop space, etc...)
2) the cobar functor from the category of differential graded coalgebras to the category of differential graded algebras
3) the rigidification functor from simplicial sets to simplicial categories
1) is a classical and important construction which appears all over through geometry, topology, and mathematical physics. 2) is a purely algebraic construction introduced by Frank Adams in the 1950's to obtain an algebraic model for the based loop space of a simply connected space, which is suitable for computations. 3) was introduced by Jacob Lurie in order to compare different models for infinity categories.
The key to relate these three functors is to introduce a cubical version of 3). Understanding these relationships reveals a few interesting consequences, for example: it tells us how to remove the simply connectedness hypothesis in Adams' theorem, we obtain a strict adjoint functor for the differential graded nerve functor and a transparent algebraic definition for infinity local systems, and we also obtain a direct proof of the fact that the chains on the based loop space of a space with Poincaré duality has a Calabi-Yau algebra structure.
Combinatorics Seminar
Tiago Novello de Brito
Pontifícia Universidade Católica do Rio de Janeiro
Discrete Line Field
Monday, September 25, 2017, 5:00pm
Ungar Room 402
Abstract: The discrete line field is our proposal for a possible discretization of the theory of line fields. The discrete object will be a Morse matching just between the vertices and edges of a cellular complex. The objective is to define the critical objects and their indices, and then show that the complex is homotopy equivalent to a cellular complex with just the critical objects.
Geometry and Physics Seminar
Constantin Teleman
University of Oxford
The Quantum GIT Conjecture
Wednesday, September 6, 2017, 5:00pm
Ungar Room 402
Abstract: For a compact symplectic manifold X with a Hamiltonian action of a compact group, one can define the gauges Gromov-Witten theory which involves integration over the moduli space of G-bundles and twisted maps to X. The space of states of this theory has a description as the equivariant quantum cohomology of X for an action of the free loop group of G. When X is Fano, with orbifold GIT quotient, this agrees conjecturally with QH*(X//G). (For G a torus and X a vector space, this is a vast generalization of Batyrev's description of QH^* of toric Fanos.) I describe a tentative outline of the proof, and the relation to the open string version of this conjecture.
Geometry and Physics Seminar
Morgan Brown
University of Miami
The Dual Complex of a Product of Degenerations
Wednesday, August 30, 2017, 5:00pm
Ungar Room 402
Abstract: To a simple normal crossing degeneration of algebraic varieties, we can associate an invariant called the dual complex, which is the intersection complex of the special fiber. In this talk I will investigate how this construction behaves under products. Unfortunately, the product of two simple normal crossings degenerations over a curve in general fails to remain snc, but belongs to the broader class of toroidal singularities. I will introduce toroidal geometry and use it to show that the dual complex of a product of semistable degenerations is PL homeomorphic to the product of the individual dual complexes.
Combinatorics Seminar
José Alejandro Samper Casas
University of Miami
Hopf Algebras in Combinatorics
(continuation)
Monday, August 28, 2017, 5:00pm
Ungar Room 402
Combinatorics Seminar
José Alejandro Samper Casas
University of Miami
Hopf Algebras in Combinatorics
(continuation of the summer seminar)
Monday, August 21, 2017, 5:00pm
Ungar Room 402
Combinatorics Seminar
Federico Castillo
University of California, Davis
Deformation Cones for Polytopes
Friday, June 23, 2017, 11:00am
Ungar Room 411
Abstract: Given a lattice polytope, the set of all polytopes having the same (or a coarsening) normal fan is a polyhedral cone. This cone has appeared in different contexts, for instance, it is closely related to the nef cone of the associated toric variety. In the case of the regular permutohedron we get the cone of submodular functions. The purpose of this talk is to survey known results and show how to compute this deformation cones in further combinatorial examples. This is joint work with Fu Liu.
Combinatorics Seminar
Rafael González D'León
University of Kentucky
The Gamma-coefficients of the Tree Eulerian Polynomials
Wednesday, June 21, 2017, 2:30pm
Ungar Room 411
Abstract: We consider the generating polynomial T_n(t) of the number of rooted trees on the set {1,2,...,n} counted by the number of descending edges (a parent with a greater label than a child). This polynomial is an extension of the descent generating polynomial of the set of permutations of a totally ordered $n$-set, known as the Eulerian polynomial. We show how this extension shares some of the properties of the classical one. In particular it has palindromic coefficients and hence it can be expressed in the the basis $\left \{ t^i(1+t)^{n-1-2i}\,\mid\, 0\le i \le \lfloor \frac{n-1}{2}\rfloor\right \}$, known as the $\gamma$-basis. We show that $\T_n(t)$ has nonnegative $\gamma$-coefficients and we present various combinatorial interpretations for them.
Geometry and Physics Seminar
Lars Andersson
Max-Planck Institute for Gravitational Physics
Hidden Symmetries and Conservation Laws
Wednesday, May 3, 2017, 5:00pm
Ungar Room 402
Abstract: Maxwell's theory of electromagnetism is a relativistic field theory on Minkowski space which is symmetric under the 15-dimensional conformal group of Minkowski space. However, it admits further less obvious symmetries including the Heaviside-Larmor-Rainich symmetry, as well as hidden symmetries associated with the 20-dimensional space of conformal Killing-Yano 2-tensors on Minkowski space. I will give some background to these facts and discuss their relation to conservation laws for the Maxwell and gravitational field on Minkowski space as well as on non-trivial geometries including the Kerr black hole spacetime.
AMS Graduate Student Chapter Seminar
Arnie Horta
National Security Agency
NSA: The Secret Life of Mathematicians
Friday, April 28, 2017, 4:00pm
Ungar Room 402
Abstract: The National Security Agency is one of the largest employers of mathematicians in the United States. In this talk, Dr. Horta will discuss his career at the NSA, job opportunities for both mathematicians and computer scientists, and what "A Day in the Life of an NSA Mathematician" is like. Also, sample mathematical applications to NSA problems will be presented, including the applications of Number Theory to Cryptography and applications of Graph Theory to Data Science.
Geometry and Physics Seminar
Professor Ina Petkova
Dartmouth College
Categorifying the Alexander Polynomial as a Reshetikhin-Turaev Invariant
Wednesday, April 26, 2017, 5:00pm
Ungar Room 402
Abstract: The Reshetikhin-Turaev construction for the standard representation of the quantum group gl(1|1) sends tangles to C(q)-linear maps in such a way that a knot is sent to its Alexander polynomial. After a brief review of this construction, I will give an introduction to tangle Floer homology -- a combinatorial generalization of knot Floer homology which sends tangles to (homotopy equivalence classes of) bigraded dg bimodules. Finally, I will discuss how to see tangle Floer homology as a categorification of the Reshetikhin-Turaev invariant. This is joint work with Alexander Ellis and Vera Vertesi.
Applied Math Seminar
Don Olson
University of Miami
Modeling Evolution in the Galapagos
Tuesday, April 25, 2017, 11:00am
Ungar Room 411
Combinatorics Seminar
Vasu Tewari
University of Washington
Labeled Binary Trees, Schur-positivity and Generalized Tamari Lattices
Monday, April 24, 2017, 5:00pm
Ungar Room 402
Abstract: Gessel introduced a multivariate formal power series tracking the distribution of ascents and descents in labeled binary trees. In addition to showing that it was a symmetric function, he conjectured that it was Schur-positive.
In this talk, I will present a proof of this conjecture which utilizes an extension of a beautiful bijection of Preville-Ratelle and Viennot concerning generalized Tamari lattices. I will subsequently discuss connections between specializations of Gessel's symmetric function and Frobenius characteristics of symmetric group actions on certain Coxeter deformations, focusing in particular on semiorder and Linial arrangements. Finally, I will discuss some potential avenues to pursue.
This is joint work with Ira Gessel and Sean Griffin.
Geometry and Physics Seminar
Professor Anatoly Libgober
National Science Foundation
Equivariant Invarians of GIT Quotients and Vanishing Cycles
Wednesday, April 19, 2017, 5:00pm
Ungar Room 402
Abstract: I will describe a relation between Hodge structure on vanishing cohomolgy of weighted homogeneous singularities and equivariant cohomology motivated by Landau-Ginzburg/Calabi Yau correspondence. Extension of this relation to correspondence between hybrid models also will be discussed.
Combinatorics Seminar
Michelle Wachs
University of Miami
On r-inversions and Symmetric Functions
Monday, April 17, 2017, 5:00pm
Ungar Room 402
Abstract: The r-inversion number is a statistic on words of length n (over the positive integers), which interpolates between the descent number (r=2) and the inversion number (r=n). We consider a symmetric function U_{n,r} that enumerates words of length n by this statistic. The symmetric function U_{n,r} is an example of an LLT polynomial. The LLT polynomials were shown to be Schur-positive by Grojnowski and Haiman by means of Kazhdan-Lusztig theory. It is an open question to give a combinatorial description of the coefficients in the Schur basis expansion. For r = 2 and r=n, such descriptions are well known. For r = 3, a description (in a more general setting) was conjectured by Haglund and was proved by Blasiak using noncommutative Schur functions and Lam's algebra of ribbon Schur operators. In this talk I will describe a more elementary proof for the r = 3 case, which uses classical RSK theory. I will also discuss results for some other cases, and a consequence involving an r-analog of the q-binomial coefficients. This is joint work with Yuval Roichman.
Geometry and Physics Seminar
Ian Shipman
Harvard University
Toward Ulrich Bundles on ACM Varieties
Wednesday, April 5, 2017, 5:00pm
Ungar Room 402
Abstract: Let X be a variety embedded in projective space. Ulrich bundles on X are those vector bundles which admit a linear resolution when viewed as sheaves on the ambient projective space. Existence of an Ulrich bundle implies that the homogeneous coordinate ring of X satisfies Lech's conjecture and that cone of cohomology tables of X agrees with that of a projective space of the same dimension. Motivated by the problem of constructing Ulrich bundles, I will describe a result relating Ulrich bundles to higher rank Brill-Noether theory on certain curves on X. Furthermore, I will explain a result that an arithmetically Cohen-Macaulay (ACM) variety always admits a reflexive sheaf whose restriction to a general one-dimensional linear section is Ulrich.
Combinatorics Seminar
Felix Gotti
University of California, Berkeley
Dyck Paths and Positroids from Unit Interval Orders
Monday, April 3, 2017, 5:00pm
Ungar Room 402
Abstract: It is well known that the number of non-isomorphic unit interval orders on [n] equals the n-th Catalan number. Combining work of Skandera and Reed and work of Postnikov, we will assign a rank n positroid on [2n] to each unit interval order on [n]. We call such positroids "unit interval positroids." Then we will give a characterization of the unit interval positroids by describing their associated decorated permutations, showing that each one must be a 2n-cycle encoding a Dyck path of length 2n.
Combinatorics Seminar
Tewodros Amdeberhan
Tulane University
Determinants in "Wonderland"
Monday, March 27, 2017, 5:00pm
Ungar Room 402
Abstract: Determinants are found everywhere in mathematics and other scientific endeavors. Their particular role in Combinatorics does not need any cynical introduction or special advertisement. In this talk, we will illustrate certain techniques which proved to be useful in the evaluation of several class of determinantal evaluations. We conclude this seminar with an open problem. The content of our discussion is accessible to anyone with "an intellectual appetite".
Applied Math Seminar
Don DeAngelis
University of Miami
Diffusion of a Population on a Landscape:
Does a Heterogeneous Landscape Lead to a Higher Population Size than a Homogeneous Landscape?
Tuesday, March 21, 2017, 11:00am
Ungar Room 411
Abstract: Single population reaction-diffusion models have shown that a population diffusing in an environment with a spatially heterogeneous carrying capacity can reach a higher total population size than when the same total carrying capacity, K(x), is distributed homogeneously as a function of distance, x (Lou 2006). A similar result had been pointed out earlier by Holt (1985) for a two-patch model with different growth rates and carrying capacities on the two patches, which he termed a "paradox". These results suggest that a higher population size can be attained by configuring the same total carrying capacity in a heterogeneous, as opposed to a homogeneous, manner.
However, it is biologically impossible to create carrying capacities and growth rates the way they are formulated in these models. Biological populations require input of energy and nutrients to survive and grow, and the populations affect those resources through exploitation. We consider a two-variable chemostat-type model of a consumer population and an exploitable resource (e.g., a limiting nutrient), in which the same total amount of resource input can be spatially distributed in any possible way across the landscape. Spatially varying carrying capacities and growth rates emerge through exploitation by the diffusing population. We proved for this model the following. (1) A diffusing population in a heterogeneous environment can exceed in size a non-diffusing population under certain conditions on the parameters. (2) However, the population size of a diffusing population, when the resource inputs are heterogeneously distributed, will always be less than (or at most equal to) the size of the population that is either diffusing or not diffusing when the same total amount of resource input is homogeneously distributed. This resolves the paradox noted by Holt (1985). Experiments using yeast (Zhang et al. submitted) corroborate these results.
Based on work by Don DeAngelis, Wei-Ming Ni, and Bo Zhang.
Combinatorics Seminar
Fabrizio Zanello
Michigan Technological University
Partition into Distinct Parts and Unimodality
Monday, March 20, 2017, 5:00pm
Ungar Room 402
Abstract: We discuss the (non)unimodality of the rank-generating function, $F_{\lambda}$, of the poset of partitions with distinct parts contained inside a given partition $\lambda$. This work, in collaboration with Richard Stanley (European J. Combin., 2015), is in part motivated by an attempt to place into a broader context the unimodality of $F_{\lambda}(q)=\prod_{i=1}^n(1+q^i)$, the rank-generating function of the ``staircase'' partition $\lambda=(n,n-1,\dots,1)$, for which determining a combinatorial proof remains an outstanding open problem.
We will present a number of results on the polynomials $F_{\lambda}$. Surprisingly, these results carry a remarkable similarity to those proven in 1990 by Dennis Stanton. His work extended, to any partition $\lambda$, the study of the unimodality of $q$-binomial coefficients --- that is, the rank-generating functions of the \emph{arbitrary} partitions contained inside given rectangular partitions.
We will also discuss some open problems and recent developments. These include a (prize-winning) paper by Levent Alpoge, who solved our conjecture on the unimodality of $F_{\lambda}$ when $\lambda$ is the "truncated staircase" $(n,n-1, \dots,n-c)$, for $n\gg c$.
Combinatorics Seminar
Laura Escobar
University of Illinois at Urbana-Champaign
Rhombic Tilings and Bott-Samelson Varieties
Monday, March 6, 2017, 5:00pm
Ungar Room 402
Abstract: Elnitsky gave an elegant bijection between rhombic tilings of 2n-gons and commutation classes of reduced words in the symmetric group on n letters. We explain a natural connection between Elnitsky's and Magyar's construction of the Bott-Samelson resolution of Schubert varieties. This suggests using tilings to encapsulate Bott-Samelson data and indicates a geometric perspective on Elnitsky's combinatorics. We also extend this construction by assigning desingularizations to the zonotopal tilings considered by Tenner. This is based on joint work with Pechenik, Tenner and Yong.
Combinatorics Seminar
Yuval Roichman
Bar Ilan University, Israel
Cyclic Descents of Standard Young Tableaux
Monday, February 27, 2017, 5:00pm
Ungar Room 402
Abstract: Permutations in the symmetric groups, as well as standard Young tableaux, are equipped with a well-established notion of descent set. The cyclic descent set of permutations was introduced by Cellini and further studied by Dilks, Petersen and Stembridge, while cyclic descents on standard Young tableaux (SYT) of rectangular shapes were introduced by Rhoades.
The existence of cyclic descent maps for SYT of all non-ribbon skew shapes was recently proved, using nonnegativity properties of Postnikov's toric Schur polynomials. The proof and its implications will be explained by Ron Adin in tomorrow's colloquium talk.
In this talk we will focus on explicit combinatorial interpretations of the concept, applications to Schur-positivity and open problems.
Based on joint works with Ron Adin, Sergi Elizalde and Vic Reiner.
Applied Math Seminar
Shigui Ruan
University of Miami
Modeling the Transmission Dynamics of Avian Influenza H7N9 Virus
Tuesday, February 21, 2017, 11:00am
Ungar Room 411
Abstract: In March 2013, a novel avian-origin influenza A (H7N9) virus was identified among human patients in China and a total of 124 human cases with 24 related deaths were confirmed by May 2013. There were no reported cases in the summer and fall 2013. However, the virus has been coming back in November every year. In fact, the second outbreak from November 2013 to May 2014 caused 130 human cases with 35 deaths, the third outbreak from November 2014 to June 2015 caused 216 confirmed human cases with 99 deaths, the fourth outbreak from November 2015 to July 2016 caused 114 confirmed human cases and 45 deaths, respectively. From November 2016 to January 2017, there have already been 304 cases with 99 deaths. In this talk, I will introduce some recent studies on modeling the transmission dynamics of the avian influenza A (H7N9) virus from birds to humans and apply our models to simulate the open data for numbers of the infected human cases and related deaths reported by the Chinese Center for Disease Control and Prevention. The basic reproduction number is estimated and sensitivity analysis of in terms of model parameters is performed. Our studies demonstrate that H7N9 virus has been well established in birds and will cause regular outbreaks in humans again in the future.
Combinatorics Seminar
Hai Long Dao
Kansas University
On Local-Global Phenomena in the Betti Tables of Stanley-Reisner Ideals
Monday, February 20, 2017, 5:00pm
Ungar Room 402
Abstract: Let I be an homogeneous ideal in a polynomial ring S over a field. The Betti table of I describes the graded minimal free resolution of I over S. When I is a Stanley-Reisner ideal of a simplicial complex C, the Betti table can be used to compute the h- and f-vectors of C. In this talk I will describe several recent results about what I call local-global phenomena in the Betti tables. Namely, information on a small part of the table forces strong result on the whole resolution, and give structural information about C such as its depth, regularity or chordality. If time permits, I will also explain the connection of these results to classical commutative algebra, and some new connections to group cohomological dimensions. The talk will be based on various joint works with Schweig-Huneke, Schweig, and Vu.
Combinatorics Seminar
Sebi Cioaba
University of Delaware
Simplicial Rook Graphs:
Algebraic and Combinatorial Properties
Friday, February 17, 2017, 4:00pm
Ungar Room 402
Abstract: A few years ago, Jeremy L. Martin and Jennifer D. Wagner introduced the simplicial rook graphs SR(d,n) as the graph whose vertices are the lattice points in the n-th dilate of the standard simplex in Rd, with two vertices adjacent if they differ in exactly two coordinates. Martin and Wagner proved that SR(3,n) has integral eigenvalues and determined other interesting properties of these graphs. In this talk, I will describe our work proving some conjectures made by Martin and Wagner as well as determining other algebraic and combinatorial facts about these graphs. This is joint work with Andries Brouwer (TU Eindhoven, The Netherlands), Willem Haemers (Tilburg University, The Netherlands) and Jason Vermette (Missouri Baptist Univ., USA).
Geometry and Physics Seminar
Orlando Alvarez
University of Miami
Emergent Gravity and H. Weyl's Volume Formula
Wednesday, February 15, 2017, 5:00pm
Ungar Room 402
Abstract: In physical theories where the energy is localized near a submanifold of Euclidean space, there is a universal expression for the energy. We derive a multipole-like expansion for the energy that has a finite number of terms, and depends on intrinsic geometric invariants of the submanifold and extrinsic invariants of the embedding of the submanifold. This universal expression is a generalization of an exact formula of Hermann Weyl for the volume of a tube. In special situations, dictated by spherical symmetry of the energy density, the expression is a generalized Lovelock lagrangian for gravity constructed using only the Lipschitz-Killing curvatures. This class of theories is interesting because there are no negative metric states. We discuss in what sense this is an emergent theory of gravity and how it is related to the local isometric embedding problem.
Combinatorics Seminar
Brittney Ellzey
University of Miami
A Directed Graph Generalization of Chromatic Quasi-symmetric Functions
Monday, February 13, 2017, 5:00pm
Ungar Room 402
Abstract: Chromatic quasisymmetric functions of labeled graphs were defined by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric functions. In this talk, we present an extension of their definition from labeled graphs to directed graphs, suggested by Richard Stanley. We show that the chromatic quasisymmetric functions of proper circular arc digraphs are symmetric functions, which generalizes a result of Shareshian and Wachs on natural unit interval graphs. The directed cycle on n vertices is contained in the class of proper circular arc digraphs, and we give a generating function for the e-basis expansion of the chromatic quasisymmetric function of the directed cycle, refining a result of Stanley for the undirected cycle. We discuss a generalization of the Shareshian-Wachs refinement of the Stanley-Stembridge e-positivity conjecture. We present our F-basis expansion of the chromatic quasisymmetric functions of all digraphs and our p-basis expansion for all symmetric chromatic quasisymmetric functions of digraphs, which extends work of Shareshian-Wachs and Athanasiadis.
Geometry and Physics Seminar
Demetre Kazaras
University of Oregon
Minimal Hypersurfaces with Free Boundary and PSC-bordism
Wednesday, February 8, 2017, 5:00pm
Ungar Room 402
Abstract: There is a well-known technique due to Schoen-Yau from the late 70s which uses (stable) minimal hypersurfaces to study the topological consequences of a (closed) manifold's ability to support a Riemannian metric with positive scalar curvature. In this talk, we describe a version of this technique for manifolds with boundary and discuss how it can be used to study bordisms of positive scalar curvature metrics.
Applied Math Seminar
Chris Cosner
University of Miami
Some Mathematical Models of Criminal Behavior
Tuesday, February 7, 2017, 11:00am
Ungar Room 411
Abstract: In recent years there has been considerable interest in developing mathematical models of socioeconomic phenomena in general and criminal activity in particular. One topic that has been treated in some detail is the spatial distribution of criminal behavior, especially burglary. It has been observed that burglaries tend to be clustered in "hotspots" with sizes and locations that do not seem to be determined in an obvious way by the geographic distribution of socio-economic factors. Two influential modeling approaches to understanding the problem of spatial distribution of crime were introduced in the papers (M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi and L. B. Chayes, A statistical model of criminal behavior, Mathematical Models and Methods in Applied Science, 18 (2008), 1249–1267) and (H. Berestycki and J.-P. Nadal, Self-organised critical hot spots of criminal activity, European Journal of Applied Mathematics 21 (2010), 371–399). Models of the first type were initially formulated as agent based models, but continuum limits were derived from those. The resulting continuum models have features somewhat similar to chemotaxis models. Models of the second type of include spatially nonlocal terms and in some cases diffusion. Both types support spatial patterns. Various versions of those models have been studied from the viewpoints of existence theory, bifurcation theory, and traveling waves. This talk will describe some results and problems related to such models.
Combinatorics Seminar
José Samper
University of Miami
On Conjectural Relatives of Matroid Polytopes
Monday, February 6, 2017, 5:00pm
Ungar Room 402
Abstract: We will present various curious similarities between shifted simplicial complexes and matroid independence complexes and provide evidence that all these similarities should be proved geometrically by extending the theory of matroid polytopes. Along the way we will pose questions, conjectures and explain some of the final goals. Based on joint work with Jeremy Martin.
Geometry and Physics Seminar
Ernesto Lupercio
Center for Research and Advanced Studies of the
National Polytechnic Institute (Cinvestav-IPN)
Quantum Toric Varieties
Wednesday, February 1, 2017, 5:00pm
Ungar Room 402
Abstract: In this talk I will talk about the theory of quantum toric virieties as developed by Katzarkov, Meersseman, Verjovsky and myself. Classical toric varieties have proved important in geometry. They are built out of tori and the combinatorics of raitonal polytopes. Our quantum toric varieties are built up out of quantum tori and the combinatorics of irrational polytopes. I will speak about their construction and also relations with the work of Kaledin, Shkolnikov and myself regarding sandpiles.
Combinatorics Seminar
Bruno Benedetti
University of Miami
Mogami Constructions of Manifolds from Trees of Tetrahedra
Monday, January 30, 2017, 5:00pm
Ungar Room 402
Abstract: A 3-ball is a simplicial complex homeomorphic to the unit ball in R^3. A "tree of tetrahedra" is a 3-ball whose dual graph is a tree. It is easy to see that every (connected) 3-manifold can be obtained from some tree of tetrahedra by recursively gluing together two boundary triangles.
The quantum physicist Tsugui Mogami has studied "Mogami manifolds", that is, those manifolds that can be obtained from a tree of tetrahedra by recursively gluing together two *incident* boundary triangles. In 1995 he conjectured that all 3-balls are Mogami. Mogami's conjecture would imply a much desired exponential bound (crucial for the convergence of certain models in quantum gravity) for the number of 3-balls with N tetrahedra. Unfortunately, we show that Mogami's conjecture does not hold.
Combinatorics Seminar
Richard Stanley
University of Miami
Massachusetts Institute of Technology
Counting with Congruence Conditions
Monday, January 23, 2017, 5:00pm
Ungar Room 402
Abstract: The archetypal result is the theorem of Lucas that the number of coefficients of the polynomial $(1+x)^n$ not divisible by a prime $p$ is $\prod(1+a_i)$, where $n=\sum a_ip^i$ is the base $p$ expansion of $n$. We will discuss numerous generalizations and analogues of this result. For example, the number of partitions of $n$ for which the number of standard Young tableaux of shape $\lambda$ is odd is equal to $2^{\sum b_i}$, where $n=\sum 2^{b_i}$ is the binary expansion of $n$ (due to I. G. Macdonald).
Combinatorics Seminar
Jim Haglund
University of Pennsylvania
LLT Polynomials and the Chromatic Symmetric Function of Unit Interval Orders
Monday, November 21, 2016, 5:00pm
Ungar Room 402
Abstract: Shareshian and Wachs have conjectured that a certain symmetric function, which depends on a Dyck path and a parameter t, has positive coefficients when expressed as a polynomial in the elementary symmetric functions. Their conjecture implies an earlier conjecture of Stanley and Stembridge. We show how some elements of the preprint of Carlsson and Mellit "A proof of the shuffle conjecture" imply that the Shareshian-Wachs symmetric function can be expressed, via a plethystic substitution, in terms of LLT polynomials, specifically LLT products of single cells. As corollaries we obtain combinatorial formulas for the expansion of Jack polynomials into the Schur basis, and also the power-sum basis. These formulas are signed, not always positive, but perhaps could be simplified. Other applications include a quick method for computing the chromatic symmetric function using plethystic operators. Based on joint work with Per Alexandersson, Greta Panova, and Andy Wilson.
Geometry and Physics Seminar
David Zurieck-Brown
Emory University
Canonical Rings of Stacky Curves
Wednesday, November 16, 2016, 5:00pm
Ungar Room 402
Abstract: We give a generalization to stacks of the classical (1920's) theorem of Petri -- we give a presentation for the canonical ring of a stacky curve. This is motivated by the following application: we give an explicit presentation for the ring of modular forms for a Fuchsian group with cofinite area, which depends on the signature of the group. This is joint work with John Voight.
Combinatorics Seminar
John Shareshian
Washington University
Subrack Lattices of Group Racks
Monday, November 14, 2016, 5:00pm
Ungar Room 402
Abstract: Let G be a finite group. A subset S of G is called a subrack if S is closed under conjugation. The set R(G) of all subracks of G is partially ordered by inclusion. With Istvan Heckenberger and Volkmar Welker of Philipps-University at Marburg, we have studied relations between the combinatorial structure of R(G) and the algebraic structure of G. I will discuss our results.
Geometry and Physics Seminar
Faramarz Vafaee
California Institute of Technology
The Prism Manifold Realization Problem
Wednesday, November 9, 2016, 5:00pm
Ungar Room 402
Abstract: The spherical manifold realization problem asks which spherical three-manifolds (equivalently, three-manifolds with finite fundamental groups) arise by surgery on knots in S^3. In recent years, the realization problem for C,T,O,I-type spherical manifolds has been solved, leaving the D-type spherical manifolds (aka prism manifolds) as the only remaining case. Every prism manifold can be parametrized as P(p,q), for a pair of relatively prime integers p>1 and q. We determine a complete list of prism manifolds P(p, q) that can be realized by positive integral surgery on knots in S^3 when q<0. The general methodology undertaken to obtain the classification is similar to that of Greene for lens spaces. The arguments rely on tools from Floer homology and lattice theory, and are primarily combinatorial in nature. This is joint work with Ballinger, Hsu, Mackey, Ni, and Ochse.
Combinatorics Seminar
Alex Lazar
University of Miami
Filtered Geometric Lattices
Monday, November 7, 2016, 5:00pm
Ungar Room 402
Abstract: In order to address some questions in tropical geometry, Mikhalkin and Ziegler introduced the notion of a filtered geometric lattice. These posets can be seen as generalizations of geometric semilattices (introduced by Wachs and Walker), which are themselves generalizations of geometric lattices.
In this talk, we will discuss some topological results of Adiprasito and Bjorner about filtered geometric lattices, as well as some open questions about these posets.
Applied Math Seminar
Mythily Ramaswamy
TIFR Centre for Applicable Mathematics, Bangalore, and Virginia Tech
Stabilization of Compressible Fluid Models
Tuesday, November 1, 2016, 12:30pm
Ungar Room 411
Abstract: Compressible fluids are modeled through Navier-Stokes equations for density and velocity. In this talk, I consider a model in a bounded interval and the stabilization (steer the system to a steady state as time goes to infinity) of the solution around constant steady states. The control acts only on the velocity. After describing the problem for simple ODE systems, I will discuss the PDE systems.
Combinatorics Seminar
Katharina Jochemko
TU Vienna, Austria
Discrete vs. Continuous Valuations: Similarities and Differences
Monday, October 31, 2016, 5:00pm
Ungar Room 402
Abstract: The prototypical valuation is presumably the volume. It has various favorable properties such as homogeneity, monotonicity and translation-invariance. In the continuous setting, valuations are well-studied and the volume plays a prominent role in many classical and structural results. In the less examined discrete setting, the number of lattice points in a polytope - its discrete volume - takes a central role. Although homogeneity and continuity are lost, some striking parallels can be drawn. In this talk, I will discuss some similarities, analogies and differences between the continuous and discrete world of translation-invariant valuations.
Geometry and Physics Seminar
Pranav Pandit
University of Vienna
Shifted Symplectic Structures and Applications
Wednesday, October 26, 2016, 5:00pm
Ungar Room 402
Combinatorics Seminar
Rainer Sinn
Georgia Tech
Positive Semidefinite Matrix Completion and Free Resolutions
Monday, October 24, 2016, 5:00pm
Ungar Room 402
Abstract: I will discuss the positive semidefinite matrix completion problem arising e.g. in combinatorial statistics and explain how we can use results in algebraic geometry to understand it better. The object linking the two different areas is the cone of sums of squares and its properties as a convex cone.
Geometry and Physics Seminar
Christopher Langdon
University of Miami
Twisted Symmetric Differentials and the Quadric Algebra
Wednesday, October 19, 2016, 5:00pm
Ungar Room 402
Abstract: It was shown by M. Schneider in the nineties that there are no symmetric differentials of degree m on projective subvarieties of low codimension even if twisted by O(k) for k<m. In this talk I will discuss the geometric nature of the extremal case k=m and its relation to the quadric hypersurfaces containing the subvariety. In particular, it is expected that these differentials are given by polynomials in the quadrics that vanish on the variety. I will give the proof for complete intersections and discuss the general case for varieties of very low codimension.
Combinatorics Seminar
Anastasia Chavez
UC Berkeley, California
The Dehn-Sommerville Relations and the Catalan Matroid
Monday, October 17, 2016, 5:00pm
Ungar Room 402
Abstract: The f-vector of a d-dimensional polytope P stores the number of faces of each dimension. When P is a simplicial polytope the Dehn-Sommerville relations condense the f-vector into the g-vector, which has length $\lceil{\frac{d+1}{2}}\rceil$. Thus, to determine the f-vector of P, we only need to know approximately half of its entries. This raises the question: Which $(\lceil{\frac{d+1}{2}}\rceil)$-subsets of the f-vector of a general simplicial polytope are sufficient to determine the whole f-vector? We prove that the answer is given by the bases of the Catalan matroid.
Geometry and Physics Seminar
Professor Nima Anvari
University of Miami
Equivariant Rho-Invariants and Instanton Homology of Torus Knots
Wednesday, October 12, 2016, 5:00pm
Ungar Room 402
Abstract: The equivariant rho-invariants are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. In this talk we discuss these rho-invariants for allere involutions on 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some involutions on Brieskorn homology spheres. As an application, we compute the Floer gradings in the singular instanton chain complex of (p, q)-torus knots with odd p and q.
CANCELLED
Geometry and Physics Seminar
Christopher Langdon
University of Miami
Twisted Symmetric Differentials and the Quadric Algebra
Wednesday, October 5, 2016, 5:00pm
Ungar Room 402
Abstract: It was shown by M. Schneider in the nineties that there are no symmetric differentials of degree m on projective subvarieties of low codimension even if twisted by O(k) for k<m. In this talk I will discuss the geometric nature of the extremal case k=m and its relation to the quadric hypersurfaces containing the subvariety. In particular, it is expected that these differentials are given by polynomials in the quadrics that vanish on the variety. I will give the proof for complete intersections and discuss the general case for varieties of very low codimension.
Applied Math Seminar
Jing Chen
University of Miami
The Importation, Establishment and Transmission Dynamics of the Mosquito-borne Disease
Tuesday, October 4, 2016, 12:30pm
Ungar Room 411
Abstract: Because of the development of the global transportation system, it is frequently found that the tourists are exposed to infections in the countries where certain infectious diseases are prevalent and the local residents are exposed to infections brought by the tourists from epidemic areas. These frequent activities lead to a possibility of international spread of infectious disease and bring threats to individuals who are geographically far from the original outbreak. Thus currently mosquito-borne diseases, including Dengue fever, Chikungunya and Zika virus, bring an emerging public health threat to some developed countries such as United States while previous attention was mainly put to the tropical countries considering the climate features and the socioeconomic conditions.
Our hypothesis is the imported cases first spread the mosquito-borne disease to local mosquitoes, which later cause local infections on humans. Based on this, we propose a mathematical model to study how these movements of humans affect the establishment and transmission dynamics of the mosquito-borne disease.
Combinatorics Seminar
Nancy Abdallah
Linköping University
Bruhat Order on Twisted Identities and KLV Polynomials
Monday, October 3, 2016, 5:00pm
Ungar Room 402
Abstract: We study the Bruhat order on the sets of twisted involution and twisted identities in a Coxeter group W equipped by an involutive automorphism. When W is the symmetric group of odd rank, we define the Kazhdan-Lusztig-Vogan polynomials indexed by elements in the set of twisted identities and we prove that they are combinatorially invariant for intervals that start with the identity. This generalizes the combinatorial invariance of the classical Kazhdan-Lusztig polynomials for lower bound intervals in a symmetric group.
This is joint work with Axel Hultman.
Geometry and Physics Seminar
Professor Carlos Simpson
Université Nice Sophia Antipolis
University of Miami
Spectral Networks and Buildings
Wednesday, September 28, 2016, 5:00pm
Ungar Room 402
Abstract: The theory of spectral networks was introduced by Gaiotto, Moore and Neitzke for WKB problems as well as for physics motivations. We relate these combinatorial geometric objects that come from spectral curves, to actions of fundamental groups on euclidean buildings. The WKB asymptotics can be encoded in an equivariant harmonic map to a building, and the spectral network is a main part of the pullback of the singular locus of the building. This is joint work with Katzarkov, Noll and Pandit.
Applied Math Seminar
Steve Cantrell
University of Miami
Fitness Based Prey Dispersal and Prey Persistence in Intraguild Predation Systems
(co-authors Robert Stephen Cantrell, King-Yeung Lam, Tian Xiang, Xinru Cao)
Tuesday, September 27, 2016, 12:30pm
Ungar Room 411
Abstract: We establish prey persistence in intraguild predation systems in bounded habitats under mild conditions when the prey disperses using its fitness as a surrogate for the balance between resource acquisition and predator avoidance. The model is realized as a quasilinear parabolic system where the dimension of the underlying spatial habitat is arbitrary.
Geometry and Physics Seminar
Morgan Brown
University of Miami
A Characterization of Toric Varieties
Wednesday, September 21, 2016, 5:00pm
Ungar Room 402
Abstract: Toric varieties are ubiquitous in algebraic geometry. They have a rich combinatorial structure, and give the simplest examples of log Calabi-Yau varieties.
We give a simple criterion for characterizing when a log Calabi-Yau pair is toric, which answers a case of a conjecture of Shokurov. This is joint work with James McKernan, Roberto Svaldi, and Runpu Zong.
Combinatorics Seminar
Alessio Sammartano
Purdue University
Blowup Algebras of Rational Normal Scrolls
Monday, September 19, 2016, 5:00pm
Ungar Room 402
Abstract: The Rees ring and the special fiber ring of a polynomial ideal I, also known as the blowup algebras of I, play an important role in commutative algebra and algebraic geometry. A central problem is to describe the defining equations of these algebras. I will discuss the solution to this problem when I is the homogeneous ideal of a rational normal scroll.
Geometry and Physics Seminar
Ken Baker
University of Miami
Asymmetric L-space Knots
Wednesday, September 14, 2016, 5:00pm
Ungar Room 402
Abstract: An L-space is a rational homology 3-sphere for which the cardinality of its first homology is the rank of its Heegaard Floer homology. It's been conjectured that the symmetry group of any knot in S^3 with a non-trivial surgery to an L-space contains an involution. We demonstrate that this is false through a general construction of the first examples of such knots for which the symmetry group is trivial.
Applied Math Seminar
Gabriel Andreguetto Maciel
University of Ottawa
Reaction-diffusion Models with Individual Movement Response to Habitat Edges
Tuesday, September 13, 2016, 12:30pm
Ungar Room 411
Abstract: How the interplay between population growth and individual movement behavior determines large scale spread and persistence in heterogeneous landscapes has been one of the central points in ecological research. Reaction-diffusion models that take into account habitat preference of individuals as a movement bias at the interface between two habitat types have been recently derived. Ovaskainen and Cornell (2003) showed that in such cases population density across the interface between patches should be discontinuous, both the level of bias and the ratio between diffusivities determining the jump in density. In this talk, I will show results for the persistence and rates of spatial spread in reaction-diffusion models that include movement response to habitat edges in patchy environments.
Combinatorics Seminar
Jay Yang
University of Wisconsin-Madison
Random Toric Surfaces and a Threshold for Smoothness
Monday, September 12, 2016, 5:00pm
Ungar Room 402
Abstract: I will present a construction of a random toric surface inspired by the construction of a random graph. With this construction we show a threshold result for smoothness of the surface. The hope is that this inspires further application of randomness to Algebraic Geometry. This talk does not require any background in Algebraic Geometry or Toric Geometry.
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
Categories and Filtrations
Wednesday, September 7, 2016, 5:00pm
Ungar Room 402
Geometry and Physics Seminar
Manuel Rivera
University of Miami
String Topology: Chain Level Transversality and Algebraic Models
Wednesday, August 31, 2016, 5:00pm
Ungar Room 402
Abstract: I will start by describing the construction of several string topology operations; these are transversal intersection type operations on the homology of the free loop space of a manifold. I will proceed with an outline of a framework (joint work with Dingyu Yang (IAS)) which allows us to work with these operations at the chain level. I will focus on a "secondary" coproduct operation for which the transversality is much more subtle than the original intersection product introduced by Chas and Sullivan. I will also explain how these operations are related with the algebraic theory of operations on Hochschild homology/cohomology of Frobenius algebras.
Applied Math Seminar
Chris Cosner
University of Miami
Spatial Population Models with Fitness Based Dispersal
Tuesday, August 30, 2016, 12:30pm
Ungar Room 411
Abstract: Traditional continuous time models in spatial ecology typically describe movement in terms of linear diffusion and advection, which combine with nonlinear population dynamics to produce semilinear equations and systems. However, if organisms are assumed to move up gradients of their reproductive fitness, and fitness is density dependent (for example logistic), the resulting models are quasilinear and may have other novel features. This talk will describe some models involving fitness dependent dispersal and some results and challenges in the analysis of such models.
Combinatorics Seminar
Hailun Zheng
University of Washington
A Characterization of Simplicial Manifolds with g 2 ≤ 2
Monday, August 29, 2016, 5:00pm
Ungar Room 402
Abstract: The celebrated low bound theorem states that any simplicial manifold of dimension ≥ 3 satisfies g 2 ≥ 0, and equality holds if and only if it is a stacked sphere. Furthermore, more recently, the class of all simplicial spheres with g 2 = 1 was characterized by Nevo and Novinsky, by an argument based on rigidity theory for graphs. In this talk, I will first define three different retriangulations of simplicial complexes that preserve the homeomorphism type. Then I will show that all simplicial manifolds with g 2 ≤ 2 can be obtained by retriangulating a polytopal sphere with a smaller g 2. This implies Nevo and Novinsky's result for simplicial spheres of dimension ≥ 4. More surprisingly, it also implies that all simplicial manifolds with g 2 = 2 are polytopal spheres.
Geometry and Physics Seminar
Will Kazez
University of Georgia
Tautness of Foliations
Wednesday, August 24, 2016, 5:00pm
Ungar Room 402
Abstract: I will give a brief overview of the role tautness plays in the study of foliations of 3-manifolds. Elementary examples will be constructed to show that notions of tautness that are equivalent for fairly smooth foliations are not equivalent in the world of less smooth foliations. These examples of "phantom tori" have implications in the study of approximations of taut foliations by contact structures. This is joint work with Rachel Roberts.
Partial Differential Equations Seminar
Andrea Scapellato
Universita di Catania, Italy
Regularity Properties of Solutions of PDE and Systems
Tuesday, June 28, 2016, 3:00pm
Ungar Room 411
View Abstract
and
Maria Alessandra Ragusa
Universita di Catania, Italy
Regularity Results for Some Classes of Higher Order Nonlinear Elliptic Systems
Tuesday, June 28, 2016, 3:45pm
Ungar Room 411
View Abstract
Probability Seminar
Nina Amini
Laboratoire des Signaux et Systèmes
Quantum Feedback Control and Filtering Problem
Thursday, May 5, 2016, 3:30pm
Ungar Room 406
Abstract: The ability to control quantum systems is becoming an essential step towards emerging technologies such as quantum computation, quantum cryptography and high precision metrology. In this talk, we consider a controlled quantum system whose finite dimensional state is governed by a discrete-time nonlinear Markov process. By assuming the quantum non-demolition (QND) measurements in open-loop, we construct a strict control Lyapunov function which is based on the open-loop stationary states. We propose a measurement-based feedback scheme which ensures the almost sure convergence towards a target state. Moreover, I discuss the estimation and filtering problem for continuous-time quantum systems which are described by continuous-time stochastic master equations.
PDE and Geometric Analysis Seminar
Piotr Chrusciel
University of Vienna
Center of Mathematical Sciences and Applications at Harvard
Mathematics of Gravitational Waves
Tuesday, April 26, 2016, 2:30pm
Ungar Room 506
Abstract: I will review the mathematical issues arising when attempting to describe gravitational radiation.
Combinatorics Seminar
Miriam Farber
Massachusetts Institute of Technology
Weak Separation, Pure Domains and Cluster Distance
Monday, April 25, 2016, 5:00pm
Ungar Room 402
Abstract: Following the proof of the purity conjecture for weakly separated sets, recent years have revealed a variety of wider classes of pure domains in different settings. In this paper we prove the purity for domains consisting of sets that are weakly separated from a pair of "generic" sets I and J. Our proof also gives a simple formula for the rank of these domains in terms of I and J. This is a new instance of the purity phenomenon which essentially differs from all previously known pure domains. We apply our result to calculate the cluster distance and to give lower bounds on the mutation distance between cluster variables in the cluster algebra structure on the coordinate ring of the Grassmannian. Using a linear projection that relates weak separation to the octahedron recurrence, we also find the exact mutation distances and cluster distances for a family of cluster variables.
This is a joint work with Pavel Galashin.
Probability Seminar
J. David Van Dyken
Department of Biology
University of Miami
Statistics of Nonlinear Biochemical Reaction Networks in Living Cells
Thursday, April 21, 2016, 3:00pm
Ungar Room 406
Abstract: Chemical reactions within cells involve sequences of random events among small numbers of interacting molecules. As a consequence, biochemical reaction networks are extremely noisy. These reactions are also non-linear, making analytical treatment of these systems difficult. I will present a method for approximating the statistics of molecular species in arbitrarily connected networks of non-linear biochemical reactions in small volumes, which I validate with stochastic simulations. I demonstrate that noise slow flux through biochemical networks with nonlinear reaction kinetics, with implications for the evolution of robustness in living cells.
Applied Math Seminar
Necibe Tuncer
Florida Atlantic University
Structural and Practical Identifiability Issues in an Immuno-epidemiological Model
Wednesday, April 20, 2016, 5:00pm
Ungar Room 506
Abstract: In this talk, I will present a mathematical model that links immunological model and epidemiological model. This model allows us to understand dynamical interplay of infectious diseases at two different scales; immunological response of the host at individual scale and the disease dynamics at population scale. Once the host is infected, it triggers the immune response which produces antigen-specific antibodies to clear the pathogen. The pathogen and antibody levels are often monitored in laboratory experiments. Can we use the data generated in the laboratory experiments to identify the parameters of the immunological model? Clearly, the parameters of the within-host immunological model have an effect on the epidemiological characteristics of disease such as reproduction number and prevalence. Epidemiological data is also available for the epidemiological model. I will present the both structural and practical identifiability issues in parameter estimation of the immuno-epidemiological model.
Geometry and Physics Seminar
Professor D. Zakharov
New York University
Symmetric Differentials on Projective Varieties
Wednesday, April 20, 2016, 5:00pm
Ungar Room 402
Abstract: A symmetric differential on a complex variety is a section of a symmetric power of the cotangent bundle. The existence of non-trivial symmetric differentials is related to the topological properties of the variety, implying that the fundamental group is large in a suitable sense. I will review some recent results on symmetric differentials, and describe a necessary and sufficient condition for a symmetric differential of rank three on a complex surface to be expressible as a product of closed holomorphic 1-forms.
Joint work with Federico Buonerba.
Combinatorics Seminar
Brittney Ellzey
University of Miami
The Chromatic Quasisymmetric Function of the Cycle
Monday, April 18, 2016, 5:00pm
Ungar Room 402
Abstract: Chromatic quasisymmetric functions were introduced by Shareshian and Wachs as a refinement of Stanley's chromatic symmetric functions. The results of Shareshian and Wachs focus primarily on incomparability graphs of natural unit interval orders. In this talk I will present my recent work on the chromatic quasisymmetric functions of other graphs, specifically the n-cycle, as well as a generalization of the n-cycle. I will give expansions of the chromatic quasisymmetric functions for these graphs in terms of Gessel's fundamental quasisymmetric basis and in terms of the power sum basis and see how these expansions compare to those obtained by Shareshian-Wachs and Athanasiadis.
Geometry and Physics Seminar
Professor Andrew Harder
University of Miami
K3 Fibered Calabi-Yau Threefolds
Wednesday, April 13, 2016, 5:00pm
Ungar Room 402
Abstract: I will describe some classification results on specific K3 fibered Calabi-Yau threefolds and how these results possibly fit in the context of mirror symmetry.
Applied Math Seminar
Professor Peng Feng
Florida Gulf Coast University
Lyme Disease, Rash and Mathematics
Wednesday, April 13, 2016, 5:00pm
Ungar Room 506
Abstract: In this talk, I will introduce the basic principles behind mathematical immunology. Then we will discuss a few temporal and spatio-temporal models that describe how our immune system responds to various pathogens. We also establish a PDE chemotaxis model for the innate response to Borrelia burgdorferi, the causative agent of Lyme disease. We illustrate the key factors that lead to the characteristic skin rash that is often associated with Lyme disease. We finish the talk with a few comments regarding modeling in immunology.
Applied Math Seminar
David Van Dyken
University of Miami
Noise-induced Slowdown in Biochemical Reaction Networks
Wednesday, April 6, 2016, 5:00pm
Ungar Room 506
Abstract: Chemical reactions within cells are inherently noisy, but little is known about whether noise affects cell function or, if so, how and by how much. In my talk I will demonstrate that noise causes a quantifiable loss in cell fitness by slowing the average rate of biosynthesis. I present an analytical framework for solving the steady-state statistics of molecular species in arbitrarily connected networks of non-linear chemical reactions in mesoscopic volumes, which I validate with extensive stochastic simulations. In general, I find that reactions obeying hyperbolic kinetics, including Michaelis-Menten reactions, experience "hyperbolic filtering" which attenuates high amplitude substrate fluctuations leading to 1) reduced average reaction flux and 2) reduced product noise. Gene expression is particularly sensitive to noise-induced slowdown because, unlike metabolic reactions, translation is not buffered by network-level feedback. Additionally, I find that translation propagates less noise than a kinetically equivalent metabolic reaction, that mRNA noise directly reduces fitness by reducing the average translation rate, and that increasing mRNA/ribosome binding affinity actually reduces protein noise even though it increases translational bursting. These phenomena are missing from previous stochastic analyses of gene expression because translation is typically modeled as linear, rather than hyperbolic.
Combinatorics Seminar
Martin Charles Golumbic
The Caesarea Rothschild Institute for Computer Science
University of Haifa, Israel
New and Old Graph Dimension Parameters
Monday, April 4, 2016, 5:00pm
Ungar Room 402
Abstract: In this talk, we will explore various intersection and containment based representations of graphs and posets along with their associated parameters. Among these are the boxicity and cubicity of graphs, the dimension and interval dimension of posets and their comparability graphs, the bending number of intersecting paths on a grid, and the grid dimension of a graph.
We will also present recent work on the new notions of the separation dimension of a graph and the induced separation dimension of a graph. One of our main aims has been to find significant interconnections between such dimensional parameters. For example, we establish bounds relating the bending number to the partial order dimension for co-comparability graphs, and relating the induced separation dimension with the separation dimension and boxicity.
Applied Math Seminar
Don DeAngelis
University of Miami
Modeling the Dynamics of Woody Plant-Snowshoe Hare Interactions with Twig Age-dependent Toxicity
Wednesday, March 30, 2016, 5:00pm
Ungar Room 506
Abstract: Modeling is used to study the effects that woody plant chemical defenses may have on population dynamics of boreal hares that feed almost entirely on twigs during the winter. The model takes into account that toxin concentration often varies with the age of twig segments. In particular, it incorporates the fact that the woody internodes of the youngest segments of the twigs of the deciduous angiosperm species that these hares prefer to eat are more defended by toxins than the woody internodes of the older segments that subtend and support the younger segments. Thus, the per capita daily intake of the biomass of the older segments of twigs by hares is much higher than their intake of the biomass of the younger segments of twigs. This age-dependent toxicity of twig segments is modeled using age-structured model equations, which are reduced to a system of delay differential equations involving multiple delays in the woody plant–hare dynamics. A novel aspect of the modeling was that it had to account for mortality of non-consumed younger twig segment biomass when older twig biomass was bitten off and consumed. Basic mathematical properties of the model are established together with upper and lower bounds on the solutions. Necessary and sufficient conditions are found for the linear stability of the equilibrium in which the hare is extinct, and sufficient conditions are found for the global stability of this equilibrium. Numerical simulations confirmed the analytical results and demonstrated the existence of limit cycles over ranges of parameters reasonable for hares browsing on woody vegetation in boreal ecosystems. This showed that age dependence in plant chemical defenses has the capacity to cause hare-plant population cycles, a new result.
Geometry and Physics Seminar
Carla Cederbaum
University of Tüebingen
On Foliations Related to the Center of Mass in General Relativity
Wednesday, March 30, 2016, 5:00pm
Ungar Room 402
Abstract: In many situations in Newtonian gravity, understanding the motion of the center of mass of a system is key to understanding the general trend of the motion of the system. It is thus desirable to also devise a notion of center of mass with similar properties in general relativity.
However, while the definition of the center of mass via the mass density is straightforward in Newtonian gravity, there is a priori no definitive corresponding notion in general relativity. We will pursue a geometric approach to defining the center of mass, using foliations by hypersurfaces with specific geometric and physical properties. I will first illustrate this approach in the (easier) Newtonian setting and then review previous work in the relativistic situation, most prominently a fundamental result by Huisken and Yau from 1996. After introducing the foliation approach to defining the center of mass, I will discuss explicit counter-examples (partially joint work with Nerz) and discuss the analytic, geometric, and physical issues they illustrate. I will then present a new approach (joint work with Cortier and Sakovich) that remedies these issues.
Combinatorics Seminar
James McKeown
University of Miami
The Combinatorics of the Waldspurger Decomposition
Monday, March 28, 2016, 5:00pm
Ungar Room 402
Abstract: In 2005 J.L. Waldspurger proved a remarkable theorem. Given a finite reflection group G, the closed cone over the positive roots is equal to the disjoint union of images of the open weight cone under the action of 1-g. When G is taken to be the symmetric group the decomposition is related to the familiar combinatorics of permutations but also has some surprising features. To see this, we give a nice combinatorial description of the decomposition.
The decomposition is not a simplicial, or even CW complex and attempts to complete it to one are problematic. It does, however, define a dual graph on n-cycles. We prove some basic facts about this graph and state a few conjectures and open problems relating to it.
Probability Seminar
Philip Ernst
Rice University
On the Volatile Correlation of Two Independent Wiener Processes
Wednesday, March 23, 2016, 3:30pm
Ungar Room 406
Abstract: In this paper, we resolve a longstanding open statistical problem. The problem is to analytically determine the second moment of the empirical correlation coefficient \beqn \theta := \frac{\int_0^1W_1(t)W_2(t) dt - \int_0^1W_1(t) dt \int_0^1 W_2(t) dt}{\sqrt{\int_0^1 W^2_1(t) dt - \parens{\int_0^1W_1(t) dt}^2} \sqrt{\int_0^1 W^2_2(t) dt - \parens{\int_0^1W_2(t) dt}^2}} \eeqn of two {\em independent} Wiener processes, $W_1,W_2$. Using tools from Fredholm integral equation theory, we successfully calculate the second moment of $\theta$ to be .240522. This gives a value for the standard deviation of $\theta$ of nearly .5. As such, we are the first to offer formal proof that two Brownian motions may be independent and yet can also be highly correlated with significant probability. This spurious correlation, unrelated to a third variable, is induced because each Wiener process is ``self-correlated'' in time. This is because a Wiener process is an integral of pure noise and thus its values at different time points are correlated. In addition to providing an explicit formula for the second moment of $\theta$, we offer implicit formulas for higher moments of $\theta$.
Combinatorics Seminar
Luca Moci
Paris 7 - Institut de Mathématiques de Jussieu, France
Higher Dimensional Colorings and Flows, Arithmetic Tutte Polynomials, and Convolution Formulae
Monday, March 21, 2016, 5:00pm
Ungar Room 402
Abstract: In a recent series of papers by various authors, the theory of colorings and flows on graphs has been extended to the higher-dimensional case of CW complexes. We will survey this theory and show how the arithmetic Tutte polynomial naturally comes into play. (Joint work with E. Delucchi).
After recalling the basic properties of this polynomial, we will show some convolution formulae and their applications to the case of CW complexes (ongoing joint work with S. Backman, A. Fink and M. Lenz).
PDE and Geometric Analysis Seminar
Dr. Dave Auckly
Kansas State University
Weird Symmetry, Nice Space
Thursday, March 17, 2016, 5:00pm
Ungar Room 411
Abstract: We will talk about joint work with Ruberman, Melvin and Kim establishing that there are nice geometric structures with weird symmetry groups. In particular we will prove that every $3$-manifold is invertible, equivariantly, $Z[\pi]$-homology cobordant to a hyperbolic manifold.
Combinatorics Seminar
Emanuele Delucchi
University of Fribourg, Switzerland
Toric Arrangements and Group Actions on Semimatroids
Thursday, March 17, 2016, 5:00pm
Ungar Room 402
Abstract: Recent work of De Concini, Procesi and Vergne on vector partition functions gave a fresh impulse to the study of toric arrangements from an algebraic, topological and combinatorial point of view. In this context, many new combinatorial structures have recently appeared in the literature, each tailored to one of the different facets of the subject. Yet, a comprehensive combinatorial framework is lacking. As a unifying structure, in this talk I will propose the study of group actions on semimatroids and of related polynomial invariants, recently introduced in joint work with Sonja Riedel. In particular, I will outline some new open problems brought to the fore by this new point of view.
Probability Seminar
Stanislav Volkov
Lund University
Asymptotic Behaviour of Some Locally-interacting Processes
Thursday, March 17, 2016, 3:30pm
Ungar Room 406
Abstract: We study locally-interacting birth-and-death processes on nodes of a finite connected graph; the model which is motivated by modelling interactions between populations, adsorption-desorption processes, and is related to interacting particle systems, Gibbs models, and interactive urn models.
Alongside with general results, we obtain a more detailed description of the asymptotic behaviour in the case of certain special graphs.
Based on a joint work with Vadim Scherbakov (Royal Holloway, University of London).
Applied Math Seminar
Don Olson
University of Miami
Chaos in the Plankton
Wednesday, March 16, 2016, 5:00pm
Ungar Room 506
Abstract: The free floating communities in the sun lit surface of the ocean, the euphotic zone, are supported by photosynthesis in the phytoplankton (P). The lower reaches of this zone are bound by the compensation depth where the photosynthetic fixation of carbon in the phytoplankton just meet their own respiratory requirements. In simple models that include P with a nutrient (N) and either a grazer (Z) or a dissolved organic pool (D) without vertical movement or diffusion, this level is characterized by a double infinity. The solutions, however, reproduce the near surface and deep nutrient and phytoplankton curves. It is the underlying nature of the upper solution going to +infinity and the lower solution to -infinity that underlie the dynamics of the lower portion of the euphotic zone. The infinities can be eliminated by adding either sinking in P or by diffusion in N. The fixed points of the system, however, still exchange stabilities at the compensation depth leading to chaotic states. Addition of more complete terms with more phytoplankton (Pi) or zooplankton grazers only lead to more complete dynamics. Observations of the deep chlorophyll maximum are in agreement with the idea that the bottom of the euphotic zone is fundamentally chaotic.
Geometry and Physics Seminar
Scott Taylor
Colby College
Neighbors of Knots in the Gordian Graph
Wednesday, March 16, 2016, 5:00pm
Ungar Room 402
Abstract: The Gordian graph is the graph with vertex set the set of knot types and edge set consisting of pairs of knots which have a diagram wherein they differ at a single crossing. Bridge number is a classical knot invariant which is a measure of the complexity of a knot. It can be refined by another, recently discovered, knot invariant known as "bridge distance". We show, using arguments that are almost entirely elementary, that each vertex of the Gordian graph is adjacent to a vertex having arbitrarily high bridge number and bridge distance. This is joint work with Ryan Blair, Marion Campisi, Jesse Johnson, and Maggy Tomova.
Combinatorics Seminar
Mark Skandera
Lehigh University
Evaluations of the Power Sum Traces at Kazhdan-Lusztig Basis Elements of the Hecke Algebra
Monday, March 14, 2016, 4:00pm
Ungar Room 402
Abstract: In 1993, Haiman studied certain functions from the Hecke algebra to Z[q] called monomial traces. He conjectured that the evaluation of these at Kazhdan-Lusztig basis elements resulted in polynomials in N[q]. A weakening of this conjecture is that the evaluations of other traces, called power sum traces, results in polynomials in N[q]. We will discuss several combinatorial interpretations of these polynomials for Kazhdan-Lusztig basis elements indexed by permutations which avoid the patterns 3412 and 4231. These interpretations come from joint work with Matthew Hyatt, and results of Athanasiadis, Shareshian, and Wachs.
Geometry and Physics Seminar
C. Robles
Duke University
Degenerations of Hodge Structure
Wednesday, March 2, 2016, 5:00pm
Ungar Room 402
Abstract: I will explain how a polarized limiting mixed Hodge structure (PLMHS) may be viewed as a degeneration of a (pure, polarized) Hodge structure. There is a notion of "polarized relation" between PLMHS that encodes information on how varieties may degenerate within a family. I will give a classification of PLMHS and their polarized relations in terms of Hodge diamonds (discrete data associated with a PLMHS).
Applied Math Seminar
Robert Stephen Cantrell
University of Miami
Resident-Invader Dynamics in Infinite Dimensional Dynamical Systems
Wednesday, March 2, 2016, 5:00pm
Ungar Room 506
Abstract: We discuss an extension of the Tube Theorem from adaptive dynamics to infinite dimensional contexts, including that of reaction-diffusion equations. This is joint work with Chris Cosner and King-Leung Lam.
Combinatorics Seminar
Cynthia Vinzant
North Carolina State University
Real Stable Polynomials, Determinants, and Matroids
Monday, February 29, 2016, 4:00pm
Ungar Room 402
Abstract: Real stable polynomials define real hypersurfaces that are maximally nested ovaloids. These polynomials appear in many areas of mathematics, including optimization, combinatorics and differential equations. In 2004, Choe, Oxley, Sokal and Wagner established a tight connection between matroids and multiaffine real stable polynomials. Branden recently used this theory and the Vamos matroid to disprove the generalized Lax conjecture, which concerns representing polynomials as determinants. I will discuss the fascinating connections between these fields and some extensions to some varieties associated to hyperplane arrangements.
Geometry and Physics Seminar
Ivan Costa e Silva
Universidade Federal de Santa Catarina, Brazil
Causal R-actions and the Rigidity of Brinkmann Spaces
Wednesday, February 24, 2016, 5:00pm
Ungar Room 402
Abstract: We study the interplay between the global causal and geometric structures of a spacetime (M,g) and the features of a given smooth R-action on M whose orbits are all causal curves, building on classic results about Lie group actions on manifolds by Palais. Although the dynamics of such an action can be very hard to describe in general, simple restrictions on the causal structure of (M,g) can simplify this dynamics dramatically. We show how this can in turn be used in some cases to constrain the global geometry of (M,g), illustrating this fact by obtaining a rigidity result for the so-called Brinkmann spacetimes.
Combinatorics Seminar
Russ Woodroofe
Mississippi State University
A Broad Class of Shellable Lattices
Monday, February 22, 2016, 5:00pm
Ungar Room 402
Abstract: Jay Schweig and I recently discovered a large class of shellable lattices. Our original motivation were the order congruence lattices of finite posets. Afterwards, we noticed that the subgroup lattices of solvable groups are also contained in the class, and indeed, the definition may be seen as a lattice-theoretic abstraction of solvable groups.
In this talk, I'll review some of the theory of supersolvable lattices, and show how to extend similar ideas to our class of lattices.
Applied Math Seminar
Shigui Ruan
University of Miami
From Malaria to Zika:
Modeling the Transmission Dynamics and Control of Mosquito-Borne Diseases
Wednesday, February 17, 2016, 5:00pm
Ungar Room 506
Abstract: Mathematical modeling is a powerful and important tool in studying the transmission dynamics and control of many infectious diseases including vector-borne diseases (such as malaria, West Nile virus, dengue, chikungunya). In this talk I will use malaria as an example to introduce some basic concepts and models in mathematical epidemiology. Then I will review some basic features about Zika virus and the status of the ongoing Zika epidemic. Finally I will briefly talk about our preliminary results on modeling the spread and control of Zika virus infection.
Geometry and Physics Seminar
Eric Woolgar
University of Alberta
Asymptotically Poincaré-Einstein Metrics
Wednesday, February 17, 2016, 5:00pm
Ungar Room 402
Abstract: I will discuss asymptotically hyperbolic manifolds and sub-types, especially Asymptotically Poincaré-Einstein (APE) manifolds. These are manifolds whose metrics admit a Fefferman-Graham type expansion near conformal infinity, such that the first few leading terms are determined by the Einstein equations. APEs appear both as time slices of asymptotically Anti-de Sitter (AdS) spacetimes and as Wick rotations of static AdS spacetimes. Among the invariants that can (sometimes) be associated to APEs are mass and renormalized volume. For APEs that represent static AdS black holes in 4 dimensions, the renormalized volume and the mass of a static slice can be related. Remarkably, this relationship, which can be established purely on geometric grounds, is the free energy relation of black hole thermodynamics. A related result is that APEs which are time-symmetric slices of globally static AdS spacetimes cannot have positive mass. Examples are provided by Horowitz-Myers geons, which are complete APEs with scalar curvature $S=-n(n-1)$ and negative mass. They serve as time-symmetric slices of so-called AdS solitons, which are Einstein spacetimes with toroidal conformal infinity. The Horowitz-Myers "positive" energy conjecture proposes that Horowitz-Myers geons minimize the mass over all APEs with scalar curvature $S\ge -n(n-1)$ and the same toroidal conformal infinity. The conjecture is an open problem, but it is possible to show that if this conjecture is true then there is a rigidity theorem. Mass minimizing metrics must be static Einstein metrics and, for $n=3$ spatial dimensions, the least-mass geon is the unique complete APE having that mass and obeying $S\ge -n(n-1)$.
Combinatorics Seminar
Matteo Varbaro
University of Genova, Italy
Dual Graphs and the Castelnuovo-Mumford Regularity of Subspace Arrangements
Monday, February 15, 2016, 5:00pm
Ungar Room 402
Abstract: In the talk I will discuss an ongoing joint work with Bruno Benedetti and Michela Di Marca on the Castelnuovo-Mumford regularity of subspace arrangements. The Castelnuovo-Mumford regularity of an embedded projective variety is an important invariant measuring its complexity. For a Gorenstein subspace arrangement, it turns out that this invariant has an amazingly simple description in terms of the dual graph of the arrangement. The goal of the talk is to discuss the concepts of dual graph (keeping in mind the motivating case of simplicial complexes) and Castelnuovo-Mumford regularity, to explain the connection we found out for Gorenstein subspace arrangements, and to provide several examples.
Geometry and Physics Seminar
Professor G. Pearlstein
Texas A&M University
Geometry of Nilpotent Cones in Hodge Theory
Wednesday, February 10, 2016, 5:00pm
Ungar Room 402
Abstract: The local monodromy of a degeneration of smooth complex projective varieties gives rise to a monodromy cone which plays a central role in constructing analogs of Mumford's toroidal compactifications for Hodge structures of arbitrary weight. In this talk, I will describe several methods for describing the possible monodromy cones which can arise in a given period domain using topological boundary components and signed Young diagrams.
Combinatorics Seminar
Yuval Roichman
Bar-Ilan University, Israel
Geometric Grid Classes and Symmetric Sets of Permutations
Tuesday, February 9, 2016, 5:00pm
Ungar Room 402
Abstract: Characterizing sets of permutations whose associated quasisymmetric function is symmetric is a long-standing problem in algebraic combinatorics. We present a general method to construct symmetric and Schur-positive sets and multisets, based on algebraic and geometric operations on grid classes. This approach produces new instances of Schur-positive sets and explains the existence of known such sets that until now were sporadic cases.
Joint with Sergi Elizalde.
Applied Math Seminar
Chris Cosner
University of Miami
Evolution of Dispersal in Spatial Population Models with Multiple Timescales
Thursday, February 4, 2016, 4:00pm
Ungar Room 506
Abstract: In many cases the timescale of dispersal for the organisms is sufficiently fast compared to timescale of their population dynamics that it is reasonable to assume the spatial distribution of the organisms is always effectively at equilibrium when viewed on the timescale of population dynamics. Starting from that assumption it is possible to construct models for population dynamics and species interaction that are based on ordinary differential equations but still contain information about the how the dispersal strategies being used by the focal populations interact with environmental heterogeneity. This talk will describe the modeling approach and discuss some applications that are related to the evolution of dispersal.
Probability Seminar
Mehdi Shadmehr
University of Miami
School of Business
Contribution Restrictions in a Global Game
Thursday, February 4, 2016, 3:30pm
Ungar Room 406
Abstract: We investigate a manager's decision to restrict the contribution of an agent with private information. We explore the link between this problem and contribution restrictions in global games with a continuum of agents and continuous actions.
Geometry and Physics Seminar
Professor M. Green
University of California, Los Angeles
From the Point of View of Moduli
What Are the Most Extreme Examples of Curves and Surfaces?
Wednesday, February 3, 2016, 5:00pm
Ungar Room 402
Abstract: What does Hodge theory looks like for these examples? We will illustrate a general set of problems about the relationship between the boundary of moduli spaces and the boundary of the period domains. This is a joint work with Griffiths, Laza, Robles.
Combinatorics Seminar
Richard Stanley
University of Miami
Massachusetts Institute of Technology
Parking Functions and their Generating Function
Monday, February 1, 2016, 4:00pm
Ungar Room 402
Abstract: A parking function is a sequence (a_1,...,a_n) of positive integers whose increasing rearrangement (b_1,...,b_n) satisfies b_i \leq i.
We first explain the connection with parking cars and give the elegant proof of Pollak that the number of parking functions of length n is (n+1)^{n-1}. The symmetric group S_n acts on parking functions of length n by permuting coordinates. This action corresponds to a symmetric function F_n with many interesting properties. We discuss properties of the generating function \sum F_n z^n and show its connection with some variants of parking functions. Finally we consider a q-analogue of the previous theory.
Probability Seminar
William Sudderth
School of Statistics
University of Minnesota
How to Control a Process to a Goal
Thursday, January 28, 2016, 3:30pm
Ungar Room 406
Combinatorics Seminar
Alex Engstroem
Aalto University, Finland
Boolean Statistics
Monday, January 25, 2016, 5:00pm
Ungar Room 402
Abstract: We introduce a new function b(x) associated to simplicial complexes that measures how versatile you can decompose it into boolean lattices. The function interpolates between three important cases: b(2) is the number of faces; b(1) is a good upper bound on the number of maximal faces; and b(0) is the minimal number of critical cells in a discrete Morse matching, which is a natural upper bound on the total dimension of cohomology. In contemporary graph theory b(1) is central, and in topological combinatorics b(2) is. We try to unify some approaches in these branches of discrete mathematics by this setup and manage for example to prove some new theorems regarding the structure of maximal independent sets in triangle-free graphs, improving on recent work by Balogh et al. In the talk we will also explain why all of the bounds mentioned above are optimal for shellable simplicial complexes.
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
Categorical Donaldson Uhlenbeck Yau Correspondence
Wednesday, January 20, 2016, 5:00pm
Ungar Room 402
Abstract: In the last two years a parallel between theory of Higgs bundles and theory of shaves of categories was created. We take this parallel to the next level giving classical correspondence a new categorical meaning. Some applications will be discussed.
Geometry and Physics Seminar
Paul Horja
University of Miami
Introduction to Theory of Algebraic Surfaces
Wednesday, January 13, 2016, 5:00pm
Ungar Room 402
Geometry and Physics Seminar
Morgan Brown
University of Miami
Introduction to Theory of Algebraic Curves
Wednesday, December 9, 2015, 5:00pm
Ungar Room 402
Probability Seminar
Carlos Bajo
University of Miami
Stochastic Models of Cancer Cell Growth
Tuesday, December 8, 2015, 4:00pm
Ungar Room 406
Abstract: Recent research suggests that cancer is a genetic disease caused by DNA alterations which induce cells to divide in uncontrolled fashion and become invasive to the organism. In the evolution of the disease, some of these pernicious cells can mutate either because they divide faster or because they become resistant to some particular treatment. Several mathematical models are presented. Is natural to investigate the distribution of the number $Z_0$ of cancer cells present in the human body at a given time and the distribution of the first time $T_m$ for this amount to reach a certain threshold, when the disease can be detected. Additionally, we shall look at $Z_{1}$, the time of the first occurrence of mutation in cancer cells. The basic model proposed is a two-type branching process in which $Z_0$ cells give birth to $Z_1$ type cells. This talk is based on the paper, "Evolutionary dynamics of tumor progression with random fitness values" by Rick Durrett, Jasmine Foo, Kevin Leder, John Mayberry and Franziska, and the survey "Branching Process Models of Cancer" written by Dr. Richard Durrett. Other discussion in reference of the paper "Modeling the Manipulation of Natural Populations by Mutagenic Chain Reaction" by Robert L. Unckless, Philipp W. Messer, Tim Connallon, and Andrew G. Clark may be address as well.
Combinatorics Seminar
Carolina Benedetti
Fields Institute
York University, Toronto
Chromatic Symmetric Functions on Simplicial Complexes
Monday, December 7, 2015, 5:00pm
Ungar Room 402
Abstract: Inspired by the theory pioneered by Stanley's chromatic symmetric function and its connection to Hopf algebras, we will see how abstract simplicial complexes can be endowed naturally with a combinatorial Hopf structure that gives rise to chromatic generating functions. Using principal specializations on these generating functions we derive certain combinatorial identities involving acyclic orientations related to the complexes. We will also discuss work in progress aiming to derive the analogous of Shareshian-Wachs' "chromatic quasisymmetric functions" for simplicial complexes.
This is joint work with J. Hallam and J. Machacek. No familiarity with Hopf algebras is required, we will give the necessary background.
Probability Seminar
Lin Xi
University of Miami
Progress Report on the Fluctuation Limit for an AIMD Model
Tuesday, December 1, 2015, 4:00pm
Ungar Room 406
Abstract: We discuss the central limit theorem for the fluctuation random field from the fluid limit (LLN) of an Additive Increase Multiplicative Decrease (AIMD) protocol used in internet traffic modeling. The random field is unusual but can be characterized inductively using polynomial test functions. These are preliminary results and a brief survey of methodology.
Combinatorics Seminar
Barbara Bolognese
Northeastern University, Boston
Dual Graphs of Projective Curves
Monday, November 30, 2015, 5:00pm
Ungar Room 402
Abstract: In 1962, Hartshorne proved that the dual graphs of an arithmetically Cohen-Macaulay scheme is connected. After establishing a correspondence between the languages of algebraic geometry, commutative algebra and combinatorics, we are going to refine Hartshorne's result and measure the connectedness of the dual graphs of certain projective schemes in terms of an algebro-geometric invariant of the projective schemes themselves, namely their Castelnuovo-Mumford regularity. Time permitting, we are also going to address briefly the inverse problem of Hartshorne's result, by showing that any connected graph is the dual graph of a projective curve with nice geometric properties.
Geometry and Physics Seminar
Professor Tyler Foster
University of Michigan
Non-Archimedean Geometry in Rank >1
Wednesday, November 18, 2015, 5:00pm
Ungar Room 402
Abstract: Recent work of Nisse-Sottile, Hrushovski-Loeser, Ducros, and Giansiracusa-Giansiracusa has demonstrated that valuation rings of rank >1 play an important role in the geometry of analytic and tropical varieties over non-Archimedean valued fields of rank 1. In this talk, I will present recent work with Dhruv Ranganathan in which we prove several foundational results on the geometry of analytic and tropical varieties over higher rank valued fields, and recent work with Max Hully in which we use rank 2 valuations to give a new, non-analytic proof of Rabinoff's theorem on the correspondence between tropical and algebraic intersection multiplicities.
Probability Seminar
Abdelaziz Alali
University of Miami (Electrical and Computer Engineering)
Reducing the Error-propagation Effect Associated with Stacking Classifiers
Wednesday, November 18, 2015, 3:30pm
Ungar Room 406
Abstract: Multi-label classification targets domains characterized by examples that may belong to more than one category at the same time. A common way to address such problems is to induce a separate classifier for each class. Thus, each classifier determines whether its respective category is relevant for a new example or not. By targeting each class independently, this technique, known as Binary Relevance (BR), assumes that classes are independent of each others, which may not necessarily be the case in all domains. For example, an image of a 'Beach' scene will likely be tagged with the concept 'Ocean' as well. Conversely, the same image is unlikely to represent the topic 'Industry'. To incorporate such class correlations in the BR framework, researchers suggest using information about the classes an example is already known to belong to as inputs in addition to the example's original attributes. Since this information is typically unknown for a previously unseen example, several approaches fill in these values using the outputs of independent classifiers such as those in the BR framework. In real-world scenarios, these outputs are prone to errors. Consequently, using them as inputs may reduce the accuracy of the classifiers. The presented work suggests two ways to address this so-called error-propagation problem. The first method reduces the dependency on the error-prone inputs by eliminating weak class dependencies from the final model. The second proposed solution is to compare the probabilistic classification confidences of the independent models with their dependent counterparts, and then choose the more confident classification. Experiments on a broad set of benchmark datasets indicate that a combination of the two approaches yields a boost in classification accuracy when compared to using the dependent models alone.
Combinatorics Seminar
Bruno Benedetti
University of Miami
Optimal Discrete Morse Vectors Are Not Unique
Monday, November 16, 2015, 5:00pm
Ungar Room 402
Abstract: In classical Morse theory, for any given manifold there is always a unique optimal Morse vector (=the vector counting the number of critical points of index 0,1,..., up to the dimension). It turns out that in Forman's discrete version of Morse theory, this is no longer the case. I will sketch how to construct a contractible 3-complex on which the 'best' discrete Morse vectors are (1,0,1,1) and (1,1,1,0), because (1,0,0,0) is out of reach.
Applied Math Seminar
Shigui Ruan
University of Miami
An Introduction to Pattern Formation in Developmental Biology and Ecology
Friday, November 13, 2015, 4:00pm
Ungar Room 402
Abstract: In 1952, Alan Turing (one of the founders of computer science) proposed a reaction-diffusion model wherein two homogeneously distributed substances, a fast-diffusing inhibitor and a slow-diffusing activator, would interact to produce stable patterns during morphogenesis. These patterns would represent regional differences in the concentrations of the two substances. Their interactions would produce an ordered structure out of random chaos. Turing's model has become one of the most important mathematical models in developmental biology.
In this talk, I will first introduce the spatial dynamics in some reaction-diffusion systems induced by Turing instability and their applications in the formation of skin patterns of some animals and fish. Then I will talk about the spatial, temporal, and spatiotemporal patterns in ecological models, in particular predator-prey models with mutual interference.
Geometry and Physics Seminar
Jeremy Van Horn-Morris
University of Arkansas
On Fillings of the Canonical Contact Structure on the Unit Cotangent Bundle of a Surface
Wednesday, November 11, 2015, 5:00pm
Ungar Room 402
Abstract: The classic example of a contact manifold is the unit sphere bundle of the cotangent bundle of a smooth manifold: $ST^*M$; and the classic example of a symplectic filling is the unit disk bundle of the cotangent bundle $DT^*M.$ It turns out that, using hard applications of holomorphic curves by McDuff, Hind and Wendl, for $Y = S^2$ or $T^2$, these are in fact the only symplectic fillings. Using tools from Li and Mak, we now can extend this list to include all surfaces, at least if we consider classifying symplectic fillings up to homotopy equivalence. This is joint work with Steven Sivek.
Combinatorics Seminar
José Samper
University of Washington, Seattle
Relaxations of the Matroid Axioms
Tuesday, November 10, 2015, 5:00pm
Ungar Room 402
Abstract: Motivated by a question of Duval and Reiner about eigenvalues of combinatorial Laplacians, we develop various generalisations of (ordered) matroid theory to wider classes of simplicial complexes. In addition to all independence complexes of matroids, each such class contains all pure shifted simplicial complexes, and it retains a little piece of matroidal structure. To achieve this, we relax many cryptomorphic definitions of a matroid. In contrast to the matroid setting, these relaxations are independent of each other, i.e., they produce different extensions. Imposing various combinations of these new axioms allows us to provide analogues of many classical matroid structures and properties. Examples of such properties include the Tutte polynomial, lexicographic shellability of the complex, the existence of a meaningful nbc-complex and its shellability, the Billera-Jia-Reiner quasisymmetric function, and many others. We then discuss the h-vectors of complexes that satisfy our relaxed version of the exchange axiom, extend Stanley's pure O-sequence conjecture about the h-vector of a matroid, solve this conjecture for the special case of shifted complexes, and speculate a bit about the general case. Based on joint works with Jeremy Martin, Ernest Chong and Steven Klee.
Applied Math Seminar
Yishu Song
University of Miami
Hydrodynamic Limit for a Supercritical Branching Process
Friday, November 6, 2015, 4:00pm
Ungar Room 402
Abstract: In 1993, Bak and Sneppen proposed a model aiming to describe an ecosystem of interacting species that evolve by mutation and natural selection. Thereafter various mathematical attempts have been made to study the model in its equilibrium. In this talk we'll investigate a variant of the Bak-Sneppen model and its hydrodynamic limit. The solution solves a heat equation with mass creation at a source inside the domain, normalized to have mass one. We discuss its representation as the average of the empirical measure of an auxiliary branching system with mass growing exponentially fast and the relationship between the stationary measure and quasi-stationarity for the auxiliary semigroup.
Probability Seminar
Hamed Amini
University of Miami
Shortest-weight Paths in Random Graphs
Tuesday, November 3, 2015, 4:00pm
Ungar Room 406
Abstract: We study the impact of random exponential edge weights on the distances in a random graph and, in particular, on its diameter. Our main result consists of a precise asymptotic expression for the maximal weight of the shortest weight paths between all vertices (the weighted diameter) of sparse random graphs, when the edge weights are iid exponential random variables. This is based on a joint work with Marc Lelarge.
Combinatorics Seminar
Brittney Ellzey
University of Miami
Power Sum Expansion of the Chromatic Quasisymmetric Functions
Monday, November 2, 2015, 5:00pm
Ungar Room 402
Abstract: Shareshian and Wachs introduced the chromatic quasisymmetric function of a graph as a refinement of Stanley’s chromatic symmetric function. In their paper, Shareshian and Wachs conjecture a formula for the expansion of the chromatic quasisymmetric function of incomparability graphs of natural unit interval orders in the power sum basis. Recently, Athanasiadis proved the conjecture by using a formula of Roichman for the irreducible characters of the symmetric group. In this talk, we will present Athanasiadis' work.
Geometry and Physics Seminar
Professor Terrence Napier
Lehigh University
A Cup Product Lemma, Bounded Geometry, and the Bochner-Hartogs Dichotomy
Wednesday, October 28, 2015, 5:00pm
Ungar Room 402
Abstract: We will consider a version of Gromov's cup product lemma in which one factor is the (1,0)-part of the differential of a continuous pluri-subharmonic function, as well as consequences for the structure of complete Kaehler manifolds with bounded geometry along levels of suitable plurisubharmonic functions. For example, we will see that the Bochner-Hartogs dichotomy holds for any connected one-ended covering of a weakly 1-complete non-compact complete Kaehler manifold; that is, either the first compactly supported cohomology with values in the structure sheaf vanishes, or there exists a proper holomorphic mapping onto a Riemann surface.
Probability Seminar
Ilie Grigorescu
University of Miami
Quasi-stationarity and the Fleming-Viot Particle System
Tuesday, October 27, 2015, 4:00pm
Ungar Room 406
Abstract: We discuss a general class of stochastic processes obtained from a given Markov process whose behavior is modified upon contact with a catalyst, from the perspective of a particle system that undergoes branching with conservation of mass (Fleming-Viot mechanism). We explain the relation of the process and its scaling limit to the existence of quasi-stationary distributions and their simulation. Non-explosion and large deviations for the soft catalyst case will be discussed if time permits. Joint work with Min Kang.
Combinatorics Seminar
Bennet Goeckner
University of Kansas
A Non-partitionable Cohen-Macaulay Complex
Monday, October 26, 2015, 5:00pm
Ungar Room 402
Abstract: In joint work with Art Duval, Caroline Klivans, and Jeremy Martin, we construct a non-partitionable Cohen-Macaulay simplicial complex. This construction disproves a longstanding conjecture by Stanley that would have provided an interpretation of h-vectors of Cohen-Macaulay complexes. Due to an earlier result of Herzog, Jahan, and Yassemi, this construction also disproves the conjecture that Stanley depth is always greater than or equal to depth. Time permitting, we will also discuss Garsia's open conjecture that every Cohen-Macaulay poset has a partitionable order complex.
Applied Math Seminar
Don DeAngelis
University of Miami
Effects of Diffusion on Total Biomass in Heterogeneous Continuous and Discrete-Patch Systems
Friday, October 23, 2015, 4:00pm
Ungar Room 402
Abstract: Theoretical models of populations on a system of two connected patches have shown that, when the two patches differ in maximum growth rate and carrying capacity, and in the limit of high diffusion, conditions exist for which the total population size at equilibrium exceeds that of the Ideal Free Distribution, which predicts that the total population would equal the total carrying capacity of the two patches. However, this result has only been shown for the Pearl-Verhulst growth function on two patches and for a single-parameter growth function in continuous space. Here we provide a general criterion for total population size, exceeding total carrying capacity for three commonly used population growth rates for both heterogeneous continuous and multi-patch heterogeneous landscapes with high population diffusion. We show that a necessary condition for this situation is that there is a convex positive relationship between the parameter for the maximum growth rate and the carrying capacity, as both vary across a spatial region. Because this relationship occurs in biological populations, the result has ecological implications.
Applied Math Seminar
Robert Stephen Cantrell
University of Miami
Evolution of Dispersal in Spatially Heterogeneous Temporally Constant Habitats
Friday, October 16, 2015, 4:00pm
Ungar Room 402
Abstract: In this talk we survey some recent results on the evolution of dispersal in spatially heterogeneous but temporally constant environments. We focus on the evolutionary advantage that arises from moving so as to match underlying heterogeneous resource patterns. Collaborators on this program of study include Lee Altenberg, Chris Cosner, Yuan Lou, Dan Ryan, Mark Lewis, Sebastian Schreiber and King-Yeung Lam.
Probability Seminar
Andreea Minca
Cornell University
A Game Theoretic Approach to Modeling Debt Capacity
Tuesday, October 13, 2015, 4:00pm
Ungar Room 406
Abstract: We propose a dynamic model that explains the build-up of short term debt when the creditors are strategic and have different beliefs about the prospects of the borrowers' fundamentals. We define a dynamic game among creditors, whose outcome is the short term debt. As common in the literature, this game features multiple Nash equilibria. We give a refinement of the Nash equilibrium concept that leads to a unique equilibrium.
For the resulting debt-to-asset process of the borrower we define a notion of stability and find the debt ceiling which marks the point when the borrower becomes illiquid. We show existence of early warning signals of bank runs: a bank run begins when the debt-to-asset process leaves the stability region and becomes a mean-fleeing sub-martingale with tendency to reach the debt ceiling. Our results are robust across a wide variety of specifications for the distribution of the capital across creditors' beliefs. (joint with J Wissel)
Combinatorics Seminar
Ivan Martino
University of Fribourg, Switzerland
Arrangements of Subspaces for Finite Groups and Their Geometrical Applications
Monday, October 12, 2015, 5:00pm
Ungar Room 402
Abstract: Given a faithful representation V of a group G one can consider the partially ordered set of conjugacy classes of stabilizer subgroups. Using this combinatorial object we proved that the motivic class of the classifying stack of every finite linear (or projective) reflection group is trivial.
This poset is a key combinatorial tool also in the study of the motivic class of the quotient variety U/G, where U is the open set of V where the group acts trivially. We discuss the study of such classes by starting from a theorem of Aluffi in the reflection groups case and we conclude by showing that a similar result holds for finite subgroups of GL_3(k) for an algebraically closed field of characteristic zero.
These results relate naturally to Noether's Problem and to its obstruction, the Bogomolov multiplier.
(Part of this is joint work with Emanuele Delucchi.)
Combinatorics Seminar
Florian Frick
Cornell University
Tverberg-type Theorems and Zero Sum Problems
Monday, October 5, 2015, 5:00pm
Ungar Room 402
Abstract: Tverberg-type theorems are concerned with the intersection pattern of faces in a simplicial complex when mapped to Euclidean space. One has to distinguish between results for affine maps (with straight faces) and continuous maps: In the topological case, number-theoretic conditions on the multiplicity of intersections play a role. We will show that most Tverberg-type results, which were believed to require proofs using involved techniques from algebraic topology, follow from a simple combinatorial reduction via the pigeonhole principle. We will construct counterexamples to the topological Tverberg conjecture by Bárány from 1976 building on recent work of Mabillard and Wagner, and we will apply similar ideas to investigate zero-sum problems in Euclidean space. Joint work with Pavle Blagojevic and Günter M. Ziegler.
Geometry and Physics Seminar
Gabriel Kerr
Kansas State University
Mirror Symmetry for C^2 - Point
Thursday, October 1, 2015, 5:00pm
Ungar Room 402
Geometry and Physics Seminar
Morgan Brown
University of Miami
Berkovich Spaces and Birational Geometry
Wednesday, September 30, 2015, 5:00pm
Ungar Room 402
Abstract: A Berkovich space is a type of analytic space associated to an algebraic variety over a field $K$ with valuation $v$, such as $\mathbb{Q}_p$ or $\mathbb{C}((t))$. These spaces are intimately connected with other areas of mathematics including tropical geometry and number theory. I will give an introduction to the theory of Berkovich spaces, and explain how connections with birational geometry can help us understand the geometry of Berkovich spaces.
Combinatorics Seminar
Michelle Wachs
University of Miami
Weighted Bond Posets and Graph Associahedra, Part 2
Monday, September 28, 2015, 5:00pm
Ungar Room 402
Abstract: This talk is a continuation of the September 21 talk.
Applied Math Seminar
Chris Cosner
University of Miami
The Reduction Principle, the Ideal Free Distribution, and the Evolution of Dispersal Strategies
Friday, September 25, 2015, 4:00pm
Ungar Room 402
Abstract: The problem of understanding the evolution of dispersal has attracted much attention from mathematicians and biologists in recent years. For reaction-diffusion models and their nonlocal and discrete analogues, in environments that vary in space but not in time, the strategy of not dispersing at all is often convergence stable within in many classes of strategies. This is related to a "reduction principle" which states that that in general dispersal reduces population growth rates. However, when the class of feasible strategies includes strategies that generate an ideal free population distribution at equilibrium (all individuals have equal fitness, with no net movement), such strategies are known to be evolutionarily stable in various cases. Much of the work in this area involves using ideas from dynamical systems theory and partial differential equations to analyze pairwise invasibility problems, which are motivated by ideas from adaptive dynamics and ultimately game theory. The talk will describe some past results and current work on these topics.
Combinatorics Seminar
Michelle Wachs
University of Miami
Weighted Bond Posets and Graph Associahedra
Monday, September 21, 2015, 5:00pm
Ungar Room 402
Abstract: We consider a weighted version of the bond lattice of a graph. This generalizes the poset of weighted partitions introduced by Dotsenko and Khoroshkin and studied in a previous paper of the authors. We show that for cordal graphs, each interval of the weighted bond poset has the homotopy type of a wedge of spheres, and we present an intriguing connection with h-vectors of graph associahedra studied by Postnikov, Reiner and Williams, and others. This is joint work with Rafael Gonzalez D'Leon.
Geometry and Physics Seminar
Ken Baker
University of Miami
New Surgeries between the Poincare Homology Sphere and Lens Spaces
Wednesday, September 2, 2015, 5:00pm
Ungar Room 402
Abstract: We exhibit an infinite family of hyperbolic knots in the Poincare Homology Sphere with tunnel number 2 and a lens space surgery and discuss the implications. Notably, this is in contrast to the previously known examples due to Hedden and Tange which are all doubly primitive.
Geometry and Physics Seminar
Professor Ludmil Katzarkov
University of Miami
Categorical Kaehler Metrics
Friday, August 28, 2015, 5:00pm
Ungar Room 411
Combinatorics Seminar
Fuzhen Zhang
Nova Southeastern University
Determinant, Permanent, Tensors and Words
Monday, April 20, 2015, 5:00pm
Ungar Room 402
Abstract: Using words of operators in tensor product we present an inequality for positive operators on Hilbert space. The proof of the main result is combinatorial. As applications of the operator inequality and by a multilinear approach, we show some matrix inequalities concerning induced operators and generalized matrix functions (including determinants and permanents as special cases).
Geometry and Physics Seminar
Carla Cederbaum
Tuebingen University
On the Center of Mass in General Relativity
Wednesday, April 15, 2015, 5:00pm
Ungar Room 402
Abstract: In many situations in Newtonian gravity, understanding the motion of the center of mass of a system is key to understanding the general "trend" of the motion of the system. It is thus desirable to also devise a notion of center of mass with similar properties in general relativity. However, while the definition of the center of mass via the mass density is straightforward in Newtonian gravity, there is a priori no definitive corresponding notion in general relativity. Instead, there are several alternative approaches to defining the center of mass of a system. We will discuss some of these different approaches for both asymptotically Euclidean and asymptotically hyperbolic systems and present some explicit (counter-)examples.
Combinatorics Seminar
Benjamin Iriarte Giraldo
Department of Mathematics
MIT
Combinatorics of Acyclic Orientations of Graphs:
Algebra, Geometry and Probability
Monday, March 30, 2015, 5:00pm
Ungar Room 402
Benjamin Iriarte Giraldo will defend his MIT Ph.D. thesis (under the direction of Richard Stanley) at the University of Miami.
Combinatorics Seminar
Michelle Wachs
University of Miami
Generalizations of Bjorner's NBC Basis for the Homology of the Partition Lattice
Monday, March 23, 2015, 5:00pm
Ungar Room 402
Abstract: Bjorner's NBC basis for the homology of the partition lattice has a very nice description in terms of fundamental cycles obtained by splitting increasing rooted trees. We present two generalizations of this basis. One of these, obtained in joint work with Rafael Gonzalez D'Leon, is for a weighted partition poset and the other, obtained in joint work with John Shareshian, is for the 1 mod k partition poset.
Geometry and Physics Seminar
Paul Kirk
Indiana University
Towards a Lagrangian-Floer Theory for Representation Spaces of Tangles
Wednesday, March 18, 2015, 5:00pm
Ungar Room 402
Abstract: We describe how to use SU(2) character varieties of fundamental groups of 3-manifolds to construct a Lagrangian-Floer theory counterpart to Kronheimer-Mrowka's singular instanton knot Floer homology.
Geometry and Physics Seminar
Professor Nikolai Saveliev
University of Miami
Spectra Sets for Inoue Surfaces
Wednesday, March 4, 2015, 5:00pm
Ungar Room 402
Combinatorics Seminar
Richard Stanley
University of Miami
Period Collapse of Ehrhart Quasipolynomials
Monday, February 9, 2015, 5:00pm
Ungar Room 402
Abstract: Let P be a convex polytope with rational vertices. For a positive integer n, let i(P,n) be the number of integer points in nP. A basic theorem of Ehrhart theory says that if p is the gcd of the denominators of all coordinates of the vertices of P, then for 0\leq j<p the function i(P,n) is a polynomial f_j(n) when n is congruent to j mod p. In some cases, however, the "quasiperiod" p can be smaller. After a general discussion we focus on the very simple case of a triangle with vertices (0,0), (a/b,0), (0,a/b), for which some surprising results hold. Most of this is joint work with Daniel Gardiner, who was motivated by symplectic geometry. We assume no prior knowledge of polytopes, Ehrhart theory, or symplectic geometry.
Geometry and Physics Seminar
Professor Patrick Brosnan
University of Maryland
Kashiwara Conjugation for Twisted D-modules
Wednesday, February 4, 2015, 5:00pm
Ungar Room 506
Abstract: In 1987, Kashiwara introduced a functor taking D-modules on a complex manifold X to D-modules on the complex conjugate of X. Moreover, he showed that this functor, which is called Kashiwara conjugation, is an (anti)-equivalence from the category of regular holonomic D-modules on X to those on the complex conjugate of X. Motivated by applications to representation theory, Barlet and Kashiwara extended this functor to modules over rings of twisted differential on generalized flag varieties. I will explain a simple way to extend the Barlet-Kashiwara result to more general rings of twisted differential operators on arbitrary complex manifolds. As some of my motivation for thinking about this comes from conjectures of Schmid and Vilonen on representation theory, I will also give some examples coming out of those conjectures.
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
Categorical Linear Systems and Oscillating Integrals
Saturday, November 15, 2014, 6:00pm
Ungar Room 411
Applied Math Seminar
Professor James Keesling
University of Florida
Modeling Queueing Networks
Friday, November 14, 2014, 4:30pm
Ungar Room 402
Abstract: Our research group in Mathematics at UF has recently come up with a simulation model of the flow of patients through an Emergency Department. The model was developed in collaboration with Adrian Tyndall, Head of Emergency Services at Shands Teaching Hospital at UF.
The model is based on current practices in providing emergency care at a university hospital. The triage of patients, the prioritizing of treatment, the waiting times for various services, and the requirements for facilities and attendants are all based on data from the ED at Shands.
The mathematics involved in creating this model can be applied in many other situations. One project is to expand the ED model to the flow of patients through the whole hospital. Another project is to develop a sophisticated model the spread of hospital acquired infections. We have developed some new techniques in analyzing queueing networks that make the theory better applicable in a wide variety of situations such as the ones mentioned above.
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
Linear Systems and Stability Mixed Structures
Sunday, November 9, 2014, 6:00pm
Ungar Room 411
Probability Seminar
William Sudderth
University of Minnesota
Some Remarks on Finitely-additive Probability
Thursday, November 6, 2014, 2:15pm
Ungar Room 411
Applied Math Seminar
Professor Xiaosheng Li
Florida International University
Inverse Boundary Value Problems with Incomplete Data
Friday, October 31, 2014, 4:30pm
Ungar Room 402
Abstract: Inverse boundary value problems arise when one tries to recover internal parameters of a medium from data obtained by boundary measurements. In many of these problems the physical situation is modeled by partial differential equations. The goal is to determine the coefficients of the equations from measurements of the solutions on the boundary. However, collecting data from the whole boundary is sometimes either not possible or extremely expensive in practice. In this talk we present the recent developments on inverse problems with incomplete data for Schroedinger types of equations in both bounded domains and unbounded domains.
Probability Seminar
Stanislav Volkov
Lund University
On Random Geometric Subdivisions
Thursday, October 30, 2014, 2:15pm
Ungar Room 411
Abstract: I will present several models of random geometric subdivisions, similar to that of Diaconis and Miclo (Combinatorics, Probability and Computing, 2011), where a triangle is split into 6 smaller triangles by its medians, and one of these parts is randomly selected as a new triangle, and the process continues ad infinitum. I will show that in a similar model the limiting shape of an indefinite subdivision of a quadrilateral is a parallelogram. I will also show that the geometric subdivisions of a triangle by angle bisectors converge (but only weakly) to a non-atomic distribution, and that the geometric subdivisions of a triangle by choosing a uniform random points on its sides converges to a flat triangle, similarly to the result of the paper mentioned above.
Combinatorics Seminar
Michelle Wachs
University of Miami
On q-gamma Positivity
Tuesday, October 28, 2014, 5:00pm
Ungar Room 402
Abstract: Gamma-positivity is a property of polynomials that implies palindromicity and unimodality. It has received considerable attention in recent times because of Gal's conjecture, which asserts gamma-positivity of the h-polynomial of flag homology spheres. Eulerian polynomials and the Narayana polynomials are examples of such h-polynomials that are known to be gamma-positive. In this talk I will present q-analogs of formulas establishing gamma-positivity of these and other polynomials. Geometric interpretations involving toric varieties and consequences such as q-unimodality will also be discussed. This is based on joint work with John Shareshian and with Christian Krattenthaler.
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
Categorical Multiplier Ideal Sheaf
Sunday, October 26, 2014, 5:00pm
Ungar Room 411
Geometry and Physics Seminar
R. Paul Horja
University of Miami
On Categorical Multiplier Ideal Sheaf
Tuesday, October 14, 2014, 5:00pm
Ungar Room 411
Applied Math Seminar
Shigui Ruan
University of Miami
Turing Instability and Hopf Bifurcation: Spatio-temporal Dynamics
Friday, October 3, 2014, 4:30pm
Ungar Room 402
Abstract: For a physical or biological system described by reaction-diffusion equations, spatial patterns can occur via Turing instability mechanism (that is bifurcation induced by the diffusion); temporal patterns can occur via Hopf bifurcation induced by the change of parameters. At the points where the Turing instability curve and Hopf bifurcation curve intersect, the model can undergo Turing-Hopf bifurcation and exhibit spatiotemporal patterns. As examples, spatial, temporal, and spatiotemporal dynamics of biological and physical systems are presented and numerical simulations are carried out to verify andillustrate the bifurcation phenomena.
Geometry and Physics Seminar
Sean Bowman
Oklahoma State University
Thin Position, Graph Clustering, and Applications
Thursday, October 2, 2014, 5:00pm
Ungar Room 506
Abstract: We describe a novel algorithm for clustering vertices of graphs. The method is inspired by the technique of thin position in low dimensional topology. We show that a version of our algorithm works well on an important real world data set from biology. This is joint work with Doug Heisterkamp, Jesse Johnson, and Danielle O'Donnol.
Applied Math Seminar
Stephen Cantrell
University of Miami
Avoidance Behavior in Intraguild Predation Communities: A Cross-Diffusion Model
Friday, September 26, 2014, 4:30pm
Ungar Room 402
Abstract: A cross-diffusion model of an intraguild predation community where the intraguild prey employs a fitness based avoidance strategy is examined. The avoidance strategy employed is to increase motility in response to negative local fitness. Global existence of trajectories and the existence of a compact global attractor are proved. It is shown that if the intraguild prey has positive fitness at some point in the habitat when trying to invade, then it will be uniformly persistent in the system if its avoidance tendency is sufficiently strong. This type of movement strategy can lead to coexistence states in which the intraguild prey is marginalized to areas with low resource productivity while the intraguild predator maintains high densities in regions with abundant resources, a pattern observed in many real world intraguild predation systems.
Functors and Complexity Seminar
Colin Diemer
University of Miami
Landau-Ginzburg Models and Secondary Polytopes II
Tuesday, September 16, 2014, 5:00pm
Ungar Room 411
Abstract: We will continue our review of recent efforts by Kapranov, Kontsevich, and Soibelman to understand categories associated to Lefschetz fibrations by means of a combinatorial deformation theory. Time permitting, will conjecture possible connections to other geometric approaches to decomposing these structures.
Applied Math Seminar
Chris Cosner
University of Miami
Spatial Population Dynamics in a Producer-Scrounger Model
Friday, September 12, 2014, 4:30pm
Ungar Room 402
Abstract: The spatial population dynamics of an ecological system involving producers and scroungers is studied using a reaction-diffusion model. The two populations move randomly and increase logistically, with birth rates determined by the amount of resource acquired. Producers can obtain the resource directly from the environment, but must surrender a proportion of their discoveries to nearby scroungers through a process known as scramble kleptoparasitism. The proportion of resources stolen by a scrounger from nearby producers decreases as the local scrounger density increases. Producer persistence depends in general on the distribution of resources and producer movement, whereas scrounger persistence depends on the ability to invade an environment when producers are at steady-state. It is found that (i) both species can persist when the habitat has high productivity, (ii) neither species can persist when the habitat has low productivity, and (iii) slower dispersal of both the producer and scrounger is favored when the habitat has intermediate productivity. This research is in collaboration with Andrew Nevai.
Functors and Complexity Seminar
Colin Diemer
University of Miami
Landau-Ginzburg Models and Secondary Polytopes
Tuesday, September 9, 2014, 5:00pm
Ungar Room 411
Abstract: We will review recent efforts by Kapranov, Kontsevich, and Soibelman to understand categories associated to Lefschetz fibrations by means of a combinatorial deformation theory. Time permitting, will conjecture possible connections to other geometric approaches to decomposing these structures.
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
Linear Systems – Old and New
Wednesday, September 3, 2014, 5:00pm
Ungar Room 506
Combinatorics Seminar
Anton Dochtermann
University of Miami
Face Rings of Cycles, Associahedra, and Standard Tableaux
Monday, April 22, 2014, 5:00pm
Ungar Room 402
Abstract: Let J n denote the quadratic monomial ideal generated by the diagonals of an n-gon (i.e. the Stanley-Reisner ideal of an n-cycle). One way to realize a free resolution of J n is to utilize a natural monomial labeling of the faces of the (dual) associahedron - the complex computing cellular homology supports a resolution of J n. This resolution is not minimal since the dual associahedron has too many high-dimensional faces.
On the other hand, some years ago Richard Stanley gave a simple bijection between the faces of the associahedron and standard Young tableaux of certain shapes. We show that the Betti numbers of J n (the ranks of the free modules in a minimal resolution) are given by the number of standard Young tableaux of certain sub-shapes. While we do not have a good description of the differentials with this basis, this does suggest a connection with the face poset structure of the associahedron.
Combinatorics Seminar
This seminar will be a mini-symposium consisting of 3 half hour talks. Tuesday, March 18, 2014, 5:00pm Ungar Room 402
Hana Kim
Sungkyunkwan University
A Short Trip to the Riordan Group
Abstract: The Riordan group is a group of infinite lower triangular matrices defined by a pair of formal power series. It has been studied for the last twenty years over a wide range of different areas. In this talk we invite you on a short trip to the Riordan group. A brief survey of results on the algebraic structure of the Riordan group will be given. We will also go through applications of Riordan group theory and explore a few of them in more detail.
and
Tommy Wuxing Cai
South China University of Technology
A Jacobi-Trudi Formula for Macdonald Functions of Rectangular Shapes
Abstract: We express Macdonald functions of rectangular shapes using vertex operators, thereby giving a generalized Jacobi-Trudi formula for them. The proof relies on a splitting formula for the q-Dyson Laurent polynomial, from which two results follow immediately: Kadell's orthogonality conjecture proved by Károlyi, Lascoux, and Warnaar, and Andrews' q-Dyson constant term conjecture proved by Zeilberger and Bressoud. This is joint work with Naihuan Jing.
and
Lili Mu
Dalian University of Technology
Sperner Theorems for Convex Families
Abstract: Sperner's theorem states that the density of the largest Sperner family (or antichain) in the boolean algebra B n is $\binom{n}{n/2}/2^n$. It is conjectured that this is an upper bound for any convex family. We provide further evidence for the conjecture by exhibiting a number of examples of convex families satisfying the bound.
Geometry and Analysis Seminar
Professor Dave Auckly
Kansas State University
Stable Equivalence of Surfaces in 4-manifolds
Monday, March 17, 2014, 2:30pm
Ungar Room 506
Abstract: It is well known that there are homeomorphic 4-manifolds that are not smoothly equivalent, that become smoothly equivalent after taking the connected sum with one copy of a special manifold. Similar behavior may be found in other geometric settings, including diffeomorphisms up to isotopy, positive scalar curvature and knotted surfaces. In this talk we will prove that there is an infinite family of spheres in a connected sum of complex projective spaces with assorted orientations, so that no two spheres in the family are smoothly equivalent, yet every pair is topologically isotopic. Furthermore we will show that the spheres become smoothly isotopic after one stabilization. We will do so with an explicit description of the spheres and the isotopies. This is joint work with Danny Ruberman, Paul Melvin, and Hee Jung Kim.
Applied Math Seminar
Professor Zhisheng Shuai
University of Central Florida
Spatially Heterogeneous Cholera Models
Friday, February 28, 2014, 5:00pm
Ungar Room 506
Abstract: Cholera was one of the most feared diseases in the 19th century, and remains a serious public health concern today. It can be transmitted to humans directly by person-to-person contact or indirectly through ingestion of contaminated water. Spatial heterogeneity of both humans and water may influence the spread of cholera. To incorporate these spatial effects, two cholera models are proposed that both include direct and indirect transmission. The first is a multi-group model and the second is a multi-patch model. New mathematical tools from graph theory are used to understand the dynamics of both heterogeneous cholera models, and to show that each model satisfies a sharp threshold property. Specifically, Kirchhoff's matrix tree theorem is used to investigate the dependence of the disease threshold on the patch connectivity and water movement (multi-patch model), and also to establish the global dynamics of both models.
Seminar on Cluster Algebras
Ludmil Katzarkov
University of Miami
Polyhedral Cones and Categories
Friday, February 28, 2014, 4:00pm
Ungar Room 506
Combinatorics Seminar
Tamar Friedmann
University of Rochester
Enumeration of Vacua of String and M-theory:
Formulas for Counting Conjugacy Classes of Elements of Finite Order in Lie Groups
Tuesday, February 25, 2014, 5:00pm
Ungar Room 402
Seminar on Cluster Algebras
Colin Diemer
University of Miami
Cluster Algebras, Character Varieties, and Cremona Groups
Friday, February 21, 2014, 4:00pm
Ungar Room 506
Combinatorics Seminar
Michelle Wachs
University of Miami
A Geometric Interpretation of an Eulerian Number Identity
Monday, February 17, 2014, 5:00pm
Ungar Room 402
Abstract: We discuss an identity of Chung, Graham and Knuth involving Eulerian numbers and binomial coefficients. We give a geometric interpretation of this identity, of a q-analog due to Chung-Graham and Han-Lin-Zeng, and of a symmetric function analog due to Shareshian-Wachs. Our interpretation involves the h-vector of the stellohedron and the representation of the symmetric group on the cohomology of the associated toric variety. This is joint work with John Shareshian.
Applied Math Seminar
Yanyu Xiao
University of Miami
Study of Some Mathematical Models on Communicable Diseases
Friday, February 14, 2014, 5:00pm
Ungar Room 402
Abstract: In the first part of this talk, I will focus on mathematical modeling of the transmission of a vector-borne disease, Malaria, considering the following three factors: 1) disease latency; 2) spatial dispersal; 3) Multiple strains. In the second part, I will talk about some statistical analysis on epidemiological data of H1N1 Influenza in 2009.
Seminar on Cluster Algebras
Ken Baker
University of Miami
Can We Obtain an Invariant of Contact Structures from Automorphisms of Cluster Algebras?
Friday, February 7, 2014, 3:30pm
Ungar Room 411
Combinatorics Seminar
Rafael Gonzalez D'Leon
University of Miami
The Combinatorial Structure behind Multibracketed Free Lie Algebras
Tuesday, February 4, 2014, 5:00pm
Ungar Room 402
Abstract: We explore a beautiful interaction between algebra and combinatorics in the heart of the free Lie algebra on n generators: The multilinear component of the free Lie algebra Lie(n) is isomorphic as a representation of the symmetric group to the top cohomology of the poset of partitions of an n-set tensored with the sign representation. Then we can understand the algebraic object Lie(n) by applying poset theoretic techniques to the poset of partitions whose description is purely combinatorial. We will show how this relation generalizes further in order to study free Lie algebras with multiple compatible brackets. In particular we obtain combinatorial bases and compute the dimensions of these modules. Part of the talk is based on joint work with M. Wachs.
Combinatorics Seminar
Professor Richard Stanley
MIT and University of Miami
Two Enumerative Tidbits
Tuesday, January 28, 2014, 5:00pm
Ungar Room 402
Abstract: The two tidbits are:
1. The Smith normal form of some matrices connected with Young diagrams
2. A Distributive lattice associated with three-term arithmetic progressions
Applied Math Seminar
Professor Stefan Siegmund
Dresden University of Technology, Germany
Why the Present Is More Complicated than the Future
Friday, January 17, 2014, 4:00pm
Ungar Room 506
Abstract: Models from e.g. fluid dynamics or systems biology often can be described by ordinary differential equations. While the long-term behavior of those models as time tends to infinity is quite well understood, the theory of transient solution behavior is still in its infancy. We present some results as well as open questions.
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
Differentiably Finite Functions and Noncommutative Mordell Lang Conjecture
Wednesday, January 15, 2014, 5:00pm
Ungar Room 402
Abstract: In this talk we will connect some classical and new results from logic, number theory, combinatorics and dynamical systems to theory of categories.
Combinatorics Seminar
Bruno Benedetti
Freie Universitaet Berlin
Discretized Morse Theory vs. Knots
Monday, December 2, 2013, 5:00pm
Ungar Room 402
Abstract: Morse theory studies smooth manifolds up to homotopy by looking at generic real-valued functions defined on them. Discrete Morse theory relies on Whitehead's "simple homotopy" theory, and it applies to triangulations of manifolds -- or, more generally, to arbitrary (regular CW) complexes. It yields a valid tool to `simplify' a complex without changing its homotopy. In this talk we plan to sketch:
(1) the relation between smooth and discrete Morse theory. (For example, how to reconstruct a smooth function from a discrete one, or how to characterize the Heegaard genus as "best discrete Morse vector" over all possible triangulations.)
(2) obstructions coming from knot theory. (For example, how to build "nasty" triangulations of the 3-sphere, over which any discrete Morse function has lots of critical faces.)
Seminar on Cluster Algebras
Ludmil Katzarkov
University of Miami
Nonrational Clusters
Monday, December 2, 2013, 4:00pm
Ungar Room 411
Seminar on Cluster Algebras
Ken Baker
University of Miami
Surfaces and Cluster Algebras
Monday, November 18, 2013, 4:00pm
Ungar Room 411
Geometry and Physics Seminar
M. Verbitsky
HSE Moscow
Kobayashi Pseudometric on Hyperkahler Manifolds
Tuesday, November 12, 2013, 5:00pm
Ungar Room 402
Seminar on Cluster Algebras
Drew Armstrong
University of Miami
The Numerology of Finite File Type Clusters
Monday, November 11, 2013, 4:00pm
Ungar Room 411
Geometry and Physics Seminar
Allison Moore
Rice University
Pretzel Knots Admitting L-space Surgeries
Wednesday, November 6, 2013, 5:00pm
Ungar Room 402
Abstract: A rational homology sphere whose Heegaard Floer homology is the same as that of a lens space is called an L-space. We will classify pretzel knots with any number of tangles which admit L-space surgeries, and discuss some questions regarding essential Conway spheres and knot Floer homology that arise from this classification.
Applied Math Seminar
Shi-Liang Wu
University of Miami
Interaction of Traveling Waves and Entire Solutions for Several Evolution Systems
Friday, November 1, 2013, 5:00pm
Ungar Room 402
Abstract: Wave propagation occurs in many applied fields such as materials science, biology and life science. In addition to the traveling waves, one can also observe interaction between different waves. Mathematically, this phenomenon can be described by the so-called wave-like entire solution that is defined for all space and time and behaves like a combination of traveling waves as $t\rightarrow-\infty$. In this talk we show the existence and various qualitative properties of wave-like entire solutions for several evolution systems. Some open problems and issues are also suggested for future research.
Geometry and Physics Seminar
Neil Hoffman
University of Melbourne
Verified Computations for Hyperbolic 3-manifolds
Wednesday, October 16, 2013, 5:00pm
Ungar Room 402
Abstract: Given a triangulated 3-manifold M a natural question is: Does M admit a hyperbolic structure?
While this question can be answered in the negative if M is known to be reducible or toroidal, it is often difficult to establish a certificate of hyperbolicity, and so computer methods have developed for this purpose. In this talk, I will describe a new method to establish such a certificate via verified computation and compare this method to existing techniques.
This is joint work with Kazuhiro Ichihara, Masahide Kashiwagi, Hidetoshi Masai, Shin'ichi Oishi, and Akitoshi Takayasu.
Seminar on Cluster Algebras
Colin Diemer
University of Miami
The Birational Geometry of Mutations (d'apres Gross, Hacking, and Keel)
Monday, October 14, 2013, 4:30pm
Ungar Room 411
Applied Math Seminar
Fan Zhang
University of Miami
Metapopulation Models with Nonlocal Dispersal
Friday, October 11, 2013, 5:00pm
Ungar Room 402
Abstract: In this work we study evolutionary stability of nonlocal dispersal strategies in metapopulation prospective. (Holt and Timothy 2000) complement the classic Levins model (Hanski 1997) by considering the environmental gradients. Our model is a continuum generated from this metapopulation model, based on integrodifferential equations, using nonlocal operators to describe colonization at each site on a finite domain. We extend the Maximum principle, Comparison theorem and study an eigenvalue problem for our model. We also study coexistence and extinction in the competition system.
Combinatorics Seminar
Rafael D'Leon
University of Miami
About the Combinatorics of Multibracketed Free Lie Algebras
Friday, October 11, 2013, 4:00pm
Ungar Room 402
Abstract: It is a classical result that the multilinear component of the free Lie algebra with n generators Lie(n) has dimension (n-1)!. It is also well known that Lie(n) is isomorphic as a representation of the symmetric group to the top cohomology of the poset of set partitions tensored with the sign representation. I will discuss how these results generalize as to consider free Lie algebras with multiple compatible brackets.
Seminar on Cluster Algebras
Monday, October 7, 2013, 4:00pm
Ungar Room 411
Applied Math Seminar
Yanyu Xiao
University of Miami
How the Latency Impacts the Disease Dynamics
Friday, October 4, 2013, 5:00pm
Ungar Room 402
Abstract: In this work, we modified the classic Ross-Macdonald model for malaria disease dynamics by incorporating latencies both for human beings and female mosquitoes. We introduced two general probability functions (P1(t) and P2(t)) to reflect the fact that the latencies differ from individuals to individuals, and investigated the impact of the latencies on disease outbreak.
Combinatorics Seminar
Anton Dochtermann
University of Miami
Cellular Resolutions via Mapping Cones
Wednesday, October 2, 2013, 5:00pm
Ungar Room 411
Abstract: Suppose I is a monomial ideal. One can iteratively obtain a free resolution of I by considering the "mapping cone" of a map of complexes associated to adding one generator at a time. Herzog and Takayama have shown that this procedure yields a minimal resolution if the ideal I "has linear quotients".
Here we consider cellular realizations of these resolutions. Extending a construction of Mermin we describe a regular CW complex that supports the HT resolution. By varying the choice of chain map we recover other known cellular resolutions and obtain combinatorially distinct complexes with interesting structure, suggesting a notion of a "space of resolutions". This is joint work with Fatemeh Mohammadi.
Seminar on Cluster Algebras
Monday, September 30, 2013, 4:00pm
Ungar Room 411
Applied Math Seminar
Lan Zou
University of Miami
Modeling the Transmission and Control of Schistosomiasis in China
Friday, September 27, 2013, 5:00pm
Ungar Room 402
Abstract: Schistosomiasis is one of the main tropic diseases. In some provinces of China, Schistosoma japonicum is endemic. There are three types of regions: (i) plain regions with waterway networks, (ii) mountainous and hilly regions, and (iii) marshland and lake regions. The transmission of the first type of regions has been eliminated. The second type has been controlled. The disease is still endemic in some regions of the third type, although it is becoming better. Based on the human-cattle-snail transmission of schistosomiasis, a model consisting of six ordinary differential equations that describe susceptible and infected human, cattle and snail subpopulations is proposed. We perform some numerical simulations and analysis for different types of regions separately.
Geometry and Physics Seminar
Professor S. Paul
University of Wisconsin
Stable Pairs and Coercive Estimates for the Mabuchi Functional II
Friday, September 27, 2013, 4:00pm
Ungar Room 402
Abstract: We show that a projective manifold is "stable" if and only if the Mabuchi Energy is proper on the corresponding space of Bergman metrics. We also show that properness implies finite automorphism group.
Background and Motivation will be provided. The talk may be of interest to people working in algebraic geometry and invariant theory as well as P.D.E.'s and differential geometry.
Geometry and Physics Seminar
Professor S. Paul
University of Wisconsin
Stable Pairs and Coercive Estimates for the Mabuchi Functional
Thursday, September 26, 2013, 3:15pm
Florida International University
DM 409
Abstract: We show that a projective manifold is "stable" if and only if the Mabuchi Energy is proper on the corresponding space of Bergman metrics. We also show that properness implies finite automorphism group.
Background and Motivation will be provided. The talk may be of interest to people working in algebraic geometry and invariant theory as well as P.D.E.'s and differential geometry.
Seminar on Cluster Algebras
Monday, September 23, 2013, 4:00pm
Ungar Room 411
PechaKucha Global Night
Ken Baker
University of Miami
3D Printing & Mathematics
Friday, September 20, 2013, 7:00pm
The LAB Miami
400 NW 26th Street
Miami, FL 33127
Abstract: 3D printing experts explore developments in the education, medical, and corporate sectors. Live demonstrations will follow after the program.
Admission is free. Everyone is welcome to attend.
Seminar on Cluster Algebras
L. Katzarkov
University of Miami
Clusters, Categories and Cremona
Friday, September 20, 2013, 4:00pm
Ungar Room 402
Seminar on Cluster Algebras
Monday, September 16, 2013, 4:00pm
Ungar Room 411
Applied Math Seminar
Shigui Ruan
University of Miami
Multi-patch Models for Vector-borne Diseases
Friday, September 13, 2013, 5:00pm
Ungar Room 402
Abstract: We develop spatial models of vector-borne disease dynamics on a network of patches to examine how the movement of humans in heterogeneous environments affects transmission. We show that the movement of humans between patches is sufficient to maintain disease persistence in patches with zero transmission. We construct two classes of models using different approaches: (i) Lagrangian that mimics human commuting behavior and (ii) Eulerian that mimics human migration. We determine the basic reproduction number R0 for both modeling approaches and study the transmission dynamics in terms of R0. We also study the dependence of R0 on some parameters such as the travel rate of the infectives.
Seminar on Cluster Algebras
Monday, September 9, 2013, 4:00pm
Ungar Room 533
Geometry and Physics Seminar
Professor K. Baker
University of Miami
New Surgeries between Lens Spaces and Non-prime Manifolds
Wednesday, September 4, 2013, 5:00pm
Ungar Room 402
Abstract: The Cabling Conjecture asserts that any knot in S^3 with a surgery to a non-prime 3-manifold must be a cabled knot. In contrast, two families of hyperbolic knots in lens spaces with non-prime surgeries have been known. Recently we have greatly generalized these families and suggested that these might form the foundation for a classification of surgeries between lens spaces and non-prime manifolds. In this talk we'll describe this generalization and discuss the construction of yet further new families of such knots.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
Categories Dynamical Systems and Cremona Groups
Wednesday, August 28, 2013, 5:00pm
Ungar Room 402
Geometry and Physics Seminar
E. Gasparim
University of Campinas
On Landau-Ginzburg Models
Friday, July 26, 2013, 4:30pm
Ungar Room 506
Abstract: Some concrete examples of Landau-Ginzburg models can be obtained from Letschetz fibrations. I will discuss various features of Hodge diamonds obtained in such examples, and how they vary according to the choices of compactification.
Applied Math Seminar
Jing Chen
University of Miami
Bifurcations in Predator-Prey Models with Seasonal Prey Harvesting
Friday, April 26, 2013, 5:00pm
Ungar Room 402
Abstract: In last three decades, some previous work is about the application of bifurcation theory to predator-prey models with a variety of functional responses. Meanwhile it is also important to understand the nonlinear dynamics of predator-prey systems with harvesting. The most common types of harvesting are constant-effort harvesting and constant-yield harvesting. Now people begin to think the seasonal harvesting which is usually described by a periodic function and investigate the change of dynamical behaviors caused by seasonal harvesting.
My model is based on the predator-prey model with Ivlev type functional response. First considering the constant-yield harvesting, we reveal far richer dynamics compared to the case without harvesting. Secondly, we will consider the seasonal prey harvesting. There is some work about seasonal harvesting in one species system. Now we are trying to give some results in this two species system.
Geometry and Physics Seminar
N. Saveliev
University of Miami
Index Theory of the de Rham Complex on End-periodic Manifolds
Wednesday, April 17, 2013, 4:00pm
Ungar Room 402
Abstract: The analytic index of the de Rham complex on a compact orientable manifold is known to equal its Euler characteristic; the same holds for manifolds with product ends, for a properly understood L2 index. We show that this is no longer true for more general manifolds with periodic ends, by providing an explicit formula for the difference between the L2 index of the de Rham complex and the Euler characteristic of the manifold in terms of topology of the end.
Geometry and Physics Seminar
Professor Lars Andersson
Albert Einstein Institute
Max Planck Institute for Gravitational Physics
Hertz Potentials and Asymptotic Behavior of Massless Fields
Wednesday, April 10, 2013, 5:00pm
Ungar Room 506
Abstract: Hertz and related potentials provide a method to construct massless spin- sfields from solutions of wave equations. In this talk I will explain recent work which gives decay estimates for massless fields using a representation in terms of Hertz potentials. The method extends to the Kerr spacetime and I will discuss some equations which arise in this context.
Combinatorics Seminar
Michelle Wachs
University of Miami
Positivity Results and Conjectures for Chromatic Quasisymmetric Functions
Tuesday, April 9, 2013, 5:00pm
Ungar Room 402
Abstract: The chromatic quasisymmetric functions, introduced by Shareshian and Wachs, are a refinement of Stanley's chromatic symmetric functions, which in turn specialize to the classical chromatic polynomials. For incompatibility graphs of natural unit interval orders the chromatic quasisymmetric functions are actually symmetric functions, and are conjecturally related to Tymoczko's representation of the symmetric group on cohomology of Hessenberg varieties and to Iwahori-Hecke algebra characters. In this talk I will present our refinement of Gasharov's Schur-positivity result for incomparability graphs of natural unit interval orders and show how we use it to obtain coefficients in the expansion of chromatic quasisymmetric functions in the power-sum basis. This is joint work with John Shareshian.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
From WKB Method to Category Theory and Their Ergodic Nature
Wednesday, April 3, 2013, 4:00pm
Ungar Room 402
Abstract: We will look at the 200-year-old procedure from the point of view of categories.
Geometry and Physics Seminar
P. Horja
University of Vienna
The Kaehler Moduli Space of a Toric Stack
Tuesday, March 26, 2013, 5:00pm
Ungar Room 402
Abstract: The Kaehler moduli space is an interesting string theoretical notion whose mathematical definition is still unsatisfactory. In this talk, I will describe an attempt to give a categorical interpretation of this moduli space in the case of toric varieties/stacks. The construction is inspired by the grade restriction rule for categories of graded modules and coherent sheaves as introduced by Herbst-Hori-Page and Ballard-Favero-Katzarkov.
Combinatorics Seminar
Jang Soo Kim
University of Minnesota
Dyck Tilings and Related Topics
Tuesday, March 19, 2013, 5:00pm
Ungar Room 402
Abstract: Recently, Kenyon and Wilson introduced a certain matrix M in order to compute pairing probabilities of a double-dimer model. They showed that the absolute value of each entry of the inverse matrix of M is equal to the number of certain Dyck tilings of a skew shape. They conjectured two elegant formulas on the sum of the absolute values of the entries in a row or a column of M^{-1}. In this talk we will see bijective proofs of the two conjectures due to Kim, Meszaros, Panova, and Wilson. In the two bijective proofs Dyck tilings correspond to increasing labelled trees and complete matchings. We will also see a connection between Dyck tilings and the (q,t)-Catalan numbers.
Combinatorics Seminar
Professor Mark Skandera
Lehigh University
Hecke Algebra Characters and Quantum Chromatic Symmetric Functions
Tuesday, March 12, 2013, 4:00pm
Ungar Room 402
Abstract: We discuss generating functions for Hecke algebra characters, formulas for the evaluation of these characters at Kazhdan-Lusztig basis elements of the Hecke algebra, and (conjecturally) related symmetric functions defined by Shareshian and Wachs. While certain posets called unit interval orders may provide the key to connecting the Hecke algebra elements and symmetric functions, we propose to describe the connection in terms of a class of directed graphs which arose implicitly in papers of Goulden-Jackson, Greene, Stanley-Stembridge, and Haiman. Using these directed graphs, we conjecture a combinatorial formula for coefficients in the power sum expansion of the quantum chromatic symmetric functions.
Geometry and Physics Seminar
Professor Oliver Fabert
University of Freiburg
Floer Theory and Frobenius Manifolds
Wednesday, March 6, 2013, 4:00pm
Ungar Room 402
Abstract: While it is known that the axioms of Gromov-Witten theory can be encoded into the geometrical notion of Frobenius manifolds, Floer theory can be viewed as a generalization of Gromov-Witten theory. In this talk I show how the Frobenius manifolds of Gromov-Witten theory generalize to Floer theory, employing the symplectic field theory of Eliashberg-Givental-Hofer. In particular, I show that the symplectic cohomology of an open symplectic manifold can be equipped with the structure of a cohomology F-manifold in the sense of Merkulov. Here the first obstruction against smoothness is given by the BV bracket on symplectic cohomology.
Applied Math Seminar
Dr. Shigui Ruan
University of Miami
Modeling Transmission Dynamics of Rabies in China
Friday, March 1, 2013, 5:00pm
Ungar Room 402
Abstract: Human rabies is one of the major public-health problems in China. The number of human rabies cases has increased dramatically in the last 15 years, partially due to the poor understanding of the transmission dynamics of rabies and the lack of effective control measures of the disease. In order to explore effective control and prevention measures we propose a deterministic model to study the transmission dynamics of rabies in China. The model consists of susceptible, exposed, infectious, and recovered subpopulations of both dogs and humans and describes the spread of rabies among dogs and from infectious dogs to humans. The model simulations agree with the human rabies data reported by the Chinese Ministry of Health. We also modify the model to include stray dogs into account and use the model to simulate the human rabies cases reported in Guangdong Province. Furthermore, we consider the seasonal effect on the transmission of rabies. Sensitivity analysis of R0 in terms of the model parameters is performed and different control and prevention measures, such as culling and immunization of dogs, are compared. Our study demonstrates that (i) reducing dog birth rate and increasing dog immunization coverage rate are the most effective methods for controlling rabies in China; (ii) large scale culling of susceptible dogs can be replaced by immunization of them; (iii) enhancing public education and awareness about rabies; and (iv) strengthening supervision of pupils and children in the summer and autumn.
Combinatorics Seminar
Carlos Bajo
University of Miami
On the Tutte-Krushkal-Renardy Polynomial for Cell Complexes
Tuesday, February 26, 2013, 5:00pm
Ungar Room 402
Abstract: Recently V. Krushkal and D. Renardy generalized the Tutte polynomial from graphs to cell complexes. We show that evaluating this polynomial at the origin gives the number of cellular spanning trees in the sense of A. Duval, C. Klivans, and J. Martin. Moreover, after a slight modification, the Tutte-Krushkal-Renardy polynomial evaluated at the origin gives a weighted count of cellular spanning trees, and therefore its free term can be calculated by the cellular matrix-tree theorem of Duval et al. In the case of cell decomposition of a sphere, this modified polynomial satisfies the same duality identity as before. We find that evaluating the Tutte-Krushkal-Renardy along a certain line is the Bott polynomial.
Geometry and Physics Seminar
Dr. Yoshihiro Fukumoto
Ritsumeikan University
Cobordism of Lens Spaces and Instantons
Wednesday, February 20, 2013, 4:00pm
Ungar Room 402
Abstract: In this talk we consider a special type of cobordisms among lens spaces. We determined the inclusion homomorphisms on integral homology from the boundary components of cobordisms under some (strong) conditions.
Let X be a cobordim among n copies of L(p,q) and n copies of L(p,-q) such that the second Betti number is zero and the inclusion homomorphism on integral homology from the boundary components is surjective. A typical example of X is the connected sum of n copies of L(p,q) x [0,1]. We show that when q = 1 the kernel of the inclusion homomorphism is of the "same form" as in the case of the typical example. In the proof we use Donaldson theory to construct a non-trivial reducible SU(2)-flat connection on X. A key point is "bubbling" phenomena of instantons on orbifolds.
The argument can be seen as an illustration of one aspect of Donaldson theory whose counterpart in Heegaard Floer theory or Seiberg-Witten theory is still missing. This is a joint work with M. Furuta.
Applied Math Seminar
Dr. Chris Cosner
University of Miami
Population Models with Nonlocal Dispersal in Discrete and Continuous Space
Friday, February 15, 2013, 5:00pm
Ungar Room 402
Applied Math Seminar
Dr. Steve Cantrell
University of Miami
Nonlinear Diffusion and Resource Matching in Population Dynamics
Friday, February 8, 2013, 5:00pm
Ungar Room 402
Geometry and Physics Seminar
Professor B. de Oliveira
University of Miami
Closed Symmetric Differentials, Foliations and Fibrations Part 2: The Proofs
Wednesday, February 6, 2013, 4:00pm
Ungar Room 402
Abstract: We will prove the two main theorems described in the previous lecture. We will start and the emphasis will be given to theorem 2.
Theorem 1: If a projective manifold has a closed symmetric 2-differential w of the 1st kind, then π 1(X) is infinite and w comes from a map from X to a quotient of an Abelian variety by a cyclic or dihedral group.
The next theorem deals with a class of closed symmetric differentials that contains differentials not of the 1st kind.
Theorem 2: If a projective manifold has a closed symmetric 2-differential of rank 2 which is decomposable as a product of two closed meromorphic 1- differentials, then the 1st Betti of X is non trivial and the Albanese dimension of X is ≥2.
Combinatorics Seminar
Drew Armstrong
University of Miami
Conjectures on Rational Catalan Numbers
Tuesday, February 5, 2013, 5:00pm
Ungar Room 402
Abstract: Given coprime positive integers a and b, there are some things we don't understand about the relationship between b^{a-1} and a^{b-1}. There are categorifications of these numbers into bigraded representations of the symmetric groups S_a and S_b, respectively; which we call "rational parking spaces". The multiplicity of the trivial character in each equals the "rational Catalan number"
Cat(a,b) = (a+b-1)! / (a!b!).
What explains the a,b-symmetry? I will describe an encoding of this theory in terms of Shi hyperplane arrangements, and state several interesting conjectures.
The discussion will be informal(!), as I will not have time to write a script.
Combinatorics Seminar
Anton Dochtermann
University of Miami
Bigraphical Arrangements
Tuesday, January 29, 2013, 5:00pm
Ungar Room 402
Abstract: Associated to any graph G, Hopkins and Perkinson recently defined a family of hyperplane arrangements that they call 'bigraphical'. Specifying various parameters of the bigraphical arrangements recover the classical Shi and interval order arrangements. Hopkins and Perkinson show that a certain 'Pak-Stanley' labeling of the regions of the bigraphical arrangement recover the G-parking functions, and furthermore that the set of labels is invariant under a notion of sliding hyperplanes. We will discuss these notions and speculate on connections to some underlying commutative algebra. Time permitting, we will discuss a generalization to higher dimensional complexes.
Geometry and Physics Seminar
Professor B. de Oliveira
University of Miami
Closed Symmetric Differentials, Foliations and Fibrations
Wednesday, January 23, 2013, 4:00pm
Ungar Room 402
Abstract: We will show that if a projective surface has a symmetric differential of degree 2 which decomposes as a product of two closed meromorphic differentials, then the fundamental group of X is infinite (in fact we will describe the geometric structures that produce such differentials).
If time permits we will explore the properties of the 2-webs (2 foliations) associated to closed symmetric differentials. Here we are thinking of results that describe when does the presence of a closed symmetric 2-differential implies the existence of a fibration (associated to a map to a curve). These results are somewhat related to the De Franchis-Castelnuovo result and the result that states that a foliation associated to a closed holomorphic 1-form is a fibration if it contains a compact leaf that it is not exceptional.
Geometry and Physics Seminars
Joshua Greene
Boston College
Strong L-spaces
Friday, November 30, 2012, 4:00pm
Ungar Room 402
Abstract: Strong L-spaces are a family of 3-manifolds with a concrete, combinatorial description. However, they owe their definition to Heegaard Floer homology. I will explain the special role they play in Heegaard Floer homology and then turn to the question of their classification. Some surprising combinatorics enters into the picture (perfect matchings and Pfaffian orientations on graphs).
Combinatorics Seminar
Michelle Wachs
University of Miami
On Bounded Regions of Hyperplane Arrangements (Continued)
Tuesday, November 27, 2012, 5:00pm
Ungar Room 402
Combinatorics Seminar
Michelle Wachs
University of Miami
On Bounded Regions of Hyperplane Arrangements
Tuesday, November 20, 2012, 5:00pm
Ungar Room 402
Geometry and Physics Seminar
Kristen Moore
Albert Einstein Institute
Evolving Hypersurfaces by Their Inverse Null Mean Curvature
Wednesday, November 14, 2012, 4:00pm
Ungar Room 402
Abstract: We introduce a new second order parabolic evolution equation where the speed is given by the reciprocal of the null mean curvature. This flow is a generalisation of inverse mean curvature flow and it is motivated by the study of black holes and mass/energy inequalities in general relativity. We present a theory of weak solutions using level-set methods and an appropriate variational principle, and outline a natural application of the flow as a variational approach to constructing marginally outer trapped surfaces (MOTS), which play the role of quasi-local black hole boundaries in general relativity.
Geometry and Physics Seminar
Carla Cederbaum
Duke University
Geometrostatics: The Geometry of Static Spacetimes in General
Tuesday, November 13, 2012, 5:00pm
Ungar Room 402
Abstract: Geometrostatics is an important subdomain of Einstein's General Relativity. It describes the mathematical and physical properties of static isolated relativistic systems such as stars, galaxies, or black holes. For example, geometrostatic systems have a well-defined ADM- mass (Chrusciel, Bartnik) and (if this is nonzero) also a center of mass (Huisken-Yau, Metzger, Huang) induced by a CMC-foliation at infinity. We will present surface integral formulas for these physical properties in general geometrostatic systems. Together with an asymptotic analysis, these can be used to prove that ADM-mass and center of mass 'converge' to the Newtonian mass and center of mass in the Newtonian limit c→∞(using Ehler's frame theory). We will discuss geometric similarities of geometrostatic and classical static Newtonian systems along the way.
Geometry and Physics Seminar
M. Verbitsky
HSE Moscow
Special Holonomy and the ADHM Construction
Monday, November 5, 2012, 5:00pm
Ungar Room 506
Abstract: I will explain the construction of instantons over P^3 via a complexified version of the ADHM construction. This gives a holomorphic connection with special holonomy on the moduli of mathematical instantons on P^3. The same geometry, called trisymplectic, appears whenever one attempts to build a complexification of a hyperkaehler manifold.
A trisymplectic structure on a complex 2n-manifold is a triple of holomorphic symplectic forms such that any linear combination of these forms has rank 2n, n or 0. We show that a trisymplectic manifold is equipped with a holomorphic 3-web and the Chern connection of this 3-web is holomorphic, torsion-free, and preserves the three symplectic forms. We construct a trisymplectic structure on the moduli of rational curves in the twistor space of a hyperkaehler manifold.
The space of moduli of mathematical instantons on CP^3 can be identified with a component on the moduli space of rational curves on a twistor space of moduli of framed instantons on CP^2. The trisymplectic structure on this space is applied to obtain the space of instantons using a new geometric reduction procedure, called trihyperkaehler reduction. This is used to prove that the space of mathematical instantons on CP^3 is smooth, settling a long-standing conjecture.
Geometry and Physics Seminar
Fabrice Debbasch
University of Paris VI
Stochastic Processes on Lorentzian Manifolds
Wednesday, October 31, 2012, 4:00pm
Ungar Room 402
Abstract: Stochastic processes on Lorentzian manifolds were first considered by Dudley in 1965. The first physically realistic Lorentzian process was proposed in 1997 by Debbasch, Mallick and Rivet. Since then, the field has been developing quickly at the interface between the physics and mathematics communities. I will review the basic ideas underlying the construction of Lorentzian processes and compare all existing processes with each other. A very robust and elegant H-theorem will be presented on this occasion. I will also discuss some important recent results which highlight connections between diffusions on Lorentzian manifolds and geometrical flows, black hole horizons and quantum walks.
Geometry and Physics Seminar
Dr. Chen-Yun Lin
University of Connecticut
On Bartnik's Construction of Prescribed Scalar Curvature
Wednesday, October 24, 2012, 4:00pm
Ungar Room 402
Abstract: Riemannian 3-manifolds with prescribed scalar curvature arise naturally in general relativity as spacelike hypersurfaces in the underlying spacetime. In 1993, Bartnik introduced a quasi-spherical construction of metrics of prescribed curvature on 3-manifolds. Under quasi-spherical ansatz, the problem is converted to the initial value problem for a semi-linear parabolic equation of the lapse function. In this talk, we consider background foliations given by conformal round metrics, and by the Ricci flow on 2-speres. We discuss conditions on the scalar curvature function and on the foliation that guarantee the solvability of the parabolic equation, and thus the existence of flat 3-matrics with a prescribed inner boundary.
Geometry and Physics Seminar
Ken Baker
University of Miami
Annular Twists and Bridge Numbers of Knots
Wednesday, October 10, 2012, 4:00pm
Ungar Room 402
Abstract: Performing +1/n and -1/n Dehn surgery on the boundary components of an annulus A in a 3-manifold M provides a homeomorphism of M similar to a Dehn twist. If a knot intersects the interior of A in an essential manner, then this twisting produces an infinite family of knots. In joint work with Gordon and Luecke, we show (under certain hypotheses) that if the bridge numbers of this family with respect to a given Heegaard surface of M are bounded, then the annulus may be isotoped to embed in the Heegaard surface. With this we construct genus 2 manifolds that each contain a family of knots with longitudinal surgeries to S^3 and unbounded genus 2 bridge number. In contrast, our earlier work gives an a priori upper bound on the genus g bridge number of a knot with a non-longitudinal S^3 surgery.
Combinatorics Seminar
Rafael S. González D'León
University of Miami
Weighted Partition Posets
Tuesday, October 2, 2012, 5:00pm
Ungar Room 402
Abstract: V. V. Dotsenko and A. S. Khoroshkin introduced an analog of the lattice of set partitions where each block in a partition is provided with a nonnegative number less than its cardinality. These partitions are said to be weighted and the poset is called the poset of weighted partitions. We will discuss some of the interesting algebraic and homological properties of this poset.
Geometry and Physics Seminar
E. Gasparim
University of Campinas
Lefschetz Fibrations on Adjoint Orbits
Monday, October 1, 2012, 2:00pm
Ungar Room 402
Abstract: I will explain how to obtain structures of Lefschetz fibrations on the contangent bundles of flag manifolds using Lie theory. This is joint work with L. Grama and L. San Martin.
Geometry and Physics Seminar
Eric Harper
McMaster University
SU(N) Casson-Lin Invariants for Links
Friday, September 28, 2012, 3:45pm
Ungar Room 402
Abstract: We will introduce a family of invariants of links in the 3-sphere using projective SU(N) representations and braid theory. We will give some examples of computations, including the theorem that the SU(2) invariant of a two-component link is the linking number between its components.
Geometry and Physics Seminar
Alan Parry
Duke University
Modeling Wave Dark Matter in Dwarf Spheroidal Galaxies
Wednesday, September 26, 2012, 4:00pm
Ungar Room 402
Abstract: Many dwarf spheroidal galaxies are some of the most dark matter dominated galaxies known. As such they are excellent test beds for dark matter theories. There has been some recent work by astrophysicists attempting to match velocity dispersion profiles predicted by particle dark matter models to the observed velocity dispersions in dwarf spheroidal galaxies with reasonable success. We compare these models to those predicted by static spherically symmetric wave dark matter and use these comparisons to obtain an estimate on the value of the constant, Upsilon, which is a fundamental component of the wave dark matter model.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
Phantoms in Geometry
Wednesday, September 19, 2012, 4:00pm
Ungar Room 402
Abstract: Recently a new notion of phantom category was introduced. We give examples and discuss possible geometric applications.
Geometry and Physics Seminar
Dr. Jiakun Liu
Princeton University
Global Regularity of Reflector Problem
Wednesday, September 12, 2012, 4:00pm
Ungar Room 402
Abstract: In this talk we study a reflector system which consists of a point light source, a reflecting surface and an object to be illuminated. Due to its practical applications in optics, electro-magnetics, and acoustic, it has been extensively studied during the last half century. This problem involves a fully nonlinear partial differential equation of Monge-Ampere type, subject to a nonlinear second boundary condition. In the far field case, it is related to the reflector antenna design problem and optimal transportation problem. Therefore, the regularity results of optimal transportation can be applied. However, in the general case, the reflector problem is not an optimal transportation problem and the regularity is an extremely complicated issue. In this talk, we give necessary and sufficient conditions for the global regularity and briefly discuss their connection with the Ma-Trudinger-Wang condition in optimal transportation. This is a joint work with Neil Trudinger.
Combinatorics Seminar
Anton Dochtermann
University of Miami
Laplacian Ideals, Arrangements, and Resolutions
Tuesday, September 11, 2012, 5:00pm
Ungar Room 402
Abstract: The lattice ideal of the Laplacian matrix of a graph G provides an algebraic perspective on the combinatorial dynamics of the Abelian Sandpile Model and the more general Riemann-Roch theory of G. The generators of this ideal form a Groebner bases with respect to a certain term order, and the associated initial ideals have nice connections to G-parking functions. We study resolutions of these initial ideals and show that, at least under certain conditions on G, a minimal free resolution is supported on the bounded subcomplex of a hyperplane section of the graphical arrangement of G. It is conjectured that these complexes also support resolutions for the Laplacian lattice ideal itself. This generalizes constructions from Postnikov and Shaprio (for the case of the complete graph) and connects to work of Manjunath and Sturmfels, and Perkinson on the commutative algebra of Sandpiles. Time permitting we will discuss some connections to the topology of generalized partition posets. This is joint work with Raman Sanyal.
Geometry and Physics Seminar
Professor Santiago Simanca
University of New Mexico
On the Shape of Representable Integral Homology Classes in a Riemannian Manifold
Wednesday, May 16, 2012, 4:00pm
Ungar Room 402
Abstract: Let (M,g) be a closed Riemannian manifold, D be a class in H k(M;Z) . If D can be represented by an embedded submanifold, we endow such a representative with the induced metric from that on M, and consider the space of all embedded representatives of D. On it we define the functional given by the squared L 2 -norm of the second fundamental form of the embedding. Its minimum of smallest volume, should it exists, provide for the canonical shape of D. We discuss this problem and some of its details in general, and focus on the particulars of the cases of P 2 with the Fubini-Study metric, or Sp(2) with suitable left-invariant metrics adapted to the fibration S 3 ↪ Sp(2) → S 7 .
Geometry and Physics Seminar
Professor Gabor Szekelyhidi
University of Notre Dame
On the Positive Mass Theorem for Manifolds with Corners
Friday, May 4, 2012, 4:00pm
Ungar Room 402
Abstract: A problem originally studied by P. Miao is whether the positive mass theorem holds on manifolds with certain singularities along a hypersurface. I will discuss an approach to this problem which uses the Ricci flow to smooth out the metric, so that one can apply the usual positive mass theorem. This allows for extending the rigidity statement in the zero mass case to higher dimensions, which was only known in the 3 dimensional case previously. This is joint work with D. McFeron.
Geometry and Physics Seminar
Dr. Jeff Jauregui
University of Pennsylvania
An Axiomatic Approach to Quasi-local Mass in General Relativity
Wednesday, April 11, 2012, 4:00pm
Ungar Room 402
Abstract: In general relativity, the notion of quasi-local mass seeks to answer the question: "how much mass is contained in a bounded region in a spacelike slice of a spacetime"? We propose a definition that is motivated by axioms and designed to be as simple as possible. This definition is related to the existence of solutions to a boundary value problem for metrics of nonnegative scalar curvature. Interestingly, it tends to vanish on static vacuum regions. Finally, we recognize this quasi-local mass as a type of product of two well-known other definitions.
Combinatorics Seminar
Drew Armstrong
University of Miami
Cluster Combinatorics
Tuesday, April 10, 2012, 5:00pm
Ungar Room 402
Abstract: Cluster algebras were invented around 2000 by Fomin and Zelevinsky through their study of total positivity in algebraic groups, and have since become extremely popular. The algebraic structure of cluster algebras can be largely reduced to a combinatorial structure called a "cluster complex", which generalizes the classical associahedron. Via this reduction, one can show that cluster algebras of "finite type" are parametrized by Dynkin diagrams (again). I will bring two zome models of associahedra to the talk, which you may inspect. (Some people think that it is impossible to build associahedra from zome tools. I discovered today that this is quite false.)
Combinatorics Seminar
Anton Dochtermann
University of Miami
Probabilistic and Topological Bounds on Chromatic Number
Tuesday, April 3, 2012, 5:00pm
Ungar Room 402
Abstract: A graph G is called "bipartite" if can be properly colored with two colors (the chromatic number of G is at most 2). Consider two alternative characterizations for this notion:
Probabilistic: G is bipartite if for any random walk W on G, the position of W at arbitrarily large time t restricts the possible starting position of W.
Topological: G is bipartite if in the "space of directed edges" of G, there's no way to walk from an edge E to the same edge with the reverse orientation.
Brightwell and Winkler introduced a generalization of the first property which they called the "warmth" of a graph, and showed that warmth is a lower bound for chromatic number. This was somewhat surprising since the warmth a graph G is defined in terms of homomorphisms from some fixed graph (whereas chromatic number is about homomorphisms intocomplete graphs). In this sense warmth also looks like the Hom complexes first introduced by Lovasz (the objects I discussed in the Grad student seminar last month), where the topological connectivity of a "space of directed edges" provides a lower bound for chromatic number. In addition, warmth and Hom complexes behave similarly with respect to certain graph operations including "foldings". However, no direct connection has been established (as far as I know). We'll discuss these concepts and present a conjectural inequality relating warmth and the topology of the Hom complexes.
Geometry and Physics Seminar
Professor Ryan Derby-Talbot
Quest University
Essential Surfaces and Dehn Filling
Wednesday, March 28, 2012, 4:00pm
Ungar Room 402
Abstract: A driving question in 3-dimensional topology is how the structure of a 3-manifold M with torus boundary can change under Dehn filling. For example, if M contains an essential surface, the surface will usually remain essential after Dehn filling, the exceptions occurring for a restricted set of filling slopes. Is the same true for the entire set of essential surfaces in M? In other words, will this set usually be preserved under Dehn filling except for a restricted set of filling slopes? We will show that the answer is yes. The techniques involved use a kind of thin position argument and some normal surface theory. This is joint work with Dave Bachman and Eric Sedgwick.
Geometry and Physics Seminar
Dr. Wilderich Tuschmann
Karlsruhe Institute of Technology
Curvature vs. Curvature Operator
Wednesday, March 21, 2012, 4:00pm
Ungar Room 402
Abstract: The talk will deal with recent results and open questions in the global geometry and topology of manifolds with nonnegative and almost nonnegative curvature and curvature operator, resp., and, in particular, describe how to distinguish these spaces from each other.
Combinatorics Seminar
Mark Skandera
Lehigh University
A Conjectured Combinatorial Interpretation for Induced Sign Characters of the Hecke Algebra
Tuesday, March 6, 2012, 5:00pm
Ungar Room 402
Abstract: Many combinatorial formulas for computations in the symmetric group S n can be modified appropriately to describe computations in the Hecke algebra H n(q), a deformation of C[S n]. Unfortunately, the known formulas for induced sign characters of S n are not among these. For induced sign characters of H n(q), we conjecture formulas which specialize at q=1 to formulas for induced sign characters of S n. We will discuss evidence in favor of the conjecture, and relations to the chromatic quasi-symmetric functions of Shareshian and Wachs.
This is joint work with Brittany Shelton of Lehigh University.
Combinatorics Seminar
Nathan Williams
University of Minnesota
Promotion and Rowmotion
Friday, March 2, 2012, 5:00pm
Ungar Room 506
Abstract: We present an equivariant bijection between two actions--promotion and rowmotion--on order ideals in certain posets. This bijection simultaneously generalizes a result of R. Stanley concerning promotion on the linear extensions of two disjoint chains and the type A case of work of D. Armstrong, C. Stump, and H. Thomas on noncrossing and nonnesting partitions. We apply this bijection to several classes of posets, obtaining equivariant bijections to various known objects under rotation.
Applied Math Seminar
John Chadam
Department of Mathematics
University of Pittsburgh
Optimal Prepayment of Mortgages
Friday, March 2, 2012, 5:00pm
Ungar Room 402
Abstract: The optimal strategy for the prepayment of fixed rate mortgages is modeled mathematically as a free boundary problem for a parabolic PDE. Basic existence and uniqueness results are summarized. Non-linear integral equations are then developed for the location of the free boundary (the risk-free interest rate below which the mortgage should be prepaid). They are used to derive a fast and accurate numerical scheme for calculating the early prepayment boundary. Finally, a simple, easily implemented analytic approximation for this boundary is obtained using asymptotic analysis. (Joint work with Xinfu Chen (Pittsburgh) and Dejun Xie (Suzhou))
Combinatorics Seminar
Gabriel Kerr
University of Miami
Fiber Polytopes
Tuesday, February 14, 2012, 5:00pm
Ungar Room 402
Abstract: Given a linear projection $P\to Q$ of convex polytopes, Billera and Sturmfels defined a third polytope $\Sigma(P\to Q)$ called the {\bf fiber polytope} of the projection. If P is an $(n-1)$-dimensional simplex and $Q$ is a convex $n$-gon then $\Sigma(P\to Q)$ is the associahedron. We will discuss the basics of this theory, and maybe more.
Combinatorics Seminar
Sergi Elizalde
Dartmouth University
Consecutive Patterns in Permutations:
Clusters, Generating Functions and Asymptotics
Friday, February 10, 2012, 5:00pm
Ungar Room 402
Abstract: A permutation p avoids a consecutive pattern q if no subsequence of adjacent entries of p is in the same relative order as the entries of q. For example, alternating permutations are those that avoid the consecutive patterns 123 and 321.
I will discuss some old and new results on the enumeration of permutations that avoid consecutive patterns. One of the main tools is the cluster method of Goulden and Jackson, based on inclusion-exclusion, which reduces the enumeration of these permutations to counting linear extensions of certain posets. For several patterns of arbitrary length, we obtain differential equations for the generating functions counting occurrences of the consecutive patterns.
I will also show that among consecutive patterns of any fixed length, the monotone pattern is easier to avoid than any non-overlapping pattern.
Combinatorics Seminar
Jim Haglund
University of Pennsylvania
The Monotone Column Permanent Conjecture and Multivariate Eulerian Polynomials
Monday, December 19, 2011, 3:00pm
Ungar Room 402
Abstract: Let B be an n by n matrix of real numbers, weakly increasing down columns. The Monotone Column Permanent Conjecture says that the permanent, of the matrix whose ij-th entry is (B)ij +z, has only real zeros, as a polynomial in z. In this talk we discuss the recent proof of this conjecture by Branden, Visontai, Wagner and the speaker. Our proof is based on the theory of stable polynomials, which are multivariate polynomials which are non-vanishing if all the variables have positive imaginary part. As a by-product of our work we obtain mutivariate stable versions of Eulerian polynomials.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
SHS and Automorphic Forms
Tuesday, November 29, 2011, 5:00pm
Ungar Room 506
Combinatorics Seminar
Michelle Wachs
University of Miami
Eulerian Numbers, Chromatic Quasisymmetric Functions and Hessenberg Varieties
Tuesday, November 22, 2011, 5:00pm
Ungar Room 402
Abstract: We consider three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of the Eulerian numbers, the one in symmetric function theory deals with a refinement of Stanley's chromatic symmetric functions, and the one in algebraic geometry deals with a representation of the symmetric group on the cohomology of the regular semisimple Hessenberg variety of type A. Our purpose is to explore some connections between these topics and consequences of these connections. This talk is based on joint work with John Shareshian.
Applied Math Seminar
Professor Igor Belykh
Georgia State University
Synchrony in Metapopulations:
The Role of Dispersal
Monday, November 21, 2011, 4:30pm
Ungar Room 402
Abstract: Many plant and animal populations have been shown to synchronize over large areas. In this talk, I will discuss synchrony in networks of food chains, composed of resource, consumer and predator populations (for example, algae, zooplankton and fish). Each community is described by the Rosenzweig-MacArthur tritrophic food chain model, and the communities interact through a network, composed of patches and migration corridors. I will present a general method to determine global stability of synchronization in ecological networks with any coupling topology. I will also demonstrate that, if only one species can migrate, the dispersal of the consumer (i.e., the intermediate trophic level) is the most effective mechanism for promoting synchronization.
Applied Math Seminar
Steve Cantrell
University of Miami
A Problem in Transitioning from Spatial to Landscape Ecology:
Perspectives from Several Modeling Formulations:
Part 2- Integro-difference and Average Dispersal Success Matrix Approaches
Friday, November 18, 2011, 5:00pm
Ungar Room 402
Abstract: In this talk we compare and contrast the predictions of some spatially explicit and implicit models in the context of a thought problem at the interface of spatial and landscape ecology. The situation we envision is a one-dimensional spatial universe of infinite extent in which there are two disjoint focal patches of a habitat type that is favorable to some specified species. We assume that neither patch is large enough by itself to sustain the species in question indefinitely, but that a single patch of size equal to the combined sizes of the two focal patches provides enough contiguous favorable territory to sustain the given species indefinitely. When the two patches are separated by a patch of unfavorable matrix habitat, the natural expectation is that the species should persist indefinitely if the two patches are close enough to each other but should go extinct over time when the patches are far enough apart. Our focus here is to examine how different mathematical regimes may be employed to model this situation, with an eye toward exploring the trade-off between the mathematical tractability of the model on one hand and the suitability of its predictions on the other. In particular, we are interested in seeing how precisely the predictions of mathematically rich spatially explicit regimes (reaction-diffusion models, integro-difference models) can be matched by those of ostensibly mathematically simpler spatially implicit patch approximations (discrete-diffusion models, average dispersal success matrix models).
Joint work with Chris Cosner (University of Miami) and William Fagan (University of Maryland).
Geometry and Physics Seminar
Ryan Budney
University of Victoria
Some Simple Triangulations
Friday, November 11, 2011, 1:30pm
Ungar Room 506
Abstract: I'll describe the story of how Thurston observed some very simple triangulations of knot and link complements in the 3-sphere. This allowed for a relatively simple way to find hyperbolic structures on such manifolds, and was a key inspiration for the Geometrization Conjecture of 3-manifolds. Ben Burton and I have recently been studying 4-dimensional triangulations and we came across an analogous triangulation for the complement of an embedded 2-sphere in the 4-sphere. While this does not lead to an amazing conjecture like Geometrization, it does lead to an interesting insight into things called Cappell-Shaneson knots, which are historically related to the smooth 4-dimensional Poincare conjecture. This is joint work with Ben Burton and Jonathan Hillman.
Geometry and Physics Seminar
R. Fedorov
Kansas State University
Generalized Langlands Correspondences
Wednesday, November 9, 2011, 4:00pm
Ungar Room 506
Combinatorics Seminar
Drew Armstrong
University of Miami
Parking Spaces
Tuesday, November 8, 2011, 5:00pm
Ungar Room 402
Abstract: There is a program called, say, "Catalan Combinatorics" that seeks to unify various kinds of combinatorics (parking functions, noncrossing/nonnesting partitions, cluster complexes/associahedra, Shi arrangements, core partitions, etc.) under the theory of reflection groups. Today I will talk about the role of parking functions in this project. The classical parking functions are well known. Given a Weyl group W with root lattice Q and Coxeter number h, Haiman generalized parking functions to the finite torus Q/(h+1)Q. In joint work with Vic Reiner and Brendon Rhoades, we have now generalized Q/(h+1)Q in two directions for any real (and maybe complex) reflection group W. We call these the "noncrossing parking space" and the "algebraic parking space". These new parking spaces are actually W × C-modules, where C is the cyclic group generated by a Coxeter element. Our Main Conjecture says that the NC parking space and the algebraic parking space are W × C-isomorphic. A uniform proof of this conjecture would solve several open problems in the subject.
Applied Math Seminar
Steve Cantrell
University of Miami
A Problem in Transitioning from Spatial to Landscape Ecology:
Perspectives from Several Modeling Formulations:
Part 1- Reaction-diffusion, Discrete-diffusion and Metapopulation Approaches
Friday, November 4, 2011, 5:00pm
Ungar Room 402
Abstract: In this talk we compare and contrast the predictions of some spatially explicit and implicit models in the context of a thought problem at the interface of spatial and landscape ecology. The situation we envision is a one-dimensional spatial universe of infinite extent in which there are two disjoint focal patches of a habitat type that is favorable to some specified species. We assume that neither patch is large enough by itself to sustain the species in question indefinitely, but that a single patch of size equal to the combined sizes of the two focal patches provides enough contiguous favorable territory to sustain the given species indefinitely. When the two patches are separated by a patch of unfavorable matrix habitat, the natural expectation is that the species should persist indefinitely if the two patches are close enough to each other but should go extinct over time when the patches are far enough apart. Our focus here is to examine how different mathematical regimes may be employed to model this situation, with an eye toward exploring the trade-off between the mathematical tractability of the model on one hand and the suitability of its predictions on the other. In particular, we are interested in seeing how precisely the predictions of mathematically rich spatially explicit regimes (reaction-diffusion models, integro-difference models) can be matched by those of ostensibly mathematically simpler spatially implicit patch approximations (discrete-diffusion models, average dispersal success matrix models).
Joint work with Chris Cosner (University of Miami) and William Fagan (University of Maryland).
Geometry and Physics Seminar
Professor Albert Fathi
Ecole Normale Supérieure de Lyon
Smooth Time Functions for Stably Causal and Non-stably Causal Manifolds
Wednesday, November 2, 2011, 4:00pm
Ungar Room 506
Abstract: Existence of smooth time functions on stably causal Lorentzian manifolds has finally been established 5/6 years ago by Bernal and Sanchez, About the same time with Antonio Siconolfi we obtained a proof that is also valid for cone structures on manifolds. Our approach uses ideas that we have developed to construct smooth subsolutions of the Hamilton-Jacobi Equation. We will first explain the ideas of this approach.
Time permitting in a second part, we will explain the current development, with Antonio Siconolfi and Pierre Pageault, where we have been able to understand how to obtain degenerate smooth time functions for a general cone structure on a manifold that gives a genuine time function on the stably causal part.
Geometry and Physics Seminar
Professor S. Kaliman
University of Miami
Flexible Varieties
Wednesday, October 19, 2011, 4:00pm
Ungar Room 506
Geometry and Physics Seminar
Professor Lan-Hsuan Huang
Columbia University
Hypersurfaces with Nonnegative Scalar Curvature and the Positive Mass Theorem
Wednesday, October 12, 2011, 4:00pm
Ungar Room 506
Abstract: Since the time of Gauss, geometers have been interested in the interplay between the intrinsic metric structure of hypersurfaces and their extrinsic geometry from the ambient space. For example, a result of Sacksteder tells us that if a complete hypersurface has non-negative sectional curvature, then its second fundamental form in Euclidean space must be positive semi-definite.
In a recent joint work with Damin Wu, we study hypersurfaces under a much weaker curvature condition. We prove that a hypersurface with nonnegative scalar curvature which is either closed or complete of finite many regular ends must be weakly mean convex. This result is optimal in the sense that the scalar curvature cannot be replaced by other k-th mean curvatures. The result and argument have applications to the mean curvature flow, positive mass theorem, and rigidity theorems.
Applied Math Seminar
Shigui Ruan
University of Miami
Within-host Dynamics of Malaria Infection with Immune Responses -
with an Introduction of Research by Some Nobel Laureates in Medicine or Physiology
Friday, October 7, 2011, 5:00pm
Ungar Room 402
Abstract: On Monday (October 3), three scientists won this year's Nobel Prizes in Medicine or Physiology for their discoveries on how the innate and adaptive phases of the immune response are activated and thereby provided novel insights into disease mechanisms. Their work has opened up new avenues for the development of prevention and therapy against infections, cancer, and inflammatory diseases. In this talk I'll introduce how both innate immunity and adaptive immunity fight again malaria infection and model the within-host dynamics of malaria infection with immune response. I will show that synchronization with regular periodic oscillations (of period 48 h) occurs in blood-stage malaria infection.
Applied Math Seminar
Chris Cosner
University of Miami
Modeling the Evolution of Conditional Dispersal in Spatially Heterogeneous Environments
Friday, September 23, 2011, 5:00pm
Ungar Room 402
Abstract: Mathematical models predict that in environments that are heterogeneous in space but constant in time, there will be selection for slower rates of unconditional dispersal, including specifically random dispersal by diffusion. However, some types of unconditional dispersal may be unavoidable for some organisms, and some organisms may disperse in ways that depend on environmental conditions. In some cases, models predict that certain types of conditional dispersal strategies may be evolutionarily stable within a given class of strategies. For environments that vary in space but not in time those strategies are often the ones that lead to an ideal free distribution of the population using them, provided that such strategies are available within the class of feasible strategies.
Problems in the evolution of dispersal have been addressed from two complementary mathematical viewpoints, namely game theory and mathematical population dynamics. This talk will describe some results and open problems from the viewpoint of spatially explicit models in population dynamics, specifically reaction-diffusion-advection models. Some of the results and problems are related to the evolutionary stability of dispersal strategies leading to an ideal free distribution and the mechanisms that might allow organisms to realize such strategies.
Geometry and Physics Seminar
Dr. Jeffrey Case
Princeton University
Quasi-Einstein Metrics and Conformal Geometry
Thursday, September 1, 2011, 4:00pm
Ungar Room 402
Abstract: Quasi-Einstein Metrics are an important class of metrics which include Einstein metrics, static metrics, and gradient Ricci solitons. Except for gradient Ricci solitons, these metrics all admit a natural formulation in conformal geometry. Moreover, this conformal formulation is reflected in many aspects of the study of gradient Ricci solitons. We will introduce and describe this conformal formulation in two contexts. First, we will use it to prove a precompactness theorem for compact quasi-Einstein metrics, yielding in particular convergence to gradient Ricci solitons. Second, we will use it to introduce the tractor calculus to quasi-Einstein metrics, which will yield some insights into the basic structure of quasi-Einstein metrics.
Geometry and Physics Seminar
R. Sazdanovic
University of Pennsylvania
Categorifications of the Polynomial Ring Z[x]
Wednesday, August 31, 2011, 4:00pm
Ungar Room 506
Abstract: We develop a diagrammatic categorification of the polynomial ring Z[x], based on a geometrically defined graded algebra, and show how to lift various operations on polynomials to the categorified setting. Our categorification satisfies a version of Bernstein-Gelfand-Gelfand reciprocity property with the indecomposable projective modules corresponding to xn and standard modules to (x-1)n in the Grothendieck ring. Generalization of this approach leads to categorification of the Chebyshev, Hermite, and other orthogonal polynomials. This is joint work with M. Khovanov.
Combinatorics Seminar
Brant Jones
James Madison University
Abacus Models for Parabolic Quotients of Affine Weyl Groups
Wednesday, August 3, 2011, 4:45pm
Ungar Room 402
Abstract: The cosets of a finite Weyl group inside the corresponding affine Weyl group have remarkable structure with connections to various objects in algebra and geometry. The abacus is a versatile combinatorial model for these cosets that originates in the work of James and Kerber for the symmetric group. We describe generalizations of this model for the affine types B, C, and D.
Workshop: Topics in Mathematical Relativity
Pengzi Miao
University of Miami
Second Variation of Wang-Yau Quasi-local Energy
Thursday, July 28, 2011, 4:00pm
Ungar Room 411
Abstract: Recently Wang and Yau have introduced a new concept of quasi-local energy associated to an admissible function on a closed spacelike two-surface. The Wang-Yau quasi-local mass is then defined as the infimum of the quasi-local energy over all admissible functions. In this talk, we provide some remarks on the second variation of this quasi-local functional.
Workshop: Topics in Mathematical Relativity
Carlos Vega
University of Miami
On the Bartnik Splitting Conjecture
Thursday, July 28, 2011, 3:00pm
Ungar Room 411
Workshop: Topics in Mathematical Relativity
José Luis Flores
Universidad de Málaga
Periodic Geodesics on Compact Lorentzian Manifolds with a Killing Vector Field
Tuesday, July 26, 2011, 3:00pm
Ungar Room 411
Abstract: In this talk we prove a compactness result for subgroups of the isometry group of a compact Lorentzian manifold with a Killing vector field which is timelike somewhere. As a consequence, we prove the existence of one timelike non self-intersecting periodic geodesic. If the Killing vector field is never vanishing, then there are at least two distinct periodic geodesics; as a special case, compact stationary manifolds have at least two periodic timelike geodesics.
Workshop: Topics in Mathematical Relativity
Dr. Didier Solis
Universidad Autónoma de Yucatán
(UM PhD, 2006)
Local Flatness in Asymptotically Flat Spacetimes
Thursday, July 21, 2011, 3:00pm
Ungar Room 411
Abstract: It is known that an asymptotically simple solution to the vacuum Einstein equations having a null line has to be isometric to Minkowski space. Here we present a result that guarantees the local flatness and 1-connectedness of vacuum solutions having a null line for the broader class of asymptotically flat and globally hyperbolic spacetimes.
Dissertation Defense
Matthew Hyatt
University of Miami
Quasisymmetric Functions and Permutation Statistics for Coxeter Groups and Wreath Product Groups
Wednesday, July 20, 2011, 10:00am
Ungar Room 301
Dissertation Defense
Daniel P. Ryan
University of Miami
Fitness Dependent Dispersal in Intraguild Predation Communities
Tuesday, July 12, 2011, 10:00am
Ungar Room 301
Geometry and Physics Seminar
Yijia Liu
University of Miami
Cycles and Sheaves: A 45 Minute Intro
Wednesday, April 27, 2011, 4:00pm
Ungar Room 506
Combinatorics Seminar
Julian Moorehead
University of Miami
Partial Matroid Decomposition Posets
Monday, April 25, 2011, 5:00pm
Ungar Room 402
Abstract: The partition lattices are a fundamental class in the theory of posets, exhibiting an array of nontrivial properties. The study of q-analogues of these lattices has yielded large families of interesting posets, though often without some degree of "niceness". In this talk, we introduce a new q-analogue and some of its properties, including a generalization to a larger family of posets.
Geometry and Physics Seminar
Professor N. Saveliev
University of Miami
An Index Theorem for End-periodic Operators
Wednesday, April 13, 2011, 4:00pm
Ungar Room 506
Abstract: I will present a new index theorem which generalizes to manifolds with periodic ends the index theorem of Atiyah, Patodi and Singer. This is a joint project with Tomasz Mrowka and Daniel Ruberman.
Combinatorics Seminar
Jeremy Martin
University of Kansas
Critical Groups of Simplicial Complexes
Monday, April 11, 2011, 5:00pm
Ungar Room 402
Abstract: The critical group of a graph G is a finite abelian group K(G) whose order is the number of spanning trees of G. We generalize the definition of the critical group from graphs to simplicial complexes. Specifically, given a simplicial complex X of dimension d, we define a family of finite abelian groups K_0(X), ..., K_{d-1}(X) in terms of combinatorial Laplacian operators, generalizing the construction of K(G). We show how to compute the groups K_i(X) explicitly as cokernels of reduced Laplacians, and prove that they are finite, with orders given by weighted enumerators of simplicial spanning trees. We describe the groups completely for the cases that X is (a) a simplicial sphere or (b) a skeleton of a simplex; the latter result uses work of M. Maxwell. If time permits, I will talk about how to interpret the critical groups in terms of higher-dimensional analogues of flows in graphs, and/or another potential interpretation as discrete analogues of Chow groups. This is joint work with Art Duval and Carly Klivans.
Combinatorics Seminar
Michelle Wachs Galloway
University of Miami
Unimodality of q-Eulerian Numbers and p,q-Eulerian Numbers, Part II
Monday, April 4, 2011, 5:00pm
Ungar Room 402
Abstract: This talk is a continuation of a seminar talk I gave last fall. First I will review the previous talk, which focused on my work with Shareshian on unimodality of q-Eulerian polynomials, and then I will present my more recent work with Henderson on the cycle type refinement of the unimodality result.
The Eulerian numbers enumerate permutations in the symmetric group S_n by their number of excedances or by their number of descents. It is well known that they form a unimodal palindromic sequence of integers. In this talk, which is based on joint work with John Shareshian and Anthony Henderson, we consider the q-analog of the Eulerian numbers obtained by considering the joint distribution of the major index and the excedance number, and the p,q-analog of the Eulerian numbers obtained by considering the multivariate distribution of the major index, descent number and excedance number. We show that the q-Eulerian numbers form a unimodal palindromic sequence of polynomials in q and the p,q-Eulerian numbers refined by cycle type form a unimodal palindromic sequence of polynomials in p and q. The proofs of these results rely on the Eulerian quasisymmetric functions introduced by Shareshian and Wachs, on symmetric and quasisymmetric function theory, and on representation theory of the symmetric group.
Geometry and Physics Seminar
Professor L. Katzarkov
University of Miami
Braid Monodromy, Floer Combinatorics and Fukaya Category
Monday, April 4, 2011, 4:00pm
Ungar Room 411
Geometry and Physics Seminar
Professor C. Diemer
University of Miami
Combinatorics of Multiplier Ideal Sheaves III
Wednesday, March 30, 2011, 4:00pm
Ungar Room 506
Geometry and Physics Seminar
Professor C. Diemer
University of Miami
Combinatorics of Multiplier Ideal Sheaves II
Thursday, March 24, 2011, 4:00pm
Ungar Room 411
Geometry and Physics Seminar
Professor Fedor Bogomolov
New York University
Strong Form of the Grothendieck Section Conjecture in Functional Case
Wednesday, March 16, 2011, 4:00pm
Ungar Room 506
Abstract: In the talk I will give a proof of the Grothendieck section conjecture in the following form. Let $f : X\to Y$ be a surjective map of projective manifolds with an irreducible generic fiber and $f_a : G_a(X)\to G_a(Y)$ the corresponding map between pro-$l$-abelian Galois groups of the algebraic closures of the fields $k(X),k(Y)$ respectively, i.e. if we denote the Galois group $Gal(\bar k(X)/k(X))$ as $G_X$ then $G_a(X)= (G_X/[G_X,G_x])_l$ where $l$ stand for maximal pro-$l$- quotient and $l\neq k(X)$If there is a rational section $s: Y\to X$ then there are associated group sections $s^a : G_a(Y)\to G_a(X)$ (usually nonunique) with $f_a s_a = id$. The problem we are trying to solve is what conditions have to be imposed on $s^a$ so that it is associated to a rational section. It is clear that geometric section $s$ provides with a possiblity to lift the group section $s^a$ to the section of for $s^g : G_Y\to G_X$ for the surjective map of the Galois groups $f_g : G_X\to G_Y$. Since we are dealing with $l$-quotient only we will also consider geometric $p$-section. The latter correspond to the sections for induced maps $f^F :X^F\to Y^F$ where $Y^F$ is model of a purely inseparable extension of $k(Y)$ and $f_F,X^F$ are induced from $f,X$ by the map $Y^F\to Y$ (which is geometrically identical map).
Theorem: Assume that the ground field $k= \bar F_p,p\neq l$ and $dim Y \geq 2$. Let $ s^a : G_a(Y)\to G_a(X)$ be a group section which image is a closed subgroup with additional property: for any pair $x,y\in G_a(Y)$ such that preimages $\tiled x,\tilde y\in (G_Y/[[G_Y,G_Y] G_Y])_l$ commute the images $s^a(x),s_a(y)$ have the same property with respect to $(G_X/[[G_X,G_X] G_X])_l$. Then there is a rational $p$-section $ s : Y^F\to X^F$ some $Y^F$ such that $s^a$ is associated to $s$.
Note that since $G_Y/[[G_Y,G_Y] G_Y])_l$ is central extesnion of $(G_X/[[G_X,G_X])_l= G_a(Y)$ the property that $\tiled x,\tilde y$ commute in $G_Y/[[G_Y,G_Y] G_Y])_l$ does not depend on $x,y$.
The proof in general functional case with $k$-algebraically closed is similar but is technically more invloved and hence is not yet completed. The initial Grothedieck conjecture states a similar correspondence for a Galois group $s': G_Y\to G_X$ and in our approach we derive the result from minimal noncommutative quotients : $(G_Y/[[G_Y,G_Y] G_Y])_l$ and $(G_X/[[G_X,G_X] G_X])_l$. We hope that the result and the method ( after some modifications) can be extended to the case of arbitrary field $k$. and may be even to the case when $Y$ is a curve over arithmetic field.
It is a joint work with Yuri Tschinkel. In essence it is a corollary of the description of commuting pairs of elements in $G_Y/[[G_Y,G_Y] G_Y])_l$ which was obtained some time ago.
Geometry and Physics Seminar
Professor Neil Hoffman
University of Texas
Hidden Symmetries, Exceptional Surgeries, and Commensurability
Tuesday, March 9, 2011, 4:00pm
Ungar Room 506
Abstract: Two manifolds are in the same commensurability class if they share a common finite sheeted cover. Commensurability classes of hyperbolic 3-manifolds have infinitely many elements, so it is appealing to find types of manifolds that are rare in a commensurability class, eg knot complements. In 2006, Reid and Walsh conjectured that there are at most three hyperbolic knot complements in a given commensurability class. Recently, Boileau, Boyer, Cebanu, and Walsh announced that the conjecture holds in the case of no hidden symmetries. After providing some of the necessary background, I will talk about obstructions to knot complements admitting hidden symmetries.
Combinatorics Seminar
Professor Mark Skandera
Lehigh University
Path Tableaux and Combinatorial Interpretations for S_n-class Functions
Monday, March 7, 2011, 5:00pm
Ungar Room 402
Abstract: Around 1991, Goulden-Jackson, Greene, Haiman, Stanley, and Stembridge studied the evaluation of S_n class functions on generating functions in Z[S_n] which are products of Kazhdan-Lusztig basis elements. This led Stembridge to prove algebraically that irreducible S_n-characters evaluate nonnegatively on the Z[S_n] generating functions, and to conjecture that related "monomial virtual characters" have the same property. We point out that the analogous result for induced sign characters, which follows from the earlier Littlewood-Merris-Watkins identity, has a nice combinatorial interpretation. Using this interpretation, we combinatorially prove special cases of the Stembridge result and conjecture. We also conjecture a combinatorial interpretation for a known q-analog of the Littlewood-Merris-Watkins identity, and relate this to Haimans q-analogs of Stembridge's result and conjecture.
This is joint work with Brittany Shelton and Sam Clearman of Lehigh University.
Geometry and Physics Seminar
Professor C. Diemer
University of Miami
Combinatorics of Multiplier Ideal Sheaves
Wednesday, March 2, 2011, 4:00pm
Ungar Room 506
Geometry and Physics Seminar
George Lam
Duke University
The Riemannian Positive Mass and Penrose Inequalities for Graphs over R^n
Tuesday, March 1, 2011, 5:00pm
Ungar Room 402
Abstract: The Riemannian positive mass theorem asserts that an asymptotically flat Riemannian manifold M with nonnegative scalar curvature R has nonnegative ADM mass, and that the mass is strictly positive unless M is isometric to flat Euclidean space. If M contains an area outer minimizing horizon, the Riemannian Penrose inequality gives a positive lower bound to the ADM mass in terms of the area of the horizon. For manifolds that are graphs over R^n, we are able to prove stronger versions of the above inequalities by bounding the ADM mass from below with an integral of the product of R and a nonnegative potential function. I will give an overview of some previously known results before discussing our approach.
Geometry and Physics Seminar
Professor G. Kerr
University of Miami
Spectra as Cohomology Theory II
Wednesday, February 23, 2011, 3:30pm
Ungar Room 506
Geometry and Physics Seminar
Professor Ernesto Lupercio
Research and Advanced Studies Center of the National Polytechnic Institute of Mexico (Cinvestav - IPN)
Winner of the 2009 Srinivasa Ramanujan Prize
Orbifolds, Ghost Loop Spaces and Twisted Sectors
Wednesday, February 23, 2011, 2:00pm
Ungar Room 411
Geometry and Physics Seminar
Professor Ernesto Lupercio
Research and Advanced Studies Center of the National Polytechnic Institute of Mexico (Cinvestav - IPN)
Winner of the 2009 Srinivasa Ramanujan Prize
Non-compact Topological Field Theories and Frobenius Structures
Tuesday, February 22, 2011, 4:00pm
Ungar Room 411
Geometry and Physics Seminar
Professor Alex Iosevich
University of Rochester
Regular Value Theorem in a Fractal Setting
Monday, February 21, 2011, 5:00pm
Ungar Room 506
Abstract: The classical regular value theorem says that if $f: X \to Y$ is an immersion, where $X,Y$ are smooth manifolds of dimension $n,m$, $n>m$, respectively, then the set $\{x \in X: f(x)=y \}$ is either empty or is an $n-m$ dimensional sub-manifold of $X$. We shall see that a suitable analog of this result is available if a manifold $X$ is replaced by a set of sufficiently large Hausdorff dimension and the function $f$ satisfies a "rotational curvature" condition. Regularity of generalized Radon transforms plays a key role. Sharpness results are based on an interplay between ideas from discrete geometry and number theory.
Geometry and Physics Seminar
Dr. Todd Oliynyk
Monash University
Relativistic Fluids in 1+1 Dimensions with a Vacuum Boundary
Wednesday, February 16, 2011, 5:00pm
Ungar Room 402
Abstract: Relativistic isentropic fluids are characterized by their density, velocity, and pressure. The evolution of these fluids is governed by the relativistic Euler equations. In regions where the density is bounded away from zero, it is known how to write the Euler equations as a symmetric hyperbolic system. This allows for the use of standard theory to guarantee the well-posedness (i.e. local existence and uniqueness of solutions) of the Euler equations. However, fluids with compact support for which the pressure and density vanish simultaneously at the boundary between the fluid and the vacuum region, the known symmetric hyperbolic formulations of the Euler equations become degenerate at the vacuum boundary, and consequently, standard existence theory no-longer applies.
Until very recently, it was a long standing open problem to prove the existence of solutions to the Euler equations with a vacuum boundary that have non-zero fluid acceleration at the boundary. Physically, these type of solutions represent bodies such as stars that can be either static, expanding, or collapsing. In 2009, first in 1+1 spacetime dimensions and subsequently 3+1 dimensions, the existence of solutions to the non-relativistic Euler equations with non-zero acceleration at the fluid vacuum boundary was established by two different groups using non-standard energy estimates combined with suitable approximation techniques. The arguments used to establish existence are technical, involved, highly original, and quite different from one another.
In this talk, I will, after first providing a introduction to the relativistic and non-relativistic equations, describe the history of the problem and describe the major developments leading up the breakthrough existence results of 2009. I will also outline a new method for establishing the existence of solutions to relativistic Euler equations that have non-zero acceleration at the vacuum boundary. In contrast to the previous existence results, mine are rather straightforward, relying only on routine computation, some elementary geometry, and standard hyperbolic theory for initial boundary value problems, while, at the same time, producing very explicit representations of the solutions that are applicable to both the relativistic and non-relativistic settings.
Geometry and Physics Seminar
Dr. Martin Li
Stanford University
Free Boundary Problem for Embedded Minimal Surfaces
Tuesday, February 1, 2011, 5:00pm
Ungar Room 402
Abstract: For any smooth compact Riemannian 3-manifold with boundary, we prove that there always exists a smooth, embedded minimal surface with (possibly empty) free boundary. We also obtain a priori upper bound on the genus of such minimal surfaces in terms of the Heegard genus of the ambient compact 3-manifold. An interesting note is that no convexity assumption on the boundary is required. In this talk, we will describe the min-max construction for the free boundary problem, and then we will sketch a proof of the existence part of the theory.
Geometry and Physics Seminar
Professor G. Kerr
University of Miami
Spectra as Cohomology Theory
Wednesday, January 19, 2011, 4:00pm
Ungar Room 402
Combinatorics Seminar
Christian Stump
Centre de Recherches Mathématiques
Université de Montréal
and
Laboratoire de Combinatoire et d'Informatique Mathématique
Université du Québec à Montréal
Moon Polyominoes, Pipe Dreams and Simplicial Spheres
Monday, November 29, 2010, 5:00pm
Ungar Room 402
Abstract: We exhibit a canonical connection between maximal (0,1)-fillings of a moon polyomino avoiding north-east chains of a given length and reduced pipe dreams of a certain permutation. Following this approach we show that the simplicial complex of such maximal fillings is a vertex-decomposable, and thus shellable, sphere. In particular, this implies a positivity result for Schubert polynomials.
Applied Math Seminar
Samares Pal
Georgia Institute of Technology
and
University of Kalyani
Nutrient, Phytoplankton, Zooplankton Interaction in an Open Marine System –
A Mathematical Study
Friday, November 12, 2010, 4:00pm
Ungar Room 402
Abstract: Here we have considered a three-component model consisting of non-toxic phytoplankton (NTP), toxin producing phytoplankton (TPP) and zooplankton (Z), where the growth of zooplankton species reduce due to toxic chemicals released by phytoplankton species. We have taken into account the competition between TPP and NTP and tried to observe its effect on the marine ecosystem, both in the presence and absence of the environmental fluctuation. We observe that competition helps in the coexistence of the species, but if the effect of competition is very high on the TPP population, it results in the planktonic bloom.
Next we have proposed a three component model consisting of dissolved limiting nutrient (N) supplied at constant rate and partially recycled after the death of plankton by bacterial decomposition, phytoplankton (P) and zooplankton (Z), where the growth of zooplankton species reduce due to toxic chemicals released by phytoplankton species. Our analysis leads to different thresholds which are expressible in terms of model parameters and determine the existence and stability of various states of the system.
On combining the above two models we have studied a third one consisting of nutrient, non-toxic phytoplankton, toxin producing phytoplankton and their predator zooplankton population in open marine system. It is observed that nutrient- phytoplankton-zooplankton interactions are very complex and situation specific. Different exciting results, ranging from stable situation to cyclic blooms may occur under different favorable conditions, which may give some insights for predictive management.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
Inverse Spectra Problem in Algebraic Geometry
Tuesday, November 9, 2010, 5:00pm
Ungar Room 402
Applied Math Seminar
Guangyu Zhao
Department of Mathematics
University of Miami
Time Periodic Traveling Wave Solutions of Reaction-diffusion Systems
Friday, November 5, 2010, 4:00pm
Ungar Room 402
Abstract: The study of traveling wave solutions for parabolic equations and systems is an area of great interest, not only in the applications of the waves themselves but also in their use in gaining a better understanding of phenomena in large domains. Typically, traveling wave solutions arise from a competition between two equilibria and describe the transition processes that appear in many areas of biology, chemistry and physics. Over the past three decades, there have been many interesting studies on (stationary) traveling wave solutions to reaction-diffusion systems for which the corresponding kinetic systems are autonomous. In this presentation, I will talk about our recent work concerning time periodic traveling waves for a periodic Lotka-Volterra competition system with diffusion. I will focus on basic problems of wave existence, uniqueness of waves, and stability of waves.
Combinatorics Seminar
Michelle Wachs Galloway
University of Miami
Unimodality of q-Eulerian Numbers and p,q-Eulerian Numbers
Monday, November 1, 2010, 5:00pm
Ungar Room 402
Abstract: The Eulerian numbers enumerate permutations in the symmetric group S_n by their number of excedances or by their number of descents. It is well known that they form a unimodal palindromic sequence of integers. In this talk, which is based on joint work with John Shareshian and Anthony Henderson, we consider the q-analog of the Eulerian numbers obtained by considering the joint distribution of the major index and the excedance number, and the p,q-analog of the Eulerian numbers obtained by considering the multivariate distribution of the major index, descent number and excedance number. We show that the q-Eulerian numbers form a unimodal palindromic sequence of polynomials in q and the p,q-Eulerian numbers refined by cycle type form a unimodal palindromic sequence of polynomials in p and q. The proofs of these results rely on the Eulerian quasisymmetric functions introduced by Shareshian and Wachs, on symmetric and quasisymmetric function theory, and on representation theory of the symmetric group.
Applied Math Seminar
Chris Cosner
Department of Mathematics
University of Miami
Reaction-diffusion-advection Models for the Effects and Evolution of Dispersal
Friday, October 29, 2010, 4:00pm
Ungar Room 402
Abstract: The dispersal of organisms is an important ecological process that can often be described mathematically in terms of diffusion and advection. The dispersal strategy that a species uses can affect its population dynamics and interactions with other species, and those in turn can impose selective pressure on dispersal strategies. Reaction-diffusion-advection models can be used to study the effects and evolution of dispersal strategies. One way to compare dispersal strategies is to construct and analyze models for competing populations that are the same in all ecological respects except their dispersal strategies. In the context of reaction-diffusion-advection models for dispersal in environments that are variable in space but constant in time this approach suggests that the effects of a given dispersal strategy depend on how well it allows a population to match its spatial distribution to the distribution of its resources. A way to understand which dispersal strategies are most likely to evolve is to study the models from the viewpoint of evolutionary stability. (A strategy is evolutionarily stable relative to a given class of strategies if a population using it cannot be invaded by any small population using any other strategy in the class.) There is evidence that evolution favors strategies that let a population match its resources perfectly. This talk will review a number of results and open questions related to those ideas.
Combinatorics Seminar
Matthew Hyatt
University of Miami
Double Feature
Monday, October 25, 2010, 5:00pm
Ungar Room 506
Abstract: The first part of the talk will be a review of the connection between representation theory and symmetric functions. In the second part we consider a colored analog of Eulerian quasisymmetric functions. Eulerian quasisymmetric functions were introduced by Shareshian and Wachs in order to obtain a $q$-analog of Euler's exponential generating function formula for the Eulerian numbers. They are defined via the symmetric group, and applying the stable and nonstable principal specializations yields formulas for joint distributions of permutation statistics. We consider the group of colored permutations and use this group to introduce colored Eulerian quasisymmetric functions. We derive a formula for the generating function of these colored Eulerian quasisymmetric functions, and show that applying the stable and nonstable principal specializations yields formulas for joint distributions of colored permutation statistics, generalizing Euler's formula.
Geometry and Physics Seminar
Professor B. de Oliveira
University of Miami
Structure Theorem for Symmetric Differentials of Rank 1
Wednesday, October 20, 2010, 4:00pm
Ungar Room 402
Abstract: The presence of holomorphic 1-forms on a compact kahler manifold $X$ implies topological properties of $X$. Moreover, from their presence also follows the existence of a holomorphic map from $X$ into a complex torus from which all the holomorphic 1-forms of $X$ are induced from. The talk gives a complete extension of this result to symmetric differentials of rank 1. This result belongs to the program whose aim is to understand the class of symmetric differentials that have a close to topological nature (symmetric differentials of rank 1 will be shown to be closed symmetric differentials).
Combinatorics Seminar
Brant Jones
James Madison University
An Explicit Derivation of the Möbius Function for Bruhat Order
Monday, October 18, 2010, 5:00pm
Ungar Room 506
Abstract: We give an explicit non-recursive complete matching for the Hasse diagram of the strong Bruhat order of any interval in any Coxeter group. This yields a new derivation of the Möbius function, recovering a classical result due to Verma. The matching is given in terms of combinatorial objects called masks that arise in Deodhar's formula for the Kazhdan–Lusztig polynomials, and has connections to Armstrong's sorting order on Coxeter groups.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
Spectra, Gaps and Chow Groups
Friday, October 15, 2010, 4:00pm
Ungar Room 506
Geometry and Physics Seminar
Professor P. Miao
University of Miami
On a Localized Riemannian Penrose Inequality
Wednesday, October 13, 2010, 4:00pm
Ungar Room 402
Abstract: Given a compact, orientable, three dimensional Riemannian manifold with boundary, we call it "a body surrounding horizons" if its boundary is the disjoint union of two pieces: the outer boundary and the horizon boundary, where the outer boundary is a topological 2-sphere and the horizon boundary is the unique minimal surface in the manifold. Such a manifold can be thought as a bounded region, surrounding the outermost apparent horizons of black holes, in a time-symmetric slice of a space-time in the context of general relativity. Physically, one expects that there exists a geometric quantity computed from the area and the mean curvature of the outer boundary that can be estimated from below by the area of the horizon boundary. In the case that the manifold is non-compact whose outer boundary is replaced by an asymptotically flat end, such an expectation then leads to the Riemannian Penrose Inequality. In this talk, we establish an inequality of this type on a body surrounding horizons whose outer boundary is metrically a round sphere. Its potential role in suggesting the right concept of quasi-local mass will also be discussed.
Combinatorics Seminar
Anton Dochtermann
Stanford University
Cellular Resolutions of Hypergraph Edge Ideals
Monday, October 4, 2010, 5:00pm
Ungar Room 506
Abstract: Given an ideal I in the polynomial ring S = k[x_1,...,x_n], a basic problem in commutative algebra is to describe a (minimal) free resolution of I. One particularly geometric method is through the construction of a 'cellular resolution', where the syzygies of I are encoded by the faces of a polyhedral (or more general CW) complex. If our ideal is square-free and generated in a fixed degree d, then its generators can be thought of as the edges of a (hyper)graph G on n vertices; this defines the edge ideal I_G. In this talk we construct a polyhedral complex X_G whose vertices encode the directed edges of G. Using basic methods of combinatorial topology we show that X_G supports a minimal cellular resolution of I_G whenever G is what we call 'cointerval'. This class of graphs corresponds to the complement of interval graphs (when d = 2), and in general strictly contains the class of hypergraphs corresponding to pure shifted complexes whose resolutions were described by Nagel and Reiner. Furthermore, the complexes X_G admit natural embeddings into certain 'mixed subdivisions' of dilated simplices, allowing us to draw some nice pictures. This is joint work with Alex Engstrom (UC Berkeley).
Combinatorics Seminar
Anatoly Libgober
University of Miami
Geometry of Arrangements of Hyperplanes and Cohomology of Orlik-Solomon Algebras
Monday, September 27, 2010, 5:00pm
Ungar Room 506
Abstract: The Orlik-Solomon algebra is a combinatorial invariant of an arrangement of hyperplanes. Part of its multiplicative structure is encoded in an invariant called the resonance variety, which has unexpected linearity properties. Calculation of resonance varieties leads to questions about non-linear geometry involving arrangements. I will survey general properties of Orlik-Solomon algebras and more recent results and open problems on resonance varities.
Applied Math Seminar
Shigui Ruan
Department of Mathematics
University of Miami
Modeling the Transmission Dynamics and Control of Hepatitis B Virus
Friday, September 24, 2010, 4:00pm
Ungar Room 402
Abstract: Hepatitis B is a potentially life-threatening liver infection caused by the hepatitis B virus (HBV) and is a major global health problem. HBV is the most common serious viral infection and a leading cause of death in mainland China. Around 130 million people in China are carriers of HBV, almost a third of the people infected with HBV worldwide and about 10% of the general population in the country; among them 30 million are chronically infected. Every year, 300,000 people die from HBV-related diseases in China, accounting for 40-50% of HBV-related deaths worldwide. Despite an effective vaccination program for newborn babies since the 1990s, which has reduced chronic HBV infection in children, the incidence of hepatitis B is still increasing in China. Based on the HBV data from China, we first propose an ordinary differential equation model to describe the transmission dynamics and prevalence of HBV infection. The model provides an approximate estimate of the basic reproduction number is 2.406 in China which indicates that hepatitis B is endemic in China and is approaching its equilibrium with the current immunization program and control measures. Taking the fact that age structure is one of the characteristics of HBV transmission, we also propose an age-structured model. By determining the basic reproduction number, we study the existence and stability of the disease-free and endemic steady state solutions of the model and explore optimal strategies for controlling the transmission of HBV.
Combinatorics Seminar
Drew Armstrong
University of Miami
The Ish Arrangement of Hyperplanes
Monday, September 20, 2010, 5:00pm
Ungar Room 506
Abstract: The Shi arrangement of hyperplanes plays an important role in the representation theory of affine Weyl groups. In type A, this arrangement is Shi(n)=\{x_i-x_j=0, x_i-x_j=1 : 1\leq i<j\leq n\}. The arrangement Shi(n) divides \R^n into (n+1)^{n-1} regions --- an interesting number, yes? --- and it has beautiful combinatorics. In this talk I will define a new hyperplane arrangement Ish(n), which I call the Ish arrangement. You will like this hyperplane arrangement. (Some of this is joint work with Brendon Rhoades.)
Geometry and Physics Seminar
Professor C. Diemer
University of Pennsylvania
Tropical Geometry, Compactifications, and Birational Geometry
Wednesday, August 25, 2010, 4:30pm
Ungar Room 402
Abstract: Tropical geometry is a collection of methods which replace algebro-geometric objects with certain polyhedral complexes and aims to give combinatorial interpretations of constructions in algebraic geometry. In this talk I'll give a (friendly) survey of the foundations of tropical geometry and some of its applications. Particular attention will be given to the "geometric tropicalization" approach of Hacking, Keel, and Tevelev which relates tropical geometry to the boundary structure of compactifications of varieties, and in turn gives combinatorial manifestations of some constructions in (log) birational geometry.
Geometry and Physics Seminar
Professor A. Libgober
University of Miami
Alexander Invariants of Fundamental Groups of the Complements to Plane Algebraic Curves
Tuesday, August 24, 2010, 4:00pm
Ungar Room 402
Abstract: This will be an introductory talk to the role and properties of Alexander polynomials and their generalizations in the context of Algebraic geometry. I also will discuss main problems and conjectures in this theory.
Combinatorics Seminar
Dr. Martina Kubitzke
Reykjavik University
Triangulations, the Associahedron and Gamma-vectors for Planar Lattices
Wednesday, July 7, 2010, 4:00pm
Ungar Room 506
Abstract: The talk is divided into two parts. In the first part we consider posets given as the product of two chains $C_k \times C_{n-k}$. We construct a special reverse lexicographic triangulation of the order polytope of $C_2 \times C_{n-2}$ which is abstractly isomorphic to the join of a simplex with the associahedron. It remains open if there is a meaningful generalization of this result to general k. In the second part of the talk we focus on Gal's conjecture in the special setup of planar lattices. It was already shown by Bränden that Gal's conjecture holds for $C_2\times C_{n-2}$. Being the lattice of order ideals of this poset planar or equivalently being the poset of width 2 we ask if the conjecture is true in this greater generality. We are able to answer this question in the affirmative and give some hints how one could proceed for posets of width at least 3. This is joint work with Kathrin Vorwerk.
Geometry and Physics Seminar
L. Katzarkov
University of Miami
How Not to Learn Minimal Model Program
Thursday, April 29, 2010, 4:00pm
Ungar Room 402
Geometry and Physics Seminar
Professor Leonid Parnovski
University College London
Integrated Density of States of Schröedinger Operators with Periodic and Almost-periodic Potentials
Wednesday, April 14, 2010, 4:00pm
Ungar Room 402
Abstract: I will discuss new results (joint with R. Shterenberg) on the asymptotic behaviour of the integrated density of states of a Schrödinger operator $H=-\Delta+b$ acting in $\R^d$ when the potential $b$ is either smooth periodic, or generic quasi-periodic (finite linear combination of exponentials), or belongs to a wide class of almost-periodic functions.
Combinatorics Seminar
Jay Schweig
University of Kansas
On Lattice Path Matroids and Polymatroids
Friday, April 9, 2010, 5:00pm
Ungar Room 506
Abstract: Lattice path matroids are an especially tractable class of transversal matroids whose bases are in correspondence with planar lattice paths. We discuss some enumerative properties of these matroids, one of which leads naturally to a related class of discrete polymatroids. We then examine these polymatroids and their toric ideals. Finally, we provide generating sets and Gröbner bases for these ideals, and discuss many possible directions for future research. No previous knowledge of matroid theory or toric ideals will be assumed.
Geometry and Physics Seminar
Professor Gordon Heier
University of Houston
On Uniformly Effective Boundedness of Shafarevich Conjecture-type
Wednesday, April 7, 2010, 4:00pm
Ungar Room 402
Abstract: The talk deals with uniformly effective versions of the classical Shafarevich Conjecture over function fields (aka Parshin-Arakelov Theorem). We will discuss the speaker's effective solution to the classical case and his recent extension to the case where the fibers are canonically polarized compact complex manifolds. In the proofs, Chow varieties play a key role.
Combinatorics Seminar
Julian Moorehead
University of Miami
On q-analogs of the k-equal Partition Lattice
Tuesday, April 6, 2010, 5:00pm
Ungar Room 506
Abstract: The ordinary k-equal partition lattice served as the original motivating example for Björner and Wachs to extend the notion of lexicographic shellability of posets from the pure case to more general nonpure cases. In this talk, we discuss the construction of a family of q-analogs to this lattice and a common edge labeling which indicates that each is a shellable poset. We also describe methods for counting falling chains in these lattices, as well as conjectures for certain special cases which greatly improve the computational time necessary to determine the total number of chains.
Geometry and Physics Seminar
Professor Shulim Kaliman
University of Miami
On the Present State of the Andersen-Lempert Theory
Wednesday, March 24, 2010, 4:00pm
Ungar Room 402
Abstract: We discuss a theory of completely integrable algebraic (resp. holomorphic) vector fields on smooth affine algebraic varieties.
Combinatorics Seminar
Brendon Rhoades
Massachusetts Institute of Technology
Cyclic Sieving and Polygon Multidissection Enumeration
Tuesday, March 23, 2010, 5:00pm
Ungar Room 506
Abstract: Let X be a finite set, C = \langle c \rangle be a finite cyclic group acting on X, \zeta be a root of unity of multiplicative order |C|, and X(q) \in \mathbb{Z}[q] be a polynomial with integer coefficients. Following Reiner, Stanton, and White, we say the triple (X, C, X(q)) exhibits the cyclic sieving phenomenon (CSP) if for any d \geq 0, the fixed point set cardinality |X^{c^d}| equals the polynomial evaluation X(\zeta^d). We prove a collection of CSPs related to the action of rotation on multidissections of polygons, i.e., dissections where edges can occur with multiplicity and boundary edges may or may not be included. Our proofs involve modelling the action of rotation via general linear group representations and use geometric realizations of finite type cluster algebras due to Fomin and Zelevinsky.
Geometry and Physics Seminar
Professor Fedor Bogomolov
New York University
Galois Groups and Birational Invariants of Functional Fields
Wednesday, March 17, 2010, 4:00pm
Ungar Room 402
Abstract: I want to discuss our joint results with Yuri Tschinkel Bloch-Kato conjecture implies that any element in tht cohomology of algebraic variety with finite coefficients after restriction to some open subariety can be induced from abelian quotient of the fundamantal group of the latter. Our theorem on the structure of the Galois groups of functional fields implies a similar result for nonramified cohomology. Namely for any element $a$ of nonramified cohomology $H^i_{nr}(Gal(\bar K/ K, Z_{l^n}, i\geq 2, K=\bar F_p(X), p\neq l)$ there is a finite topological quotient $G^c$ of $Gal(\bar K/ K)$ such that $a$ is induced from a nonramified element $b$ of $H^i_{nr}(G^c, Z_{l^n}$. Here $G^c$ is a finite group which is a central extension of an abelian group. It has the following geometric interpretation: there exists a rational map $f :X \to \prod P^i/ A$ and a nonramified element $b\in H^i_{nr}(G^c, Z_{l^n}$ such that $f^*(b)= a$.
Combinatorics Seminar
Matthew Hyatt
University of Miami
Signed Eulerian Quasisymmetric Functions
Tuesday, March 9, 2010, 5:00pm
Ungar Room 506
Abstract: We introduce signed Eulerian quasisymmetric functions, which are an extension of the Eulerian quasisymmetric functions introduced by Shareshian and Wachs. We define them via the hyperoctahedral group, or group of signed permutations, and we compute their generating function. A central part of this computation is a so called tri-colored necklace bijection, which is an extension of the bi-colored necklaces appearing the work of Shareshian and Wachs, which is in turn an extension of techniques introduced by Gessel and Reutenauer. By applying certain ring homomorphisms to our formula for the generating function, we obtain results for the joint distribution of certain signed permutation statistics. Some of these results are new, although one is a special case of a joint distribution previously computed by Foata and Han, but here an alternate proof is given.
Combinatorics Seminar
Rafael S. Gonzalez D'Leon
University of Miami
On the Half-plane Property and the Tutte-group of a Matroid
Tuesday, March 2, 2010, 5:00pm
Ungar Room 506
Abstract: A matroid has the weak half-plane property (WHPP) if there exists a stable polynomial with support equal to the set of bases of the matroid. If the polynomial can be chosen with all nonzero coefficients equal to one then the matroid has the half-plane property (HPP). We describe a systematic method that allows us to reduce the WHPP to the HPP for large families of matroids. This method makes use of the Tutte-group of a matroid. We prove that no projective geometry has the WHPP and that a binary matroid has the WHPP if and only if it is regular.
Geometry and Physics Seminar
Jesse Johnson
Oklahoma State University
Axiomatic Thin Position and Applications
Thursday, February 18, 2010, 4:00pm
Ungar Room 402
Abstract: The notion of "thin position" has been a powerful tool for understanding surfaces in 3-manifolds and knot complement. However, it has been defined and applied in a number of different ways that are related more in spirit than in details. I will describe an axiomatic framework that allows one to define exactly what is meant by thin position, and which leads to a toolbox of methods that can be used in a number of different settings.
Combinatorics Seminar
Benjamin Braun
University of Kentucky
Nowhere-Harmonic Colorings of Graphs
Wednesday, February 17, 2010, 5:00pm
Ungar Room 506
Abstract: Proper vertex colorings of a graph are related to its boundary map, also called its signed vertex-edge incidence matrix. The vertex Laplacian of a graph, a natural extension of the boundary map, leads us to introduce nowhere-harmonic colorings and analogues of the chromatic polynomial and Stanley's theorem relating negative evaluations of the chromatic polynomial to acyclic orientations. Further, we discuss some examples demonstrating that nowhere-harmonic colorings are more complicated from an enumerative perspective than proper colorings. Our primary tool for these investigations is the theory of "inside-out polytopes," developed by M. Beck and T. Zaslavsky, and the theory of Ehrhart quasi-polynomials for rational polytopes. This is joint work with Matthias Beck of San Francisco State University.
Geometry and Physics Seminar
Professor I. Hambleton
McMaster University
Conjugation Spaces and 4-manifolds
Wednesday, February 17, 2010, 4:00pm
Ungar Room 402
Abstract: A conjugation space is a space X with involution, where the cohomology mod 2 of the fixed set is the same as the cohomology of the space after doubling dimensions. The first example is X = complex projective space, with the involution given by complex conjugation. In the talk I will describe the relation between smooth conjugation 4-manifolds and knotted surfaces in mod 2 homology 4-spheres. This is joint work with Jean-Claude Hausmann.
Geometry and Physics Seminar
Professor I. Itenberg
Université de Strasbourg
On Real Determinantal Quartics
Wednesday, February 10, 2010, 4:00pm
Ungar Room 402
Abstract: Let A_0, A_1, A_2, and A_3 be real symmetric 4 x 4 matrices. One can associate to these four matrices a spectral surface in the three dimensional complex projective space CP^3 (the set of points (x_0 : x_1 : x_2 : x_3) in CP^3 such that the determinant of the matrix x_0 A_1 + x_1 A_1 + x_2 A_2 + x_3 A_3 is zero) and a spectrahedron in the three dimensional real projective space RP^3 (the set of points (x_0 : x_1 : x_2 : x_3) in RP^3 such that the matrix x_0 A_1 + x_1 A_1 + x_2 A_2 + x_3 A_3 is semidefinite).
In general, the spectral surface considered has 10 double points. We show that the boundary of the spectrahedron cannot contain more than 8 doubles points of the spectral surface. The proof is based on a study of period spaces of real K3-surfaces.
Combinatorics Seminar
Michelle Wachs
University of Miami
Eulerian Quasisymmetric Functions and Cyclic Sieving
Tuesday, February 9, 2010, 5:00pm
Ungar Room 506
Abstract: Certain q-analogs of classical combinatorial numbers exhibit the curious phenomenon of evaluating to a positive integer when q is set equal to an nth root of unity. A stronger phenomenon called the cyclic sieving phenomenon (of Reiner, Stanton and White) is exhibited when these positive integers can be interpreted as the number of fixed points of an element of a cyclic group acting on a set whose size is equal to the classical combinatorial number.
In this talk I will present an instance of the cyclic sieving phenomenon involving a q-analog of the Eulerian numbers and their cycle type refinement. The main tool in proving this result is the Eulerian quasisymmetric functions introduced a few years ago in joint work with Shareshian.
This is joint work with Bruce Sagan and John Shareshian.
Geometry and Physics Seminar
Dima
University of Miami
Motives
Thursday, February 4, 2010, 2:00pm
Ungar Room 547
Combinatorics Seminar
Drew Armstrong
University of Miami
Reduced Decompositions of Permutations
Tuesday, February 2, 2010, 5:00pm
Ungar Room 506
Abstract: Consider the group of permutations of {1,2,\ldots, n}, which is generated as a Coxeter group by the adjacent transpositions (i,i+1). The reduced S-decompositions for a permutation \pi are the ways of writing \pi as a product of the fewest adjacent transpositions. A nice result gives a bijection from the (essentially different) reduced S-decompositions of the longest permutation to *rhombic tilings of a regular 2n-gon*.
We will describe an analogous result for the reduced T-decompositions of a permutation (using all transpositions, not just the adjacent ones). We will give a bijection from the (essentially different) reduced T-decompositions of the long cycle to *quadrangulations of a regular 2n-gon*.
We will note some striking similarities between these two results.
Geometry and Physics Seminar
Alexandr Usnich
Ludmil Katzarkov
Dima
University of Miami
On the Work of Lurie
Tuesday, February 2, 2010, 4:00pm
Ungar Room 547
Geometry and Physics Seminar
Ludmil Katzarkov
University of Miami
Hodge Structures and Spectra
Tuesday, February 2, 2010, 2:00pm
Ungar Room 547
Geometry and Physics Seminar
Alexandr Usnich
University of Miami
A infty Categories
Monday, February 1, 2010, 11:00am
Ungar Room 547
NSF-CSMS Project Industry-Liaison Seminar
Michael Goldberg
President and CEO
Flamingo Software
A Career in Software Development, Web-Based Systems
Wednesday, January 27, 2010, 5:00pm
Ungar Room 402
Abstract: Over 40 years ago, Michael Goldberg started a Miami software company called FDP (Financial Data Planning), employing many UM mathematics and computer science students over the years and eventually becoming the leading provider of software for the insurance and pension industries. After selling FDP, Michael started another company 5 years ago called Flamingo Software, specializing in web-based systems for insurance and financial services companies. Find out what a career in software development can be like and what it takes to run a successful software company.
Geometry and Physics Seminar
Professor A. Dvorsky
University of Miami
Realizations of Minimal Representation of O(p,q)
Wednesday, December 9, 2009, 3:00pm
Ungar Room 402
Abstract: The "smallest" unitary representation of a non-compact simple Lie group is a surprisingly rich object with many interesting analytic properties. Unlike the metaplectic representation of Sp(2n), which was studied extensively, the minimal representation for the orthogonal groups O(p,q) has not been analyzed in comparable detail. We will discuss the explicit models for this representation, constructed recently by Kobayashi and Orsted, and introduce some applications of these models.
Geometry and Physics Seminar
Jeremy Van Horn-Morris
American Institute of Mathematics
Symplectic Fillings of Contact Manifolds
Friday, December 4, 2009, 3:30pm
Ungar Room 402
Abstract: Work of Loi and Piergalini as well as Akbulut and Ozbagci allows us to create symplectic fillings of contact 3-manifolds from factorizations of the monodromy of a compatible open book decomposition. Recent results of Wendl complete this correspondence: every symplectic filling comes from such a factorization. We'll explain this correspondence in more detail and give some example applications including the rational blowdown operation and the uniqueness of symplectic fillings of certain Lens spaces. Some of this work is joint with Tom Mark and Hasaaki Endo and some is joint with Olga Plamenevskaya.
Geometry and Physics Seminar
Professor L. Katzarkov
University of Miami
Spectra of Categories and Applications to Low Dimensional Topology
Tuesday, December 1, 2009, 5:00pm
Ungar Room 402
Abstract: In this talk we will define the notion of spectrum of category and will compute it on the example of some Fukaya categories. Other applications will be discussed.
Applied Math Seminar
Dr. Orou Gaoue
ITME Post Doc
Department of Biology
University of Miami
Modeling the Impact of Non-timber Forest Product Harvest in Variable Environments
Friday, November 20, 2009, 4:30pm
Ungar Room 402
Abstract: Harvesting wild plants for non-timber forest products is an important source of income, food and medicine for millions of people around the world. Over-exploitation of these plant resources may lead to species extinction and impair their availability for future use by people who depend on them for their livelihoods. Yet, our knowledge of the way harvesting some non-timber forest products may affect population dynamics is still limited. I will use the case study of Khaya senegalensis (Meliaceae) foliage and bark harvest by indigenous Fulani people in Africa, to demonstrate that harvesting reduces population growth rate even further if environmental conditions vary stochastically. I will show how using harvest-specific elasticity analysis provides in-depth understanding of the management options that are available to mitigate the negative effects of harvest at the population level. I suggest that the temporal sequence of harvest intensity matters when modeling the impact of wild plant harvest.
Geometry and Physics Seminar
Professor N. Saveliev
University of Miami
Seiberg-Witten Equations and End-periodic Dirac Operators
Wednesday, November 18, 2009, 4:00pm
Ungar Room 402
Abstract: Let X be a smooth spin 4-manifold with homology of S^1 x S^3. In our joint project with Tom Mrowka and Daniel Ruberman, we study the Seiberg-Witten equations on X. The count of their solutions, called the Seiberg-Witten invariant of X, depends on choices of Riemannian metric and perturbation. A similar dependency issue is resolved in dimension 3 by relating the jumps in the Seiberg-Witten invariant to the spectral flow of the Dirac operator; the resulting invariant is then the Casson invariant. In dimension 4, we use Taubes' theory of end-periodic operators to relate the jumps in the Seiberg-Witten invariant to the index theory of the Dirac operator on a manifold with periodic end modeled on the infinite cyclic cover of X. The resulting invariant is then a smooth invariant of X whose reduction is the Rohlin invariant. Some calculations and applications of this invariant will be discussed.
Applied Math Seminar
Professor Gaetano Zampieri
Dipartimento di Informatica
Universita di Verona
A Class of Integrable Hamiltonian Systems and Weak Lyapunov Stability
Friday, November 13, 2009, 4:30pm
Ungar Room 402
Abstract: The aim of the talk is to introduce a class of Hamiltonian autonomous systems which are completely integrable and their dynamics is described in all details. In particular we show explicit examples of Hamiltonian systems with an unstable equilibrium where the eigenvalues of the linearization are imaginary and no motion is asymptotic to the equilibrium in the past, namely no solution has the equilibrium as limit point as time goes to minus infinity.
Combinatorics Seminar
Professor Eric Gottlieb
Rhodes College
A Combinatorial Optimization Problem from Genomics
Friday, November 13, 2009, 3:00pm
Ungar Room 506
Abstract: Biologists often wish to locate the gene controlling for a specific feature in a given species. One approach is to use recombinant inbred lines (RILs) from that species. RILs are homozygous, with genetic material alternating between a parent having the trait in question and a parent not having the trait. The break points in the genetic contributions from the parents occur at different points in different RILs. Biologists typically select a (usually large) subset of the RILs that visually appears to have sufficiently varied break points to ensure that the location of the controlling gene can be resolved by comparing which RILs have the trait with the parental contribution at each gene.
Unfortunately, this subjective approach does not guarantee the ability of the selected subset to resolve the gene location as well as the full set of RILs. In addition, the experiments that must be performed to determine whether a given RIL has the trait in question can be intensive with respect to time, money, and laboratory space. For this reason, it is desirable to minimize the size of the set of RILs selected for analysis. The typical approach makes little or no effort to select a smallest set.
We describe a Mathematica program we have written to find sets of RILs that are as small as possible subject to the constraint of being able to resolve the location of any gene the full set of RILs can resolve.
This is joint work with Jonathan Fitz Gerald, Department of Biology, Rhodes College.
Applied Math Seminar
Professor Neil Johnson
Department of Physics
University of Miami
Insurgent Wars, Pandemics, Global Emissions and Market Crises:
One Model Fits All?
Friday, November 6, 2009, 4:30pm
Ungar Room 402
Abstract: For complex real-world problems, it seems that there are (at least) as many models in the literature as there are researchers in the field. In this seminar, I will attempt the opposite approach: One model, stretched in various directions, to encompass four major issues. The model is a coalescence-fragmentation model in which clusters are continually playing the 'El Farol' bar attendance game. In certain limits, analytic solutions are obtainable which seem to capture the stylized statistical facts of each of these problems. Generalizations of the model, and their implications in each real-world scenario, are discussed.
Geometry and Physics Seminar
Professor Bruno de Oliveira
University of Miami
Closed Symmetric Differentials of Degree 2 and the Geometry of Complex Surfaces
Wednesday, November 4, 2009, 4:00pm
Ungar Room 402
Abstract: It is well understood how holomorphic differential p-forms reflect the topology of a given complex manifold. On the other hand, little is known about the relationship between the topology and the algebra of holomorphic symmetric differentials of complex manifolds. In this talk we will give results about the impact of the presence of closed symmetric 2-differentials on the topology and geometry of complex surfaces.
