Colloquia

Ugur G. Abdulla
Okinawa Institute of Science and Technology

Removability of the Fundamental Singularity for the Heat Equation and its Measure-theoretical, Topological and Probabilistic Consequences

Wednesday, November 20, 2024, 5:30pm
Ungar Room 402

Abstract: The major problem in the Analysis of PDEs is understanding the nature of singularities of solutions to the PDEs reflecting the natural phenomena. In this talk, I will present new Wiener-type criteria for the removability of the fundamental singularity for the parabolic PDEs. The criteria characterize the uniqueness of boundary value problems with singular data, reveal the nature of the parabolic measure of the singularity point, asymptotic laws for the conditional Markov processes, and criteria for thinness in minimal-fine topology. The talk will be oriented to a general audience including non-expert faculty and graduate students.

Jane Heffernan
York University, Toronto, Canada

Multi-Scale Modelling: Immunity

Tuesday, November 5, 2024, 5:00pm
Ungar Room 402

Abstract: Immunity is both presented as a characteristic of an individual, and of a population. Immunity is generated in individuals through vaccination and/or infection, and it is quantified in populations to determine If herd immunity (or a specific level of wanted immunity in the population) can be achieved or maintained. In this talk we will provide an overview of models of immunity at both scales. Mathematical models of antibody and memory cell generation, maintenance, and waning (decay) in-host will be introduced as well as epidemiological models with detailed immunity distributions. The dynamics of COVID-19, measles and influenza immunity generation and decay from vaccination and infection in individuals and populations will be highlighted.

Yasuhuro Takeuchi
Aoyama Gakuin University, Japan

Bifurcation Analysis in a Tumor-Immune System Interaction Model

Thursday, September 26, 2024, 5:00pm
Ungar Room 402

Abstract: We study a tumor-immune system interaction model with dendritic cell therapy and immune response delay. Our results reveal that the delay can destabilize the stable positive equilibrium through Hopf bifurcation. Furthermore, the direction and stability of Hopf bifurcation are derived. When there is a cusp, we show that it is a BT singularity for any delay and the delay model also undergoes BT bifurcation. Finally, numerical simulations are presented to illustrate the theoretical results.

References:

1. Dynamical behavior of combinational immune boost against tumor, M. Nagata, Y. Furuta, Y. Takeuchi and S. Nakaoka, Japan J. Ind. App. Math., Vol.32, No.3, 2015, 759-770.
2. Mathematical modeling and analysis of combinational immune boost for tumor elimination, N. Nakada, M. Nagata, Y. Takeuchi, and S. Nakaoka, Citation: AIP Conference Proceedings 1723, 2016, 020002.
3. Bifurcation analysis in a tumor-immune system interaction model with dendritic cell therapy and immune response delay, Y. Zhang, L. Xie, Y. Dong, J. Huang, S. Ruan, and Y. Takeuchi, SIAM Journal on Applied Mathematics, SIAM J Appl. Math., Vol. 83, No.5, 2023, 1892-1914.

Ruobing Zhang
Princeton University

Metric Geometry of Einstein Manifolds and Generalizations

Monday, May 6, 2024, 5:00pm
Ungar Room 402

Abstract: This lecture concerns the metric Riemannian geometry of Einstein manifolds. We will exhibit the rich geometric/topological structures of Einstein manifolds and specifically focus on the structure theory of moduli spaces of Einstein metrics. My recent works center around the intriguing problems regarding the compactification of the moduli space of Einstein metrics, which tells us how Einstein manifolds can degenerate. We will introduce recent major progress and propose several open questions in the field.

Rafal Komendarczyk
Tulane University

Linking and Knotting in Volume Preserving Flows

Thursday, March 28, 2024, 5:00pm
Ungar Room 402

Abstract: The topology of orbits in volume-preserving flows is of special interest in the areas of fluid dynamics and plasma physics. The well-known helicity invariant, which measures an asymptotic average linking number of the orbits of the flow, provides an estimate for the 𝐿²-energy of the field. Invariants of flows, derived from various knot and link invariants, are of special interest and have been studied extensively by researchers. In this talk, I will present an overview of the subject and its connections to other areas of low-dimensional topology and geometry.

Philip Hanlon
Distinguished Presidential Scholar, University of Miami
President Emeritus, Dartmouth College

The Amazing Eulerian Idempotents

Monday, March 18, 2024, 5:00pm
Ungar Room 402

Abstract: The Eulerian idempotents, sitting inside the group algebra of the symmetric group, are a set of pairwise orthogonal idempotents that sum to the identity. They are remarkable for the broad range of contexts within which they arise, from homological algebra to card shuffling to chromatic polynomials of graphs. In this lecture, we will introduce the Eulerian idempotents, discuss some of their algebraic properties and survey some of their many applications.

Lisa Fauci
Pendergraft Nola Lee Haynes Professor, Tulane University
2023 Inductee to the National Academy of Sciences

Modeling Choices in the Biofluids of Locomotion:
Two Case Studies

Friday, March 8, 2024, 5:00pm
Ungar Room 402

Abstract: At the macroscale, the undulatory swimming gait of a fish emerges from the coupling of neural signaling, passive body properties and muscle mechanics with the surrounding fluid environment. At the microscale, unicellular and colonial protists use flagella to swim and to create a water current transporting bacterial prey for capture. These are two examples of systems in biological fluid dynamics that we have modeled in collaboration with experimentalists. We will discuss the choices made in model development and the computational frameworks used. In particular, will discuss coordinated laboratory experiments and simulations that investigate the swimming and feeding performance of individual choanoflagellates carrying bacterial prey, as well the hydrodynamic performance of colonies of hundreds of these cells. At the macroscale, we will present a closed-loop model that examines the hypothesis that amplified body-sensing feedback could restore effective locomotion in lampreys with spinal injuries.

This talk is based on joint works with F. Delarue, M. Shkolnikov, X. Zhang and Y. Guo.

Sergey Nadtochiy
Illinois Institute of Technology

Physical Solution to Stefan Equation

Friday, March 1, 2024, 3:00pm
Ungar Room 402

Abstract: This talk is concerned with the probabilistic methods for solving Stefan free-boundary PDEs. The latter equations appear in many models of fundamental physical and biological processes, such as: phase transition (i.e., melting/freezing), phase segregation (e.g., aging of alloys), crystal growth, interaction among neurons, etc. Despite their importance, to date, there exists no general existence and uniqueness theory for such equations due to the potential singularity of the solutions. Recently, the probabilistic methods, based on the analysis of associated mean-field particle systems and McKean-Vlasov equations, were successfully used to tackle the mathematical challenges that could not be addressed by the classical analytic methods, yielding the notion a physical solution of Stefan equation and its well-posedness. I will present an overview of the existing results, including the recently obtained proof of well-posedness of Stefan equation with surface tension under radial symmetry.

This talk is based on joint works with F. Delarue, M. Shkolnikov, X. Zhang and Y. Guo.

Eran Nevo
Einstein Institute of Mathematics
Hebrew University Jerusalem

Rigidity, Expansion and Polytopes

Thursday, February 29, 2024, 5:00pm
Ungar Room 402

Abstract: Given a graph G and an embedding of its vertices in R^d, what continuous motions of the vertices preserve all edge lengths? Clearly all motions induced by an isometry of R^d do, these are the trivial motions; are there any others? If the answer is NO for all (equivalently, for one) generic embedding, G is called d-rigid. What are the d-rigid graphs? This problem has been extensively studied since the 70s, and is still widely open for d≥3. It is studied mainly from algebraic geometry and combinatorial points of view. Variants of it, especially in dimensions 2 and 3, are of importance also beyond mathematics, e.g. in structural engineering, computational biology and more.

I will focus on a quantitative version of rigidity via spectral analysis of the related stiffness matrix, including the construction of "rigidity expanders", generalizing expander graphs. Higher dimensional notions of rigidity and stiffness matrices, and their relation to the study of polytopes, will be addressed too.

Lionel Levine
Cornell University

Math for AI Safety

Monday, February 19, 2024, 5:00pm
Ungar Room 402

Abstract: AI holds great promise and, many believe, great peril. What can mathematicians contribute to ensuring that promise is fulfilled, and peril avoided? I'll highlight research directions relevant to AI safety in many different fields of math. I'll also touch on the question of how educators can empower students to succeed in a world pervaded by AI, and to shape that world for the better: After all, many of today's math majors will be tomorrow's AI engineers. This talk will be interactive, and I'll mostly supply questions rather than answers, so please come prepared to debate!

Karola Mészáros
Cornell University

Log-concavity of the Alexander Polynomial

Friday, February 16, 2024, 5:00pm
Ungar Room 402

Abstract: The central question of knot theory is that of distinguishing links up to isotopy. The first polynomial invariant of links devised to help answer this question was the Alexander polynomial (1928). Almost a century after its introduction, it still presents us with tantalizing questions, such as Fox's conjecture (1962) that the absolute values of the coefficients of the Alexander polynomial Δ_L(t) of an alternating link L are unimodal. Fox's conjecture remains open in general with special cases settled by Hartley (1979) for two-bridge knots, by Murasugi (1985) for a family of alternating algebraic links, and by Ozsváth and Szabó (2003) for the case of genus 2 alternating knots, among others. We settle Fox's conjecture for special alternating links. We do so by proving that a certain multivariate generalization of the Alexander polynomial of special alternating links is Lorentzian. As a consequence, we obtain that the absolute values of the coefficients of Δ_L(t), where L is a special alternating link, form a log-concave sequence with no internal zeros. In particular, they are unimodal. This talk is based on joint work with Elena Hafner and Alexander Vidinas.

Benjamin Braun
University of Kentucky

Ehrhart Theory and Lattice Simplices

Thursday, November 9, 2023, 5:00pm
Ungar Room 402

Abstract: The Ehrhart h*-polynomial of a lattice polytope is an important and well-studied invariant in geometric and algebraic combinatorics. However, many fundamental questions regarding these polynomials remain unanswered. For lattice simplices, the h*-polynomial reflects both arithmetic and geometric structure, and these objects form a surprisingly rich family of polytopes from an Ehrhart-theoretic perspective. In this talk, I will provide a survey of examples, theorems, and questions regarding both h*-polynomials and local h*-polynomials for lattice simplices.

Marcelo Disconzi
Vanderbilt University

General-relativistic Viscous Fluids

Thursday, October 19, 2023, 5:00pm
Ungar Room 402

Abstract: In this talk, we will review some recent developments at the intersection of mathematics and physics regarding theories of relativistic fluids with viscosity. The discovery of the quark-gluon plasma that forms in heavy-ion collision experiments provides a unique opportunity to study properties of matter under extreme conditions, as the quark-gluon plasma is the hottest, smallest, and densest fluid known to humanity. Studying the quark-gluon plasma also provides a window into the earliest moments of the universe, since microseconds after the Big Bang the universe was filled with matter in the form of the quark-gluon plasma.

For more than two decades, the community has intensely studied the quark-gluon plasma with the help of a rich interaction of experiments, theory, phenomenology, and numerical simulations. From these investigations, a coherent picture has emerged, indicating that the quark-gluon plasma behaves essentially like a relativistic liquid with viscosity.

More recently, state-of-the-art numerical simulations strongly suggested that viscous and dissipative effects can also have non-negligible effects on gravitational waves produced by binary neutron star mergers. But despite the importance of viscous effects for the study of such systems, a robust and mathematically sound theory of relativistic fluids with viscosity is still lacking. This is due, in part, to difficulties in preserving causality upon the inclusion of viscous and dissipative effects into theories of relativistic fluids.

In this talk, we will survey the history of the problem and report on a new approach to relativistic viscous fluids that addresses these issues. This talk is intended for a general Math and Physics audience.

Juhi Jang
University of Southern California

Dynamics of Gaseous Stars

Wednesday, October 11, 2023, 5:00pm
Ungar Room 402

Abstract: In astrophysical fluid dynamics, stars are considered as isolated fluid masses subject to self-gravity. A classical model of a self-gravitating Newtonian star is given by the gravitational Euler- Poisson system, while a relativistic star is modeled by the Einstein-Euler system. In the talk, I will review some recent progress on the local and global dynamics of Newtonian star solutions, and discuss mathematical construction of self-similar gravitational collapse of Newtonian stars including Larson-Penston solution for the isothermal stars, Yahil solution for polytropic stars, which show the existence of smooth initial data that lead to finite time collapse, characterized by the blow-up of the star density. If time permits, I will also discuss the relativistic analogue of Larson-Penston solutions and formation of naked singularities for the Einstein-Euler system.

Igor Pak
University of California, Los Angeles

Combinatorial Inequalities

Monday, May 1, 2023, 5:00pm
Ungar Room 402

Abstract: In the ocean of combinatorial inequalities, two islands are especially difficult. First, Mason's conjectures say that the number of forests in a graph with k edges is log-concave. More generally, the number of independent sets of size k in a matroid is log-concave. These results were established just recently, in a remarkable series of papers by Huh and others, inspired by algebro-geometric considerations.

Second, Stanley's inequality for the numbers of linear extensions of a poset with value k at a given poset element, is log-concave. This was originally conjectured by Chung, Fishburn and Graham, and famously proved by Stanley in 1981 using the Alexandrov–Fenchel inequalities in convex geometry.

In the first part of the talk we will survey these and other combinatorial inequalities. I will then briefly discuss our new framework of combinatorial atlas which allows one to give elementary proofs of the two results above, and extend them in several directions. The talk is aimed at the general audience.

Christos Mantoulidis
Rice University

Generic Regularity of Minimizing Hypersurfaces in Dimensions Up to 10

Thursday, April 27, 2023, 5:00pm
Ungar Room 402

Abstract: A classical problem in geometric analysis and the calculus of variations is to find least-area submanifolds spanning a prescribed boundary. Deep results in geometric measure theory developed between the 1950s and the 1970s guarantee that for every smooth compact (n-2)-dimensional submanifold L in R^n there exists a smooth compact hypersurface-with-boundary that spans L with least area – provided n <= 7. When n >= 8, some boundaries L do not bound smooth least-area hypersurfaces. In this talk, I will discuss recent joint work with O. Chodosh and F. Schulze where we prove that the non-existence of smooth minimizers is Baire non-generic if n <= 10. This improves upon the analogous 1980's result for n=8 by R. Hardt and L. Simon.

Jianhong Wu
York University, Canada

Global and Generic Dynamics of Nonlinear Systems with Embedded Predator-Prey Feedback Cycles

Tuesday, April 25, 2023, 5:00pm
Ungar Room 402

Abstract: Many nonlinear systems with embedded predator-prey interactions have the order-preserving monotonicity with respect to a rank-k cone (with k>1). We show such systems enjoy the limit-set trichotomy properties (including potential heteroclinic cycles), and the restricted semiflow on any given omega-set can be characterized by a Lipschitz-continuous flow on the k-dimensional Euclidean space. In the case where k=2, we can also obtain the generic Poincare-Bendixson theorem, and this can be used to establish the generic periodicity of high-dimensional epidemic systems with non-standard nonlinear instance functions. The talk is based on a series of studies in collaboration with Lirui Feng and Yi Wang, some of these results have been reported in SIAM J. Math. Anal. (2017), J. Differential equations (2021) and SIAM J. Applied Math. (2022), and many more to come.

Yanyan Li
Rutgers University

Symmetry of Hypersurfaces and the Hopf Lemma

Tuesday, April 18, 2023, 5:00pm
Ungar Room 402

Abstract: A classical theorem of A.D. Alexandrov says that a connected compact smooth hypersurface in Euclidean space with constant mean curvature must be a sphere. We give exposition to results on symmetry properties of hypersurfaces with ordered mean curvature and associated variations of the Hopf Lemma. Some open problems will be discussed.

Dr. Bryna Kra
Northwestern University

Infinite Patterns in Large Sets of Integers

Friday, March 31, 2023, 5:00pm
Ungar Room 402

Abstract: Resolving a conjecture of Erdős and Turan from the 1930s, in the 1970s Szemerédi showed that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Soon thereafter, Furstenberg used Ergodic Theory to give a new proof of this result, leading to the development of combinatorial ergodic theory. These tools have led to uncovering new patterns that occur in any sufficiently large set of integers, but until recently all such patterns have been finite. Based on joint work with Joel Moreira, Florian Richter, and Donald Robertson, we discuss recent developments for infinite patterns, including the resolution of conjectures of Erdős.

Inkyung Ahn
Korea University, Korea

Population Models with Fokker-Planck-type Diffusions Incorporating Perceptual Constraints of Species in a Habitat with Spatial Heterogeneity

Tuesday, February 14, 2023, 5:00pm
Ungar 528B

Abstract: In this talk, we examine the predator-prey models in a spatially heterogeneous region where the dispersal of predators is non-uniform, a process known as prey-induced dispersal (PYID). We incorporate the perceptual constraints of species in our model and examine how PYID affects the dynamics and coexistence of the system. Our analysis is based on a Holling-type II functional response model under no-flux boundary conditions. We analyze the local stability of the semi-trivial solution in models with PID and linear dispersal in the absence of predators. Furthermore, we investigate the local and global bifurcation from the semi-trivial solution in models with two different dispersals. Our results show that if a predator's satisfaction with the prey density is higher than a certain level, it may not be beneficial in terms of their fitness. Meanwhile, if predators change their motility when they are appropriately satisfied with the amount of prey, they will obtain a survival advantage. Additionally, if time allows, we discuss competition models with starvation-driven diffusions (SDD) and perceptual constraints of species.

Dr. René Mboro
University of Miami/IMSA

On Rationality Questions for Cubic Hypersurfaces

Wednesday, May 4, 2022, 5:00pm
Ungar Room 402

Abstract: Over the past two centuries, rationality questions and especially rationality of complex cubic hypersurfaces have been one of the driving topic of the development of algebraic geometry. In this talk, we will give an overview of the state of our knowledge about the geometry of cubic hypersurfaces of dimension up to 5.

Dr. Alan Hastings
University of California at Davis

Transient Dynamics:
The Key to Ecological Understanding

Friday, April 15, 2022, 5:00pm
Ungar Room 402

Abstract: Much of classical ecological theory is focused on the long-term behavior of ecological models yet the time scales of ecological dynamics are such that a focus on asymptotic behavior is likely misguided. Ecological conclusions would likely change, but additionally much mathematical theory focuses on asymptotic behavior as well. I will begin with some of my much older work that suggests the importance of transients and some of the challenges. I will then focus on more recent work, most of which has been done with a wonderful group of colleagues from a working group that began at NIMBioS and has continued. We have given a rough classification of features that produce transients similar to approaches for understanding dynamical systems, examined implications for management, and examined transients in systems where stochasticity is important.

Dr. Alexander Volberg
Michigan State University
Hausdorff Center for Mathematics, Bonn University

Enflo's Problem and Beyond

Friday, March 25, 2022, 5:00pm
Ungar Room 402

Abstract: Harmonic analysis on Hamming cube became prominent because of its connections with theoretical computer science and, in particular, with quantum algorithms. It turns out that it is an important tool in solving some old conjectures.

Enflo's problem deals with Poincaré type inequality for functions of Rademacher random variables. The problem was solved recently by P. Ivanisvili, R. Van Handel and myself. However, this type of Poincaré inequality is only one out of the whole scale of inequalities filling the gap between Pisier's inequality and singular integral inequalities on Hamming cube. It is important and interesting a) to understand what are the sharp constants in these results b) what is the stock of Banach spaces X such that the results hold for functions of Rademacher random variables with values in X. I will explain this and the relations with the Ribe program and Bourgain's discretization theorem.

Dr. Stefan Czimek
Brown University

The Characteristic Gluing Problem of General Relativity

Monday, December 13, 2021, 11:30am
Ungar Room 402

Abstract: The Einstein equations describe the dynamics of space-time in general relativity. It is well-known that – analogous to the case of Maxwell's equations – initial data for the Einstein equations needs to satisfy constraint equations. One approach to study the rigidity and flexibility of the Einstein equations is by considering gluing problems for initial data. The so-called spacelike gluing problem for initial data on slices of constant time has been intensively studied by Riemannian geometers. In this talk I will present recent work with S. Aretakis and I. Rodnianski, where we introduce the so-called characteristic gluing problem for initial data along light cones (i.e., characteristic hypersurfaces for the Einstein equations). The characteristic gluing problem is fundamentally different from the spacelike problem and displays novel rigidity and flexibility features. We moreover show how to apply our characteristic gluing to prove gluing constructions for spacelike initial data. Towards the end of the talk, I will discuss future directions.

Dr. Zhongshan An
University of Connecticut

Geometric Boundary Conditions in General Relativity

Thursday, December 9, 2021, 5:00pm
Ungar Room 402

Abstract: In general relativity, spacetime metrics satisfy the Einstein equations, which are a system of nonlinear partial differential equations. When the spacetime has nonempty boundary, it is natural to ask what geometric boundary conditions are well-posed for the Einstein equations. The study of geometric boundary conditions not only gives rise to interesting geometric PDE problems, but also plays an important role in the study of quasi-local mass of compact spacetimes in general relativity.

In this talk we will discuss geometric boundary conditions for the vacuum Einstein equations as a hyperbolic system, and also boundary conditions for the stationary vacuum Einstein equations as an elliptic system. Then we will talk about applications of these geometric boundary value problems in the construction of quasi-local mass in general relativity.

Dr. Da Rong (Daren) Cheng
University of Waterloo

Existence of Constant Mean Curvature Surfaces

Monday, December 6, 2021, 5:00pm
Ungar Room 402

Abstract: Constant mean curvature (CMC) surfaces are two-dimensional analogues of curves with constant geodesic curvature. They originated with the work of Young, Laplace and Gauss on capillary surfaces, and have since appeared in models of many other physical phenomena. Geometrically, CMC surfaces are critical points of the area functional subject to a volume constraint. The simplicity of this variational characterization and the variety of contexts in which CMC surfaces arise makes it interesting to establish a satisfactory existence theory, the pursuit of which has been a major impetus for the development of the calculus of variations. In this talk, I'll discuss some classical results and recent developments on the variational construction of CMC surfaces.

Dr. Demetre Kazaras
Duke University

How Does Mass Affect the Geometry of Space?

Monday, November 29, 2021, 5:00pm
Ungar Room 402

Abstract: In General Relativity, the ADM mass of an isolated gravitational system is a geometric invariant measuring the total mass due to matter and other fields present in spacetime. The celebrated Positive Mass Theorem of Schoen-Yau and Witten states that this invariant is non-negative and vanishes only for flat spacetime. In recent work, we showed how to compute ADM mass in 3 spatial dimensions by studying harmonic functions. Now we will use this formula to consider the following question: How flat is an "asymptotically flat" space with very little total mass? The existence of wormholes and gravity wells make this question subtle. We make progress on this problem and partially confirm conjectures made by Huisken-Ilmanen and Lee-Sormani.

Dr. Anastasiia Tsvietkova
Rutgers University, Newark

NP-hard Problems Naturally Arising in Knot Theory

Monday, March 29, 2021, 5:00pm
Online

Abstract: Low-dimensional topology and geometry has many problems with an easy formulation, but a hard solution. Despite our intuitive feeling that these problems are "hard", lower or upper bounds on algorithmic complexity are known only for some of them. Recently, several problems that lie at the heart of classical knot theory were shown to be NP-hard, which means it is unlikely that they could be solved in polynomial time. We will discuss some of such problems: unlinking and splitting by crossing changes; link equivalence by Reidemeister moves; detecting sublinks with various properties.

We are starting Zoom at 4:45 pm to give those faculty unable to schedule a one-on-one with the speaker time to meet and greet prior to the talk.

Dr. Anastasiia Tsvietkova
Rutgers University, Newark

Intrinsic Geometry, Topology and Complexity of 3-manifolds

Friday, February 19, 2021, 5:00pm
Online

Abstract: Every 3-manifold can be canonically decomposed into pieces, and each piece has a certain geometric structure (Geometrization Theorem). Thus, on a global scale, one can match topological information for the manifold with the respective geometry. However, on a local scale, i.e. intrinsically, the interplay between geometric, topological, and combinatorial properties of a 3-manifold often is not well understood. In this talk, we will focus on one topic in this framework: how the number of surfaces embedded in a 3-manifold is related to the hyperbolic volume and combinatorial "complexity" (number of crossings or number of tetrahedra) of a 3-manifold.

While results of this nature are of interest in low-dimensional geometry and topology, the obtained insight canhelpto establish connections with other areas of mathematics. Among these areas are quantum topology, differential geometry, representation theory, number theory and computational complexity theory. If the time allows, we will briefly discuss some of these connections.

Dr. Ying Wang
University of Oklahoma

An Underground Oil Recovery Model

Monday, February 15, 2021, 5:00pm
Online

Abstract: In this talk, I will discuss an underground oil recovery model whichincludes a third-order mixed derivatives term resulting from the dynamic effects in the pressure difference between the two phases. Analytic study on the computational domain reduction will be provided. A variety of numerical examples will be given. They show that the solutions may have many different saturation profiles depending on the initial conditions, diffusion parameter, and the third-order mixed derivatives parameter. The results are consistent with the study of traveling wave solutions and their bifurcation diagrams.

Ernesto Lupercio
Center for Research and Advanced Studies of the National Polytechnic Institute (Cinvestav-IPN)

An Introductory Mini Course into Quantum Toric Geometry: Lecture II

Joint IMSA & ICMS HSE Event
Friday, August 28, 2020, 9:00am
Online

Abstract: We will introduce the foundations of Quantum Toric Geometry as developed by Katzarkov, Lupercio, Meersseman and Verjovsky, Quantum toric geometry is a generalization of toric geometry where irrational fans correspond to non-commutative spaces called quantum toric varieties. As non-commutative spaces, Quantum toric varieties are to usual toric varieties what the Quantum torus is to the usual torus.

Lecture II
     1. LVM theory
     2. Quantum GIT
     3. Moduli spaces of toric varieties

References: It is useful to be familiar with toric varieties (for example the book of Fulton).

Artan Sheshmani
Harvard University/IMSA

Atiyah Class and Sheaf Counting on Local Calabi-Yau 4 Folds

Joint IMSA & ICMS HSE Event
Friday, August 28, 2020, 8:00am
Online

Abstract: We discuss Donaldson-Thomas (DT) invariants of torsion sheaves with 2 dimensional support on a smooth projective surface in an ambient non-compact Calabi Yau fourfold given by the total space of a rank 2 bundle on the surface. We prove that in certain cases, when the rank 2 bundle is chosen appropriately, the universal truncated Atiyah class of these codimension 2 sheaves reduces to one, defined over the moduli space of such sheaves realized as torsion codimension 1 sheaves in a noncompact divisor (threefold) embedded in the ambient fourfold. Such reduction property of universal Atiyah class enables us to relate our fourfold DT theory to a reduced DT theory of a threefold and subsequently then to the moduli spaces of sheaves on the base surface. We finally make predictions about modularity of such fourfold invariants when the base surface is an elliptic K3.

Tony Yue YU
Laboratoire de Mathématiques d'Orsay

Frobenius Structure Conjecture and Moduli of Calabi-Yau Pairs

Joint IMSA & ICMS HSE Event
Friday, August 28, 2020, 7:00am
Online

Abstract: I will explain the Frobenius structure conjecture of Gross-Hacking-Keel in mirror symmetry, and an application towards the moduli space of Calabi-Yau pairs. I will show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a simple way, a mirror family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form.

The structure constants of the algebra are constructed via counting non-archimedean analytic disks in the analytification of U. Furthermore, I will introduce a generalization of the Gelfand-Kapranov-Zelevinsky secondary fan, and show that the mirror family admits a natural compactification and extension over the toric variety associated to the secondary fan, which generalizes the families previously studied by Kapranov-Sturmfels-Zelevinsky and Alexeev in the toric case.

We conjecture that this gives rise to a (nearly uni) versal family of polarized Calabi-Yau pairs (embedded in the moduli space of KSBA stable pairs), and has a surprising consequence that such moduli space is unirational. We prove the stability in dimension two. This is based on arXiv:1908.09861 joint with S. Keel, and arXiv:2008.02299 joint with Hacking and Keel.

Ernesto Lupercio
Center for Research and Advanced Studies of the National Polytechnic Institute (Cinvestav-IPN)

An Introductory Mini Course into Quantum Toric Geometry: Lecture I

Joint IMSA & ICMS HSE Event
Thursday, August 27, 2020, 9:00am
Online

Abstract: We will introduce the foundations of Quantum Toric Geometry as developed by Katzarkov, Lupercio, Meersseman and Verjovsky, Quantum toric geometry is a generalization of toric geometry where irrational fans correspond to non-commutative spaces called quantum toric varieties. As non-commutative spaces, Quantum toric varieties are to usual toric varieties what the Quantum torus is to the usual torus.

Lecture I
     1. Introduction
     2. Stacks and non-commutative spaces
     3. The quantum torus
     4. Quantum toric varieties

References: It is useful to be familiar with toric varieties (for example the book of Fulton).

Artan Sheshmani
Harvard University/IMSA

Stable Higher Rank Flag Sheaves on Surfaces

Joint IMSA & ICMS HSE Event
Thursday, August 27, 2020, 8:00am
Online

Abstract: We study moduli space of holomorphic triples f: E_{1}–>E_{2}, composed of (possibly rank > 1) torsion-free sheaves (E_{1}, E_{2}) and a holomorphic map between them, over a smooth complex projective surface S. The triples are equipped with Schmitt stability condition. We prove that when Schmitt stability parameter becomes sufficiently large, the moduli space of triples benefits from having a perfect relative and absolute obstruction theory in some cases (depending on rank of holomorphic torsion-free sheaf E_{1}).

We further generalize our construction to higher-length flags of higher rank sheaves by gluing triple moduli spaces, and extend the earlier work, with Gholampur and Yau, where the obstruction theory of nested Hilbert schemes over the surface was studied. Here we extend the earlier results to the moduli space of flags E_{1}––>E_{2}—>...––>E_{n}, where the maps are injective (by stability). There is a connection, by wallcrossing, between the theory of such higher rank flags, and the theory of Higgs pairs on the surface, which provides the means to relate the flag invariants to the local DT invariants of threefold given by a line bundle over the surface, X :=Tot(L –> S).

Tony Yue YU
Laboratoire de Mathématiques d'Orsay

Frobenius Structure Conjecture and Moduli of Calabi-Yau Pairs

Joint IMSA & ICMS HSE Event
Thursday, August 27, 2020, 7:00am
Online

Abstract: I will explain the Frobenius structure conjecture of Gross-Hacking-Keel in mirror symmetry, and an application towards the moduli space of Calabi-Yau pairs. I will show that the naive counts of rational curves in an affine log Calabi-Yau variety U, containing an open algebraic torus, determine in a simple way, a mirror family of log Calabi-Yau varieties, as the spectrum of a commutative associative algebra equipped with a multilinear form.

The structure constants of the algebra are constructed via counting non-archimedean analytic disks in the analytification of U. Furthermore, I will introduce a generalization of the Gelfand-Kapranov-Zelevinsky secondary fan, and show that the mirror family admits a natural compactification and extension over the toric variety associated to the secondary fan, which generalizes the families previously studied by Kapranov-Sturmfels-Zelevinsky and Alexeev in the toric case.

We conjecture that this gives rise to a (nearly uni) versal family of polarized Calabi-Yau pairs (embedded in the moduli space of KSBA stable pairs), and has a surprising consequence that such moduli space is unirational. We prove the stability in dimension two. This is based on arXiv:1908.09861 joint with S. Keel, and arXiv:2008.02299 joint with Hacking and Keel.

Sergey Fomin
Univerity of Michigan

Morsifications and Mutations

Friday, March 1, 2019, 5:00pm
Ungar Room 402

Abstract: I will discuss a new and somewhat mysterious connection between singularity theory and cluster algebras, more specifically between the topology of isolated singularities of plane curves and the mutation equivalence of quivers associated with their morsifications. This is joint work with Pavlo Pylyavskyy, Eugenii Shustin, and Dylan Thurston.

Michael Larsen
Indiana University

The Circle Method in Algebraic Geometry

Monday, May 7, 2018, 5:00pm
Ungar Room 402

Abstract: Certain counting problems in group theory can be formulated either in terms of varieties over finite fields, or (dually) in terms of irreducible character values. By comparing the two points of view, one can either use geometry to give character estimates or (what I will mostly talk about) character estimates to prove theorems in geometry.

Ayelet Lindenstrauss
Indiana University

Cohomology Theories and Topological Hochschild Homology

Monday, May 7, 2018, 3:30pm
Ungar Room 402

Abstract: Cohomology theories assign to every space a graded abelian group, satisfying appropriate axioms. Particularly nice ones have a product, that is: they assign to every space a graded ring. It turns out that such theories can be described in terms of ring spectra. Modern constructions let us define actual products that are both associative and unital on the ring spectra that correspond to multiplicative cohomology theories, which is harder than one would think due to some "flabbiness" in the definition of the ring spectra.  Using these products, though, we can look at the ring spectra of discrete rings (corresponding to the usual cohomology of a space with coefficients in that ring) and replicate homological algebra constructions on the discrete rings with topological constructions on the corresponding ring spectra. The integers are an initial object in the category of discrete unital rings; in the category of ring spectra, that role is played by the sphere spectrum. So even if we are only interested in understanding discrete rings, doing homological algebra with their ring spectra over the sphere spectrum turns out to give new and interesting constructions.

Topological Hochschild homology is the ring spectrum version of Hochschild homology, with tensor products taken over the sphere spectrum rather than over the integers. It is a finer invariant of discrete rings, and the Dennis trace map from algebraic K-theory to Hochschild homology can be applied to ring spectra and thus factors through Topological Hochschild homology. I will discuss the topological Hochschild homology of rings including number rings (joint with Ib Madsen) and maximal orders in simple algebras over the rationals (joint with Henry Chan).

Professor Takayuki Hibi
Osaka University

Reflexive Polytopes

Monday, April 30, 2018, 5:00pm
Ungar Room 402

Abstract: A lattice polytope P of dimension d in the d-dimensional euclidean space is called reflexive if the origin is contained in the interior of P and if the dual polytope of P is again a lattice polytope. For example, the triangle in the euclidean plane with the vertices (-1,-1), (-1,2) and (2,-1) is reflexive. It turns out that reflexive polytopes play an important role in various areas of mathematics. One of the questions in combinatorics is how to construct reflexive polytopes in natural ways. In my talk, in the frame of Gröbner bases, a technique to yield reflexive polytopes will be discussed. No special knowledge will be required to understand my talk.

James McKernan
University of California, San Diego

Symmetries of Polynomials

Thursday, April 26, 2018, 5:00pm
Ungar Room 402

Abstract: Symmetries of polynomials are closely connected to the geometry of the variety of zeroes of the polynomials. Varieties come in three types and all three types have very different symmetry groups. We review some results, both old and new, which place bounds on the size of the symmetry group.

Dr. Boris Botvinnik
University of Oregon

Conformal Geometry and Topology of Manifolds

Thursday, April 19, 2018, 5:00pm
Ungar Room 402

Abstract: I will start with basics on conformal geometry, we will discuss the Einstein-Hilbert functional and Yamabe problem. Then I plan to discuss the problem of existence of metrics with positive scalar curvature for simply connected spin manifolds. At the end I would like to describe some recent results on the space of metrics with positive scalar curvature.

Mary Ann Horn
Professor and Chair, Department of Mathematics
Applied Mathematics and Statistics
Case Western Reserve University

Using Mathematical Modeling to Understand the Role of Diacylglycerol (DAG) as a Second Messenger

Thursday, April 12, 2018, 5:00pm
Ungar Room 402

Abstract: Diacylgylcerol (DAG) plays a key role in cellular signaling as a second messenger. In particular, it regulates a variety of cellular processes and the breakdown of the signaling pathway that involves DAG contributes to the development of a variety of diseases, including cancer. A mathematical model of the G-protein signaling pathway in RAW 264.7 macrophages downstream of P2Y6 activation by the ubiquitous signaling nucleotide uridine 5'-diphosphate is presented. The primary goal is to better understand the role of diacylglycerol in the signaling pathway and the underlying biological dynamics that cannot always be easily measured experimentally. The model is based on time-course measurements of P2Y6 surface receptors, inositol trisphosphate, cytosolic calcium, and with a particular focus on differential dynamics of multiple species of diacylglycerol. When using the canonical representation, the model predicted that key interactions were missing from the current pathway structure. Indeed, the model suggested that to accurately depict experimental observations, an additional branch to the signaling pathway was needed, whereby an intracellular pool of diacylglycerol is immediately phosphorylated upon stimulation of an extracellular receptor for uridine 5'-diphosphate and subsequently used to aid replenishment of phosphatidylinositol. As a result of sensitivity analysis of the model parameters, key predictions can be made regarding which of these parameters are the most sensitive to perturbations and are therefore most responsible for output uncertainty. (Joint work with Hannah Callender, University of Portland, and the H. Alex Brown Lab, Vanderbilt.)

Iosif Polterovich
Professor, Universite de Montreal
Canada Research Chair in Geometry and Spectral Theory

Sloshing, Steklov, and Corners

Thursday, April 5, 2018, 5:00pm
Ungar Room 402

Abstract: The sloshing problem is a Steklov type eigenvalue problem describing small oscillations of an ideal fluid. We will give an overview of some latest advances in the study of Steklov and sloshing spectral asymptotics, highlighting the effects arising from corners, which appear naturally in the context of sloshing. In particular, we will discuss the proofs of the conjectures posed by Fox and Kuttler back in 1983 on the asymptotics of sloshing frequencies in two dimensions. We will also outline an approach towards obtaining sharp asymptotics for Steklov eigenvalues on polygons. The talk is based on a joint work with M. Levitin, L. Parnovski and D. Sher.

Professor Fu Liu
University of California, Davis

Ehrhart Positivity

Tuesday, April 3, 2018, 5:00pm
Ungar Room 402

Abstract: The Ehrhart polynomial counts the number of lattice points inside dilation of an integral polytope, that is, a polytope whose vertices are lattice points. We say a polytope is Ehrhart positive if its Ehrhart polynomial has positive coefficients. In the literature, different families of polytopes have been shown to be Ehrhart positive using different techniques. We will survey these results in the first part of the talk, after giving a brief introduction to polytopes and Ehrhart polynomials.

Through work of Danilov/McMullen, there is an interpretation of Ehrhart coefficients relating to the normalized volumes of faces. In the second part of the talk, I will discuss joint work with Castillo in which we try to make this relation more explicit in the case of regular permutohedra. The motivation is to prove Ehrhart positivity for generalized permutohedra. If time permits, I will also discuss some other related questions.

Robert Bryant
Phillip Griffiths Professor of Mathematics
Duke University

On Self-Dual Curves

Friday, March 2, 2018, 5:00pm
Fieldhouse at the Watsco Center

Abstract: An algebraic curve in the projective plane (or, more generally in a higher dimensional projective space) is said to be 'self-dual' if it is projectively equivalent to its dual curve (after, possibly, an automorphism of the curve). Familiar examples are the nonsingular conics (or, more generally, rational normal curves in higher dimensions) and the 'binomial curves' y^a = x^b, but there are many more such curves, even in the plane.

I'll survey some of the literature on these curves, particularly in the plane and 3-space, and some of what is known about their classification and moduli, including their connection with contact curves in certain contact 3-folds, some of which are singular. I'll also provide what appear to be some new examples of these curves.

Michael Li
Applied Math Institute
University of Alberta

Nonidentifiability Issue in Parameter Estimation of Differential Equation Models

Thursday, February 22, 2018, 2:00pm
Ungar Room 402

Abstract: Transmission models for infectious diseases using differential equations are frequently confronted with disease data. For instance, these type of models are used to analyze HIV surveillance data, for the assessment of HIV epidemics and estimation of true burden of HIV in terms of incidence, prevalence and the size of undiagnosed HIV positive population. Parameter estimation when fitting the model to data is a key step of the modelling approach, and can often be complicated by the the presence of nonidentifiable parameters. Nonidentifiability results in multiple, often infinitely many, parameter values for which the model fit the data equally well, while different choices of these best-fit parameter values can produce very different model predictions for unobserved quantities of public health interest.

In this talk, I will begin by discussing various notions of nonidentifiability: structural vs practical,local vs global, etc, and different methods to detect and diagnose nonidentifiability. Then I will present a new mathematical approach to study the issue of nonidentifiability based on singular value decomposition and variance decomposition. I will then illustrate our approach with a case study of HIV estimation using an ordinary differential equation model. Some open questions for infinite dimensional models such as delay differential equation and partial differential equation models will also be discussed.

Dr. Simon Gindikin
Rutgers University

Curved Version of the Radon Inversion Formula

Thursday, February 15, 2018, 5:00pm
Ungar Room 402

Abstract: 100 years ago Radon published his famous formula for the reconstruction of functions on the plane through their integrals along lines. Is it possible, to replace in this construction lines with different curves? There are only known just a few such examples, such as geodesicsor horocycles on the hyperbolic plane. These formulas usually look similar to the Radon's formula. We give a universal reconstruction formula, as a closed differential form on the manifold of all curves, whose restriction on different cycles of curves gives specific examples of inversion formulas for curves. It is possible to interpret this construction as areal Cauchy integral formula.

Professor Yipeng Jing
Academician, The Chinese Academy of Sciences
Shanghai Jiao Tong University

Probing the Cosmic Expansion with Large Scale Galaxy Surveys:
Prospects and Challenges

Tuesday, January 30, 2018, 5:00pm
Ungar Room 402

Abstract: The Universe is found to be expanding in an accelerating phase. Mysterious dark energy is one possible solution to explain the acceleration, and modifying the gravity theory is another. In order to find out what is driving the cosmic accelerating expansion, one has to first answer the question whether the General Relativity is still valid on the cosmic scales. Astronomers are carrying out this test by observing many millions galaxies. I will introduce the basic information of this observation and summarize the current status. I will also introduce the future observations, and finally outline theoretical and observational challenges.

Professor Ziv Ran
University of California, Riverside

Filling Groovy:
The Goodness of Generic Projections

Thursday, December 14, 2017, 5:00pm
Ungar Room 402

Abstract: The classical process of projection amounts to selecting a linear subspace W from a given vector space V of functions on a geometric object X. This is analogous to vision, where W is the 2-dimensional space of coordinate functions on a retina. What information is lost by passing from V to W when W is selected randomly among subspaces of given dimension?

We will describe some progress on a version of this problem in complex algebraic geometry.

Steven Lu
Université du Québec à Montréal

The Structure of Semi-hyperbolic Projective Algebraic Manifolds

Thursday, April 27, 2017, 5:00pm
Ungar Room 402

Abstract: S. Kobayashi coined the term hyperbolic for a compact complex manifold M without nontrivial holomorphic images of C and conjectured the positivity of the canonical bundle of M. In particular M would be projective if true. But the conjecture is still wide open for projective manifolds beyond dimension two.

A spectacular advance in this direction is the resolution in the projective case by D. Wu-S.T. Yau (Invent. 2016) of the differential geometric analog of the conjecture, due to S.T. Yau. The analog pertains to compact Kähler manifolds with negative holomorphic curvature and the said advance resolves in particular the abundance conjecture, a key conjecture for the classification of algebraic varieties, for such a manifold.

In this talk, I will mainly focus on a recent joint paper with G. Heier, B. Wong and F.Y. Zheng that offers a structure theorem for projective Kähler manifolds with negative holomorphic curvature, assuming the abundance conjecture. The analysis involves a careful study of the rank of the said curvature, and offers relationships to the global abundance problem.

Dr. Nancy Rodriguez
University of North Carolina

On the Obstruction and Propagation of Entire Solutions to a Non-local Reaction Diffusion Equation with a Gap

Thursday, April 13, 2017, 5:00pm
Ungar Room 402

Abstract: In this talk I will discuss the propagation properties of a bistable spatially heterogeneous reaction-diffusion equation where the diffusion is generated by a jump process. Here the spatial heterogeneity is due to a small region with decay. First, I will focus on the existence and uniqueness of a "generalized transition front". Then I will give some partial results about propagation and obstruction of the transition front. Throughout the talk I will point out many interesting differences between the non-local and local reaction-diffusion equations.

Pengfei Guan
McGill University

Isometric Embeddings, Geometric Inequalities and Nonlinear PDEs

Wednesday, April 12, 2017, 5:00pm
Ungar Room 402

Abstract: In 1950s, Nirenberg and Pogorelov solved the classical Weyl problem regarding the isometric embedding of positively curved compact surface in to R^3. Solution to Weyl's problem is crucial to the definition of the Brown-York quasi local mass in general relativity in 1990s, and also play key role in the recent works of Liu-Yau and Wang-Yau. The development brings renewed focus on the Weyl problem of isometric embedding of surfaces to general 3D ambient space. We will discuss elliptic PDEs involved the problem and recent work on this type of nonlinear equations. We also discuss new proof of isoperimetric type of inequalities using parabolic PDEs. The talk is try to illustrate some beautiful interaction of nonlinear PDE, differential geometry and general relativity. It is accessible for general mathematical audience.

Dr. Alexander Kiselev
Rice University

Enhancement of Biological Reactions by Chemotaxis

Thursday, April 6, 2017, 5:00pm
Ungar Room 402

Abstract: Many reactions and processes in nature take place in fluid and in presence of both fluid flow and chemotaxis - directed motion of cells or species guided by attractive (or repulsive) chemical. One example of such process is broadcast spawning by corals, the way corals reproduce. Models of this process based on pure reaction-diffusion tend to dramatically underestimate the fertilization success rate. I will discuss a simplified 2D single equation model which incorporates fluid flow and chemotaxis effects. In the framework of this model built on the basis of the well known Keller-Segel equation, the role of chemotaxis turns out to be crucial. In the presence of a sufficiently strong chemotaxis, even weakly coupled reaction can lead to high fertilization rate on a fixed time scale.

If time permits, I will discuss some progress in a more sophisticated model which involves a system of two equations. Novel mathematical tools used in this work involve sharp convergence to equilibrium estimates for a class of Fokker-Planck operators with logarithmic-type potentials.

Anders Björner
KTH Royal Institute of Technology, Sweden

Around Codimension One Embeddings

Tuesday, March 7, 2017, 5:00pm
Ungar Room 402

Abstract: Being drawable in the plane without intersecting edges is a very important and much studied graph property. Euler observed in 1752 that planarity implies a linear upper bound on the number of edges of a graph (which otherwise is quadratic in the number of vertices). Several ways of characterizing planar graphs have been given during the previous century.

Planarity is, of course, a special case of a general notion of embedding a simplicial d-complex into real k-space. The k=d+1 and k=2d cases are of particular interest in higher dimensions, since they both generalize planarity. Embedding a space into some manifold is a much studied question in geometry/topology. For instance, van Kampen showed that in the k=2d case there is a very useful cohomological obstruction to embeddability.

Higher-dimensional embeddability has been studied also from the combinatorial point of view, in a tradition inspired by Euler. In this talk I will survey a few topics from the combinatorial study of embeddings, such as bounds for the number of maximal faces and algorithmic questions. I will end with mention of some joint work with A. Goodarzi concerning an obstruction to k=d+1 embeddings.

The talk will not presuppose previous familiarity with the topic.

Simon A. Levin
Princeton University

Collective Motion, Collective Decision-making, and Collective Action

Friday, March 3, 2017, 4:00pm
Ungar Room 402

Abstract: There exists a rich history of research on the mathematical modeling of animal populations. The classical literature, however, is inadequate to explain observed spatial patterning, or foraging and anti-predator behavior, because animals actively aggregate. This lecture will begin from models of animal aggregation, the role of leadership in collective motion and the evolution of collective behavior, and move from there to implications for decision-making in human societies. Ecological and economic systems are alike in that individual agents compete for limited resources, evolve their behaviors in response to interactions with others, and form exploitative as well as cooperative interactions as a result. In these complex-adaptive systems, macroscopic properties like the flow patterns of resources like nutrients and capital emerge from large numbers of microscopic interactions, and feedback to affect individual behaviors. I will explore common features of these systems, especially as they involve the evolution of cooperation in dealing with public goods, common pool resources and collective movement across systems; Examples and lessons will range from bacteria and slime molds to groups to insurance arrangements in human societies and international agreements on environmental issues.

Ron Adin
Bar Ilan University, Israel

Cyclic Descents, Toric Schur Functions and Gromov-Witten Invariants

Tuesday, February 28, 2017, 5:00pm
Ungar Room 402

Abstract: Descents of permutations have been studied since Euler. This notion has been vastly generalized in several directions, and in particular to the context of standard Young tableaux (SYT). More recently, cyclic descents of permutations were introduced by Cellini and further studied by Dilks, Petersen and Stembridge. Looking for a corresponding notion for SYT, Rhoades found a very elegant solution for rectangular shapes.

In an attempt to extend this concept, explicit combinatorial definitions for two-row and certain other shapes have been found, implying the Schur-positivity of various quasi-symmetric functions. In all cases, the cyclic descent set admits a cyclic group action and restricts to the usual descent set when the letter $n$ is ignored. Consequently, the existence of a cyclic descent set with these properties was conjectured for all shapes, even the skew ones.

This talk will report on the surprising resolution of this conjecture: Cyclic descent sets do exist for nearly all skew shapes, with an interesting small set of exceptions. The proof applies nonnegativity properties of Postnikov's toric Schur polynomials and a new combinatorial interpretation of certain Gromov-Witten invariants. We shall also comment on issues of uniqueness.

Joint with Sergi Elizalde, Vic Reiner and Yuval Roichman.

Enrico Arbarello
Università degli Studi di Roma "La Sapienza"

Hyperplane Sections of K3 Surfaces

Friday, February 3, 2017, 5:00pm
Ungar Room 402

Abstract: K3 surfaces and their hyperplane sections play a central role in algebraic geometry. This is a survey of the work done during the past five years to characterize which smooth curves lie on a K3 surface. Related topics will be discussed. These are joint works with a combination of the following authors: Andrea Bruno, Gavril Farkas, Edoardo Sernesi and Giulia Saccà.

Yuri Tschinkel
New York University

Arithmetic and Geometry of Fano Varieties

Thursday, February 2, 2017, 5:00pm
Ungar Room 402

Abstract: I will discuss recent advances in the theory of Fano varieties over nonclosed fields (joint with B. Hassett, A. Kresch, and A. Pirutka).

Dr. Tye Lidman
North Carolina State University

Left-orderability and Three-manifolds

Thursday, January 26, 2017, 5:00pm
Ungar Room 402

Abstract: A group is called left-orderable if it can be given a left-invariant total order. We will discuss the question of when the fundamental group of a three-manifold Y is left-orderable. Orderability is known to be related to certain topological aspects of Y, such as the surfaces which sit inside it. We will discuss a conjectural relationship between left-orderability and the solutions to a certain nonlinear PDE on Y.

Professor Luc Vinet
Centre de Recherches Mathématiques, Montreal

The Bannai-Ito Algebra in Many Guises

Joint Math-Physics Colloquium
Thursday, January 26, 2017, 4:00pm
Physics Conference Room, 3rd Floor

Abstract: This talk will offer a review of the Bannai-Ito algebra and of its higher rank extension. It will first be explained that it is in a Schur-Weyl duality with the super algebra osp(1,2). Its occurrence as the symmetry algebra of the Dirac-Dunkl equation will be discussed and its relation to orthogonal polynomials will also be presented.

John Morgan
Stony Brook University

Spin Manifolds of Dimensions at Most 4

Wednesday, January 25, 2017, 5:00pm
Ungar Room 402

Abstract: Low dimensional spin manifolds are interesting objects with close connections to quadratic forms. The first part of the talk will provide an overview of these manifolds and their invariants. The second part of the talk will use the understanding of low dimensional spin manifolds to give a detailed description of invariants that for each topological space X detect all 3- and 4-dimensional spin manifolds mapping to X up to spin bordism.

Dr. Pierre Magal
University of Bordeaux, France

Final Size of an Epidemic for a Two Group SIR Model

Thursday, December 15, 2016, 4:00pm
Ungar Room 402

Abstract: In this talk we consider a two-group SIR epidemic model. We study the final size of the epidemic for each sub-population. The qualitative behavior of the infected classes at the earlier stage of the epidemic is described with respect to the basic reproduction number. Numerical simulations are also preformed to illustrate our results.

Dr. Andrea Tellini
École des Hautes Études en Sciences Sociales

Enhancement of Fisher-KPP Propagation through Lines and Strips of Fast Diffusion

Wednesday, December 14, 2016, 5:00pm
Ungar Room 402

Abstract: In this talk I will present some systems of reaction-diffusion equations which take into account the presence of diffusion (and/or reaction) heterogeneities in some regions of the domain. The motivation is the modelling of the invasion of an environment by a species whose individuals can move (and/or reproduce) faster in some regions of the habitat. I will describe the dynamics of such systems and focus on the qualitative properties of the propagation speed, showing in particular when it is larger than in the case of a homogeneous environment. These are joint works with H. Berestycki, L. Rossi and E. Valdinoci.

Dr. Xi Huo
York University

Zika Outbreaks in a Highly Heterogeneous Environment: Insights from Dynamical Modelling

Monday, December 12, 2016, 5:00pm
Ungar Room 402

Abstract: Zika virus is in the family of Flaviviridae , and is often transmitted to human by Aedes aegypti, a common vector for transmitting several tropical fevers, including dengue and chikungunya. The environmental heterogeneity and intervention strategies of Zika spread also involve seasonality, co-circulation of other vector-borne diseases, and demographic structures of the mosquito population. We have been developing a variety of dynamical models to understand the transmission dynamics with focus on different aspects of environmental heterogeneities. We first consider the co-infection and co-circulation of dengue and Zika and their implication of dengue vaccination program for Zika control in the presence of experimentally reported antibody-dependent enhancement. We then consider the impact of heterogeneity of vector demographics on the initial outbreak rate and outbreak potential using age-structured partial differential equation systems and calculating the relevant threshold using non-linear semigroup theory and spectral theory. We also examine both numerically and analytically the mechanisms for potential nonlinear oscillations using the global bifurcation theory in delay differential equations.

Dr. William Christopher Strickland
University of North Carolina, Chapel Hill

Modeling Invasive Dispersal at Multiple Scales

Thursday, December 8, 5:00pm
Ungar Room 402

Abstract: Biological invasions represent an interesting challenge to model mathematically. Landscape heterogeneity, non-local and temporally dependent spreading mechanisms, coarse data, and the presence of long-distance transportation connections are but a few of the complications that can greatly affect our understanding of invasive spread. In this talk, I will look at dispersal from a multi-scale perspective in an attempt to address some of these challenges.

To begin, I will introduce a generalization of Mollison’s stochastic contact birth process (J R Stat Soc 39(3):283, 1977) which is robust to non-local distribution kernels and heterogeneity in the landscape. By interpreting the quantity of interest as species occurrence probability rather than population size, I will describe how this process may also be approximated and simulated deterministically, using niche modeling tools to characterize landscape heterogeneity. Adding to this is a method for considering the effects of a disease-vector transportation network, which can unwittingly transport a biological invader to distant sites. Finally, I will shift focus to the intial stages of an invasion and concentrate on the local and mesoscale by considering the intentional release of a parasitoid wasp biocontrol agent. Results indicate that the fluid physics of air above the landscape likely plays a critical role in the dispersal process.

Numerical results will be included throughout the talk, including simulations for the cheatgrass (bromus tectorum) invasion in Rocky Mountain National Park and the initial spread of parasitoid wasps (Eretmocerus hayati) during a biocontrol introduction.

Dr. Jan Sbierski
University of Cambridge

Strong Cosmic Censorship and the Wave Equation in the Interior of Black Holes

Wednesday, December 7, 2016, 5:00pm
Ungar Room 402

Abstract: The Einstein equations admit a locally well-posed initial value problem. However, there are explicit black hole solutions of the Einstein equations for which global uniqueness fails. The strong cosmic censorship conjecture in general relativity states that global uniqueness should hold generically -- which implies the expectation that small perturbations of the initial data for the above black hole solutions should give rise to a spacetime which is globally uniquely determined. In this talk I will explain how this motivates the study of the wave equation in the interior of black holes, present an overview of previous results, and discuss a recent instability result, obtained in collaboration with Jonathan Luk, in more detail.

Dr. Jessica Lin
University of Wisconsin-Madison

Stochastic Homogenization for Reaction-Diffusion Equations

Monday, December 5, 2016, 5:00pm
Ungar Room 402

Abstract: One way of modeling phenomena in "typical" physical settings is to study PDEs in random environments. The subject of stochastic homogenization is concerned with identifying the asymptotic behavior of solutions to PDEs with random coefficients. Specifically, we are interested in the following: if the random effects are microscopic compared to the lengthscale at which we observe the phenomena, can we predict the behavior which takes place on average? For certain models of PDEs and under suitable hypotheses on the environment, the answer is affirmative. In this talk, I will focus on the stochastic homogenization for reaction-diffusion equations with both KPP and ignition nonlinearities. In the large-scale-large-time limit, the behavior of typical solutions is governed by a simple deterministic Hamilton-Jacobi equation modeling front propagation. Such models are relevant for predicting the evolution of a population or the spread of a fire in a heterogeneous medium. This talk is based on joint work with Andrej Zlatos.

Dr. Verónica Quítalo
CoLab UT Austin-Portugal Program

Systems of Partial Differential Equations Arising from Population Dynamics and Neuroscience:
Free Boundary Problems as a Result of Segregation

Friday, December 2, 2016, 5:00pm
Ungar Room 402

Abstract: In this talk we will motivate and present our recent results on two phase free boundary problems arising from population dynamics. We will focus on systems with fully nonlinear diffusion and local interaction, and linear systems with a (nonlocal) long-range interaction. In the long-range model, the growth of a population at a point is inhibited by the other populations in a full area surrounding that point. This will force the populations to stay at distance one from each other in the limit configuration. So for the first time is obtained a free boundary problem with a gap of no-man's land between the regions where the populations exist. This is a joint work with Luis Caffarelli and Stefania Patrizi.

We will also present briefly some models of differential equations arising from neuroscience and share our current research on propagation of activity in the brain. We will motivate the need to incorporate the "volume" conductivity as well as of the neurons on a model. This is a joint work with Aaron Yip, Zoltan Nadasdy and Silvia Barbeiro.

Dr. Nathan Totz
University of Massachusetts Amherst

Global Flows with Invariant Measures for a Family of Almost Inviscid SQG Equations

Thursday, December 1, 2016, 5:00pm
Ungar Room 402

Abstract: We present a new result, joint with Andrea Nahmod, Natasa Pavlovic, and Gigliola Staffilani, in which very low regularity flows are constructed globally in time almost surely for a family of modified SQG equations using a Gibbs measure; the resulting flows leave this Gibbs measure invariant. The family of equations we treat is formed by adding a small amount of smoothing to the active scalar of the standard inviscid SQG equation. We find that global solutions can be constructed almost surely for any nonzero amount of smoothing.

Dr. Michele Coti-Zelati
University of Maryland

Deterministic and Stochastic Aspects of Fluid Mixing

Monday, November 28, 2016, 5:00pm
Ungar Room 402

Abstract: The process of mixing of a scalar quantity into a homogeneous fluid is a familiar physical phenomenon that we experience daily. In applied mathematics, it is also relevant to the theory of hydrodynamic stability at high Reynolds numbers - a theory that dates back to the 1830's and yet only recently developed in a rigorous mathematical setting. In this context, mixing acts to enhance, in certain senses, the dissipative forces. Moreover, there is also a transfer of information from large length-scales to small length-scales vaguely analogous to, but much simpler than, that which occurs in turbulence. In this talk, we focus on the study of the implications of these fundamental processes in linear settings, with particular emphasis on the long-time dynamics of deterministic systems (in terms of sharp decay estimates) and their stochastic perturbations (in terms of invariant measures).

Ryan Hynd
University of Pennsylvania

Partial Differential Equations in Finance

Thursday, August 25, 2016, 5:00pm
Ungar Room 402

Abstract: Starting with the Black & Scholes equation, I will give a tour of partial differential equations arising in various financial models. My goal is to emphasize what is mathematically interesting about each equation and how solutions are used in applications.

Dr. Daozhou Gao
Shanghai Normal University

The Importance of Synchrony in Mass Drug Administration

Friday, July 8, 2016, 4:30pm
Ungar Room 402

Abstract: Mass drug administration (MDA), a strategy in which all individuals in a population are subject to treatment without individual diagnosis, has been recommended by the World Health Organization for controlling and eliminating several neglected tropical diseases. In this talk, I will present some results arising from mass treatment of trachoma with azithromycin. In the first part, we compare three typical drug distribution strategies (regardless of health status): constant treatment, impulsive synchronized MDA, and impulsive non-synchronized treatment. We show that synchronized and constant strategies are respectively the most and least effective treatments in disease control. Elimination through synchronized treatment is always possible when adequate drug efficacy and coverage is fulfilled and sustained. In the second part, the optimal seasonal timing of mass administration of azithromycin for maximum antimalarial benefit has been established. This is joint work with Thomas M. Lietman and Travis C. Porco.

Dr. Wan-Tong Li
Lanzhou University, China

Nonlocal Effects and Nonlocal Dispersal

Thursday, July 7, 2016, 4:30pm
Ungar Room 402

Abstract: This talk is concerned with some aspects of nonlocal dispersal equations. It consists of three parts. In part one, I will present some relations between local (random) and nonlocal dispersal problems. In part two, I will report our recent results on traveling waves and entire solutions of nonlocal dispersal equations. Part III is devoted to some problems on traveling waves and entire solutions of nonlocal dispersal equations.

Professor Guo Lin
Lanzhou University, China

Traveling Wave Solutions of Evolutionary Models without Monotonicity

Monday, May 9, 2016, 4:00pm
Ungar Room 402

Abstract: This talk is concerned with the traveling wave solutions of evolutionary systems including delayed reaction-diffusion systems and integrodifference equations. Even if the general monotone conditions fail, the existence of traveling wave solutions is studied by generalized upper and lower solutions. The asymptotic behavior is established by the idea of contracting rectangles. In the study of classical Lotka-Volterra competitive systems, we obtain the existence of nonmonotone traveling wave solutions, which weakly confirms the conjecture by Tang and Fife [ARMA, 1980].

Dr. Dmitri Vassiliev
University College London

Spectral Asymptotics for First Order Systems

Thursday, April 14, 2016, 5:00pm
Ungar Room 402

Abstract: In layman's terms a typical problem in this subject area is formulated as follows. Suppose that our universe has finite size but does not have a boundary. An example of such a situation would be a universe in the shape of a 3-dimensional sphere embedded in 4-dimensional Euclidean space. And imagine now that there is only one particle living in this universe, say, a massless neutrino. Then one can address a number of mathematical questions. How does the neutrino field (solution of the massless Dirac equation) propagate as a function of time? What are the eigenvalues (energy levels) of the particle? Are there nontrivial (i.e. without obvious symmetries) special cases when the eigenvalues can be evaluated explicitly? What is the difference between the neutrino (positive energy) and the antineutrino (negative energy)? What is the nature of spin? Why do neutrinos propagate with the speed of light? Why are neutrinos and photons (solutions of the Maxwell system) so different and, yet, so similar?

The speaker will approach the study of first order systems of partial differential equations from the perspective of a spectral theorist using techniques of microlocal analysis and without involving geometry or physics. However, a fascinating feature of the subject is that this purely analytic approach inevitably leads to differential geometric constructions with a strong theoretical physics flavour.

Alexander Volberg
Michigan State University

Rectifiability of Harmonic Measure

Wednesday, April 13, 2016, 4:00pm
Ungar Room 402

Abstract: In a recent multi-authored paper by J. Azzam, S. Hofmann, J.-M. Martell, S. Mayboroda, M. Mourgoglou, X. Tolsa and myself the following result is proved:

For arbitrary open set in any \R^d such that its boundary carries harmonic measure w and for any Borel set $E$, such that $H^{d-1}(E)<\infty, if harmonic measure w|E is absolutely continuous with respect to H^{d-1}|E, then it is rectifiable.

This result solves a long standing conjecture of Chris Bishop and generalizes the result of N. Makarov obtained for d=2 and for simply connected open sets obtained in the early 80's.

We will also make an overview of harmonic measure results of the last 30 years.

Weiyi Zhang
University of Warwick

J-holomorphic Subvarieties

Friday, March 18, 2016, 5:45pm
Ungar Room 402

Abstract: In this talk, we discuss the J-holomorphic subvarieties in a 4-dimensional symplectic manifold. We will start by showing that a subvariety of a complex rational surface in an exceptional rational curve class could have higher genus components. On the other hand, such exotic phenomenon won't happen for (any tamed almost complex structures on) ruled surfaces.

We will then show a general cone theorem as in Mori theory and determine the moduli space of a spherical class in a rational or ruled surface, which could be decomposed as linear systems.

Professor Fedor Bogomolov
New York University

Comparing Torsion Points of Elliptic Curves

Friday, March 18, 2016, 4:45pm
Ungar Room 402

Abstract: In this talk we consider subset of $P^1_{E}\subset P^1$ which are the images of torsion points of complex elliptic curve $E$ under projection defined as a quotient map for standard involution. The corresponding map $E\to P^1$ and hence $P^1_{E}\subset P^1$ is uniquely defined modulo projective transform on $P^1$. It is known that the intersection of subsets $P^1_{E}, P^1_{E'}$ are finite apart from the case when they coincide. However the question is whether there is universal constant bounding this intersection. This problem is somewhat related to Serre conjecture on the image of Galois action on the torsion points. I am going to discuss some results concerning the matter. In particular we show that the intersection contains exactly one point for some families of elliptic curves, that there is an infinite set of pairs $E,E'$ where the intersection consists of $10$ points and also provide a example with $14$ intersection points.

This is joint work with Hang Fu.

Dr. Phillip Griffiths
Institute of Advanced Study, Princeton
University of Miami

The Hodge Conjecture

Friday, February 12, 2016, 5:00pm
Ungar Room 402

Abstract: The Hodge conjecture is one of the Millennium Prize problems. In this talk I will discuss the questions:

• What is it?
• Why is it interesting?
• What is known about it?
• Why has there been almost no progress in solving it?

As little background as possible will be assumed, and what is needed will be informally explained.

Based on joint work with Mark Green, Radu Laza and Colleen Robles.

Ilya Kachkovskiy
University of California, Irvine

Almost Commuting Operators

Tuesday, January 26, 2016, 5:00pm
Ungar Room 402

Abstract: The classical question of whether a pair of matrices or operators with small commutator is close to a commuting pair dates back to von Neumann, Rosenthal, and Halmos. In particular, a dimension-uniform version of this question for Hermitian matrices was open until 1993 when Lin proved the existence of a uniform distance estimate using deep C*-algebraic arguments. Lin's method didn't allow to obtain any quantitative results besides existence. In 2001, Davidson and Szarek conjectured that the distance is bounded by some universal constant times the square root of the norm of the commutator.

In this talk, I will explain the main ideas of the proof of Davidson and Szarek's conjecture (joint work with Y. Safarov), including the infinite-dimensional version.

I will also talk about the relation with Brown-Douglas-Fillmore theorem, and the case of the Hilbert-Schmidt norm for matrices. If time permits, I will discuss some work in progress that includes possible generalization to the case of unitary matrices (where an additional topological obstruction arises), and some applications to mathematical physics.

Dr. Antoine Choffrut
University of Warwick

The Analysis of Partial Differential Equations:
A Case Study with the Euler Equations of Fluid Mechanics

Thursday, January 21, 2016, 5:00pm
Ungar Room 402

Abstract: In this talk I will give an overview of my work on the incompressible Euler equations. It can be categorized in two classes: the use of convex integration to prove h-principles, and structure theorems for two-dimensional stationary flows. These are two very different types of results. However, I will try to highlight a unifying principle in this body of work.

Mihaela Ifrim
University of California, Berkeley

Two Dimensional Water Waves in Holomorphic Coordinates

Thursday, January 14, 2016, 5:00pm
Ungar Room 402

Abstract: This is joint work with Daniel Tataru, and in parts with Benjamin Harrop-Griffits and John Hunter. My talk is concerned with the infinite depth water wave equation in two space dimensions, with either gravity or surface tension. I will also make some remarks on the finite depth case, and on the infinite depth case in which constant vorticity and only gravity are assumed. We consider this problem expressed in position-velocity potential holomorphic coordinates. Viewing this problem(s) as a quasilinear dispersive equation, we develop new methods which will be used to prove enhanced lifespan of solutions and also global solutions for small and localized data. For the gravity water waves there are several results available; they have been recently obtained by Wu, Alazard-Burq-Zuily and Ionescu-Pusateri using different coordinates and methods. In the capillary water waves case, we were the first to establish a global result (two months later, Ionescu-Pusateri also announced a related result). Our goal is improve the understanding of these problems by providing a single setting for all the above cases, and presenting simpler proofs. The talk will try to be self contained.

Mimi Dai
University of Illinois at Chicago

Regularity for the 3D Navier-Stokes Equations and Related Problems

Tuesday, January 12, 2016, 5:00pm
Ungar Room 402

Abstract: As one of the most significant problems in the study of partial differential equations arising in fluid dynamics, Leray's conjecture in 1930's regarding the appearance of singularities for the 3-dimensional (3D) Navier-Stokes equations (NSE) has been neither proved nor disproved. The problems of blow-up have been extensively studied for decades using different techniques. By using a method of wavenumber splitting which originated from Kolmogorov's theory of turbulence, we obtained a new regularity criterion for the 3D NSE. The new criterion improves the classical Prodi-Serrin, Beale-Kato-Majda criteria and their extensions. Related problems, such as the well/ill-posedness, will be discussed as well.

Dr. Eduardo Sontag
Rutgers University

Qualitative Features of Transient Responses
A Case Study: Scale-invariance

Thursday, December 17, 2015, 4:00pm
Ungar Room 402

Abstract: An ubiquitous property of sensory systems is "adaptation": a step increase in stimulus triggers an initial change in a biochemical or physiological response, followed by a more gradual relaxation toward a basal, pre-stimulus level. Adaptation helps maintain essential variables within acceptable bounds and allows organisms to readjust themselves to an optimum and non-saturating sensitivity range when faced with a prolonged change in their environment. It has been recently observed that some adapting systems, ranging from bacterial chemotaxis pathways to signal transduction mechanisms in eukaryotes, enjoy a remarkable additional feature: scale invariance or "fold change detection" meaning that the initial, transient behavior remains approximately the same even when the background signal level is scaled.

I will review the biological phenomenon, and formulate a theoretical framework leading to a general theorem characterizing scale invariant behavior by equivariant actions on sets of vector fields that satisfy appropriate Lie-algebraic non-degeneracy conditions. The theorem allows one to make experimentally testable predictions, and I will discuss the validation of these predictions using genetically engineered bacteria and microfluidic devices, as well their use as a "dynamical phenotype" for model invalidation. I will conclude by briefly engaging in some wild and irresponsible speculation about the role of the shape of transient responses in immune system self/other recognition and in evaluating the initial effects of immunotherapy.

Dr. Phillip Griffiths
Institute of Advanced Study, Princeton

Hodge Theory and Moduli

Wednesday, September 9, 2015, 5:00pm
Ungar Room 402

Abstract: This talk will give an overview of the connection between the above two topics in the first non-classical case of H-surfaces.

Based on joint work with Mark Green, Radu Laza and Colleen Robles.

Professor Wayne Lawton
University of Western Australia

Multivariate Prediction and Spectral Factorization

Thursday, September 3, 2015, 5:00pm
Ungar Room 402

Abstract: Prediction theory studies stationary random functions on ordered groups. Time series are functions on the integer group and characterized by classical harmonic analysis results such as Szego's spectral factorization theorem. Images are functions on higher rank groups. We use results about entire functions and ergodic theory to derive new results for these functions.

Dr. Dimitri Gurevich
Valenciennes University, France

From Quantum Groups to Noncommutative Geometry

Tuesday, May 5, 2015, 4:00pm
Ungar Room 402

Abstract: Since creation theory of Quantum Groups numerous attempts to elaborate an appropriate differential calculus were undertaken. Recently, a new type of Noncommutative Geometry has been obtained on this way. Namely, we have succeeded in introducing the notions of partial derivatives on the enveloping algebras U(gl(m)) and constructing the corresponding de Rham complexes. All objects arising in our approach are deformations of their classical counterparts. In my talk I plan to introduce some basic notions of the theory of Quantum Groups and to exhibit possible applications of thistype Noncommutative Geometry to quantization of certain dynamical models.

Dr. Phillip Griffiths
Institute for Advanced Study, Princeton

Hodge Theory: Some History and a Possible Future Direction

Tuesday, April 21, 2015, 5:00pm
Ungar Room 402

Abstract: Hodge theory is a central part of algebraic geometry. The subject has a long and rich history and continues today to be at the forefront of much current work. This talk will discuss some of its history and one possible, less well known, possible area for future work. In brief: where did the subject come from, and where might it be heading?

Professor Yuval Roichman
Bar-Ilan University

Fine Sets and Schur Positivity

Tuesday, April 14, 2015, 5:00pm
Ungar Room 402

Abstract: Characters of symmetric groups and Iwahori-Hecke algebras may be evaluated by signed (and slightly weighted) enumerations of various classes of permutations, such as conjugacy classes, Knuth classes and inverse descent classes. We propose an abstract framework for such classes, which we call "fine sets".

It will be shown that fine sets can be characterized by Schur positivity of the associated quasi-symmetric functions. The proof involves asymmetric Walsh-Hadamard type matrices. Applications include the equivalence of classical theorems of Lusztig-Stanley and Foata-Schutzenberger.

Time allowing, some very recent developments will be described -- in particular, a generalization to type B and applications to grid classes and derangements, related to results of Desarmenien-Wachs and others.

Based on joint works with R. Adin, C. Athanasiadis and S. Elizalde.

Professor Alexei Kovalev
University of Cambridge

8-dimensional Manifolds with Holonomy Spin(7)

Monday, April 6, 2015, 4:00pm
Ungar Room 402

Abstract: Spin(7) is one of the "exceptional" holonomy groups of Riemannian metrics appearing in Berger's classification. Riemannian manifolds with holonomy Spin(7) occur in dimension 8, are Ricci-flat and have parallel spinor fields. We construct examples of asymptotically cylindrical Riemannian 8-manifolds with holonomy group Spin(7). To our knowledge, these are the first such examples. This leads to a new, connected sum construction of compact holonomy-Spin(7) manifolds from asymptotically cylindrical pairs. We show in examples that the construction produces a "pulling-apart" deformation of compact Spin(7)-manifolds previously given by Joyce, as well as topologically new Spin(7)-manifolds.

Professor Xiuxiong Chen
Stony Brook University

On the Kaehler Ricci Flow

Friday, March 27, 2015, 5:00pm
Ungar Room 402

Abstract: There is a long standing conjecture on Kaehler Ricci flow on Fano manifolds that the Ricci flow converges sub-sequentially to a Kaehler Ricci solution with at most codimension 4 singularities, with perhaps a different complex structure (the so-called "Hamilton-Tian conjecture"). In this lecture, we will outline a proof of this conjecture. This is a joint work with Bing Wang.

Professor Mario Milman
Florida Atlantic University

Uncertainty Inequalities in Metric Measure Spaces

Friday, March 20, 2015, 5:00pm
Ungar Room 402

Abstract: I will show an extension of the classical uncertainty inequalities to connected metric measure spaces. The idea is to exploit the connection with isoperimetry. I will introduce a new class of weights (*isoperimetric weights*) and prove a new local Poincare inequality formulated in terms of the isoperimetric profile. A reinterpretation of the notion of *isoperimetric weights* in terms of function spaces provides a different method to attack the problem and also allows for explicit computation of these weights for log concave measures and other model geometries. I will try to explain all the terms in the abstract, provide examples and show connections with classical inequalities by Hardy and Strichartz.

Professor King-Leung Lam
Ohio State University

On Global Dynamics of Competitive Systems in Ordered Banach Spaces

Tuesday, March 17, 2015, 5:00pm
Ungar Room 402

Abstract: A well-known result in [Hsu-Smith-Waltman, Trans. AMS (1996)] states that in a competitive semiflow defined on the product of two cones in respective Banach spaces, one of the following outcomes is possible for the two competitors: either there is at least one stable coexistence steady state, or else one of the exclusion states attracts all trajectories initiating in the order interval bounded by the two exclusion states. However, none of the exclusion states can be globally asymptotically stable if we broaden our scope to the entire positive cone. In this talk, we discuss two sufficient conditions that guarantee, in the absence of coexistence steady states, the global asymptotic stability of one of the exclusions states. Our results complement the counter example mentioned in the above paper and are frequently applicable in practice.

Dr. Alexander Volberg
Michigan State University

Why the Oracle May Not Exist:
Ergodic Families of Jacobi Matrices,
Absolute Continuity without Almost Periodicity

Thursday, March 5, 2015, 5:00pm
Ungar Room 402

Abstract: We will explain the recent solution of Kotani's problem pertinent to the existence/non-existence of "oracle" (almost periodicity) for the ergodic families of Jacobi matrices (discrete Schroedinger operators). Kotani suggested that such families are subject to the following implication: if family has a non-trivial absolutely continuous spectrum (this happens almost surely) then almost surely it consists of almost periodic matrices (hence the possibility to predict the future by the past). Kotani proved an important positive result of this sort. Recently independently Artur Avila and Peter Yuditskii-myself disproved this conjecture of Kotani (by two different approaches). We will show the hidden singularity that defines when such Kotani's oracle exists or not.

S.R.S. Varadhan
Frank J. Gould Professor of Science
Courant Institute at New York University

Large Deviations for Brownian Occupation Times, Revisited

Friday, February 20, 2015, 5:00pm
Ungar Room 402

Abstract: Brownian motion on R^d is not positive recurrent and the Large Deviation estimate holds only in a weak form. To get a strong version a compactification is needed. In the application of interest we need a translation invariant compactification and that is carried out.

Dr. Lingjiong Zhu
University of Minnesota

Self-Exciting Point Processes

Tuesday, February 17, 2015, 5:00pm
Ungar Room 402

Abstract: Self-exciting point processes are simple point processes that have been widely used in neuroscience, sociology, finance and many other fields. In many contexts, self-exciting point processes can model the complex systems in the real world better than the standard Poisson processes. We will discuss the Hawkes process, the most studied self-exciting point process in the literature. We will talk about the limit theorems and asymptotics in different regimes. Extensions to Hawkes processes and applications to finance will also be discussed.

Dr. Karim Adiprasito
Institute for Advanced Study

Toric Chordality

Monday, February 16, 2015, 5:00pm
Ungar Room 402

Abstract: We put the fundamental graph-theoretic notion of chordality into a proper context within the weight algebra of McMullen and framework rigidity. Moreover, we generalize some of the classical results of graph chordality to this context, including the fundamental relation to the Leray property.

Our main focus is the relation of higher chordality to the Hard-Lefschetz Theorem for simplicial polytopes of Saito, McMullen and others. Homological chordality allows us to state a powerful Quantitative Lower Bound Theorem which relates the "defect" to a chordal complex to the g-numbers of the same polytope, thereby providing an immediate combinatorial interpretation of the g-numbers and, more refinedly, the cohomology classes of the associated projective toric variety, in terms of induced subcomplexes of the simplicial complex. While most of our results follow quite easily once we established stress spaces as the right setting for the study of higher chordality, we provide a central propagation theorem for toric chordality. As an application, we resolve a variety of interesting questions in combinatorics of polytopes.

Dr. Hamed Amini
Swiss Federal Institute of Technology (EPFL), Lausanne

Systemic Risk in Financial Networks

Thursday, February 12, 2015, 5:00pm
Ungar Room 402

Abstract: After the last financial crisis, the monitoring and managing of systemic risk has emerged as one of the most important concerns for regulators, governments and market participants. In this talk, we propose two complementary approaches to modeling systemic risk.

In the first approach, we model the propagation of balance-sheet or cash-flow insolvency across financial institutions as a cascade process on a network representing their mutual exposures. We derive rigorous asymptotic results for the magnitude of contagion in a large financial network and give an analytical expression for the asymptotic fraction of defaults, in terms of network characteristics. We also introduce a criterion for the resilience of a large inhomogeneous financial network to initial shocks that can be used as a tool for monitoring systemic risk. Using an equilibrium approach, in the second part of the talk, we apply a coherent systemic risk measure to examine the effects on systemic risk and liquidation losses of multilateral clearing via a central clearing counterparty (CCP). We provide sufficient conditions in terms of the CCP's fee and guarantee fund policy for a reduction of systemic risk.

Dr. Jin Hyuk Choi
Carnegie Mellon University

Taylor Approximation of Incomplete Radner Equilibrium Models

Tuesday, February 10, 2015, 5:00pm
Ungar Room 402

Abstract: We will first explain the notion of Radner equilibria by using the one-period binomial model. We will see how the complete market assumption simplifies the mathematical structure. In the next part we will consider the setting of exponential investors and uncertainty governed by Brownian motions. Here we will show the existence of an equilibrium for a general class of incomplete models. Finally, we will show that the general incomplete equilibrium can be approximated by a tractable equilibrium stemming from exponential-quadratic models.

Dr. Volker Schlue
University of Toronto

Non-existence of Time-periodic Dynamics in General Relativity

Monday, February 2, 2015, 5:00pm
Ungar Room 402

Abstract: General Relativity presents us with a theory for the dynamics of space and time itself. In this lecture I will revisit the expectation that in contrast to Newton's theory of gravity there does not exist time-periodic motion. In particular, I will present a recent result that establishes purely on the basis of knowing the gravitational waves emitted from any self-gravitating system, that any time-periodic space-time is necessarily "stationary", that is time-independent. Moreover, I will elaborate on novel unique continuation results for hyperbolic partial differential equations that play an important role in the proof, and have been obtained jointly with Spyros Alexakis and Arick Shao.

Dr. Alejandro Morales
University of California, Los Angeles

Matrices over Finite Fields, Polytopes, and Symmetric Functions Related to Rook Placements

Wednesday, January 28, 2015, 5:00pm
Ungar Room 402

Abstract: Permutation matrices are fundamental objects in combinatorics and algebra. Such matrices can be viewed as placements of non-attacking rooks on a square board. If we restrict the support of the matrix we obtain rook placements that in computational complexity are an example of problems where finding a solution is easy but the enumeration is difficult. In this talk we look at various recent results where rook placements appear.

The first instance is counting invertible matrices over a finite field of size q with support that avoids some entries. The number of such matrices is a q-analogue of rook placements but this number may not be a polynomial in q (Stembridge). We give polynomial formulas when the forced zeroes are in the diagonal and on the inversions of some permutations. We also relate the latter to the zoo of combinatorial objects in Postnikov's study of the positive Grassmannian. (Joint works with A. Klein, J. Lewis, R. Liu, G. Panova, S. V Sam, Y. Zhang.)

The second instance is as vertices of the Chan-Robbins-Yuen polytope, a face of the Birkhoff polytope of doubly stochastic matrices. This polytope has a remarkable volume formula proved by Zeilberger. We discuss three variants of this polytope. (Joint works with K. Meszaros, B. Rhoades and J. Striker.)

The third instance is as coefficients in relations among Stanley's chromatic symmetric function and Shareshian-Wachs chromatic quasisymmetric functions on a Catalan family of graphs. (Ongoing work with M. Guay-Paquet and F. Saliola.)

Dr. Antoine Choffrut
University of Edinburgh

Rigidity and Flexibility for the Euler Equations:
The h-principle and the Method of Convex Integration

Tuesday, January 27, 2015, 5:00pm
Ungar Room 402

Abstract: Certain problems in Differential Geometry have long been observed to feature a dichotomy "rigidity vs. flexibility". Rigidity refers to the fact that smooth solutions are unique, while flexibility means that the set of solutions with low regularity can be extremely large. With the groundbreaking work of De Lellis and Szekelyhidi on the incompressible Euler equations of fluid mechanics, it came as an enormous surprise that such a dichotomy may also hold for problems from Mathematical Physics. Flexibility results of interest to us fall under the name of h-principles and are established using the method of convex integration. In this talk I will give a bird's eye view of these notions and explain a motivating conjecture due to Onsager. I will conclude with future problems which deserve special research efforts.

Dr. Angela Hicks
Stanford University

Combining the Classical and the Rational:
Re-expressing Rational Shuffle Conjecture

Friday, January 23, 2015, 5:00pm
Ungar Room 402

Abstract: Recent results have place the classical shuffle conjecture in the context of an infinite family of conjectures about parking functions in any rectangular lattice. The classical case describes three traditional statistics which have varying degrees of complication: area, dinv, and pmaj. Combining the work of many authors in several fields gives two of these statistics in the more general case. After introducing the conjectures, this talk will cover how these statistics can be expressed in a manner consistent with the current classical literature and why this is important. Additionally, this talk will include a conjectured generalization of the third statistic, pmaj, in a number of new cases and time permitting, explain its connection to the sandpile model.

Dr. Nicolas Addington
Duke University

Recent Developments in Rationality of Cubic 4-folds

Tuesday, January 20, 2015, 5:00pm
Ungar Room 402

Abstract: The question of which cubic 4-folds are rational is one of the foremost open problems in algebraic geometry. I'll start by explaining what this means and why it's interesting; then I'll discuss three approaches to solving it (including one developed in the last year), my own work relating the three approaches to one another, and the troubles that have befallen each approach.

Dr. Kristin Shaw
Technical University of Berlin

Tropical Curves and Surfaces

Friday, January 16, 2015, 5:00pm
Ungar Room 402

Abstract: Tropical geometry is a relatively new field of mathematics which provides us with piecewise linear models for classical geometry. Often these models encode properties of the original objects in a more combinatorial form. The first major application of the field to classical geometry came by way of Mikhalkin's correspondence theorem between planar tropical and complex curves. This correspondence has served to answer questions in enumerative geometry; how many curves of a given degree and genus satisfy an appropriate set of constraints? Various other applications have been to moduli problems, linear series on curves, intersection theory, and computations of Hodge numbers. Tropical geometry has also been suggested as a explanation for mirror symmetry.

Abstract tropical varieties in higher dimensions have not been studied to the extend of curves. Applications of tropical geometry to classical geometry often rely on establishing correspondences between the two worlds. In this talk we will see that upon leaving the plane for other surfaces or varieties of higher dimension these correspondences are harder to establish and sometimes do not exist. We will explain how intersection theory and tropical homology can be used to study curves and also tropical Picard groups in surfaces and show how operations known as tropical modification and summation can construct and help study interesting examples of tropical surfaces.

Dr. Morgan Brown
University of Michigan

Rationality of Algebraic Varieties

Thursday, January 15, 2015, 5:00pm
Ungar Room 402

Abstract: An algebraic variety is called rational if it has a parametrization by rational functions which is one-to-one almost everywhere. Determining whether a variety is rational or not is an old and very difficult problem, even when that variety is a smooth hypersurface in projective space. I will give a brief history of this problem, and explain some of the modern techniques for showing that varieties are irrational, including how certain aspects of the derived category might reflect rationality.

Dr. Bruno Benedetti
Free University of Berlin

Diameter of Polytope Graphs

Tuesday, January 13, 2015, 5:00pm
Ungar Room 402

Abstract: "Discrete" geometry focuses on geometric objects that have facets, corners, vertices, rather than on objects with a uniform smooth structure. A standard result is Balinski's theorem, "the graph of every convex d-dimensional polytope is d-connected". But given a d-polytope with n facets, how many edges do we have to walk along (at most), if we want to go from a vertex to another? This "combinatorial diameter" question has relevance in optimization, being related to the simplex algorithm. Hirsch conjectured in 1957 that the answer should be "at most n-d", but the conjecture was disproved by Santos in 2010. I will explain two recent positive results:

(1) The Hirsch conjecture holds for all flag polytopes. The proof uses methods from differential geometry. (Joint work with K. Adiprasito.)

(2) The notion of dual graph naturally extends to projective curves, or to arbitrary algebraic varieties. Surprisingly, a broader version of Balinski's theorem still holds. (Joint work with M. Varbaro, and work in progress with B. Bolognese.)

Dr. Xing Liang
Chinese University of Science and Technology

Spreading Speeds of Monostable Parabolic Equations in Periodic Habitat with Free Boundary

Wednesday, December 17, 2014, 4:30pm
Ungar Room 402

Abstract: In this talk, we will show the existence of the speading speeds of monostable parabolic equations in the spatially and temporally periodic habitats. We main focus on the case where the equations is not linearly determinate.

Dr. James Keesling
Department of Mathematics
University of Florida

An Agent Based Microsimulation Model for the Spread of Citrus Greening

Thursday, November 13, 2014, 5:00pm
Ungar Room 402

Abstract: The citrus industry is a pillar of Florida's economy. It is hard to imagine that it may disappear in short order. However, due to the devastating effects of Citrus Greening, this may very well be the case.

Citrus Greening (or Huang Long Bing) is a disease of citrus caused by a bacterium (Candidatus Liberibacter asiaticus). It is spread by a psyllid, Diaphorina citri. In this talk we will describe a simulation model for the spread of this disease. The model is based on a detailed analysis of the biology of the psyllid vector. Current work expanding and applying the model is being funded by the Citrus Research and Development Foundation as part of the Psyllid Shield Project.

Professor John Stillwell
University of San Francisco

What Does "Depth" Mean in Math?
History, Foundations, and Logic

Thursday, November 6, 2014, 5:00pm
Witten Learning Center 160

Abstract: Every mathematician believes that certain theorems are "deep," but the concept of depth does not have a formal definition. By looking at some famous theorems, ancient and modern, we will study some candidates for "depth" at various levels, particularly the undergraduate level. With these examples in hand we hope to discuss whether any concepts of logic now available can give "depth" a precise meaning.

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Anatoly Libgober
University of Illinois at Chicago

Landau-Ginzburg/Calabi-Yau and McKay Correspondences for Elliptic Genus

Monday, October 27, 2014, 5:00pm
Ungar Room 506

Abstract: I will discuss elliptic genus of singular varieties and its extension to Witten's phases of N=2 theories. In particular McKay correspondence for elliptic genus will be described. As one of applications I will show how to derive relations between elliptic genera of Calabi-Yau manifolds and related Witten phases using equivariant McKay correspondence for elliptic genus.

Dr. Vyacheslav Shokurov
Johns Hopkins University

Weak Kawamata Conjecture

Wednesday, October 15, 2014, 5:00pm
Ungar Room 506

Abstract: The talk will explain the finiteness of weak log canonical models related by bounded flops.

Dr. Kimihiko Motegi
Nihon University, Tokyo, Japan

Twisted Families of L-space Knots

Wednesday, September 10, 2014, 5:00pm
Ungar Room 402

Abstract: A knot in the 3-sphere is called an L-space knot if it admits a nontrivial Dehn surgery yielding an L-space, which is a generalization of a lens space from the algebraic viewpoint of Heegaard Floer homology. Given an L-space knot K, can we obtain an infinite family of L-space knots by twistings of K along a suitably chosen unknotted circle? We will discuss this question in the case where K admits a Seifert surgery, and give a sufficient condition on such an unknotted circle. If K is a torus knot, then we have an unknotted circle c such that twistings along c produce an infinite family of hyperbolic, L-space knots. In particular, for the trivial knot we can take infinitely many such unknotted circles. We also demonstrate that there are infinitely many hyperbolic, L-space knots with tunnel number greater than one, each of which arises from a trefoil knot by alternate twistings along two unknotted circles.

Dr. Ryan Hynd
Department of Mathematics
University of Pennsylvania

Option Pricing in the Large Risk-aversion, Small-transaction-cost Limit

Thursday, August 21, 2014, 4:00pm
Ungar Room 402

Abstract: We discuss an alternative to the well-known Black-Scholes option pricing model and characterize the asking price of a risk-averse seller, in the limit of small transaction costs. The resulting mathematical problem is one of asymptotic analysis of partial differential equations and the real challenge involves a particular eigenvalue problem.

Professor Andrea Nahmod
University of Massachusetts, Amherst

Randomization and Long Time Dynamics in Nonlinear Evolution PDE

Tuesday, April 29, 2014, 5:00pm
Ungar Room 402

Abstract: The field of nonlinear dispersive equations has undergone significant progress in the last twenty years thanks to the influx of tools and ideas from nonlinear Fourier and harmonic analysis, geometry and analytic number theory, into the existing functional analytic methods. This body of work has primarily focused on deterministic aspects of wave phenomena and answered fundamental questions such as existence and long time behavior of solutions, in various regimes. Yet there remain some important obstacles and open questions.

A natural approach to tackle some of them, and one which has recently seen a growing interest, is to consider certain evolution equations from a non-deterministic point of view (e.g. the random data Cauchy problem, invariant measures, etc.) and incorporate to the deterministic toolbox, powerful but still classical tools from probability as well.

Such approach goes back to seminal work by Bourgain in the mid 90's where global well-posedness of certain periodic Hamiltonian PDEs was studied in the almost sure sense via the existence and invariance of their associated Gibbs measures.

In this talk we will explain these ideas, describe some recent work and future directions with an emphasis on the interplay of deterministic and probabilistic approaches.

Professor Ian Hambleton
McMaster University

Manifolds and Symmetry

Wednesday, April 23, 2014, 5:00pm
Ungar Room 402

Abstract: This will be survey talk about connections between the topology of a manifold and its group of symmetries. I will illustrate this theme by discussing finite group actions on compact manifolds, such as spheres or products of spheres, and infinite discrete groups acting properly discontinuously on non-compact manifolds, such as products of spheres and Euclidean spaces.

Dr. Andras Nemethi
Hungarian Academy of Sciences

Lattice Cohomology and the Geometric Genus of Surface Singularities

Monday, April 14, 2014, 5:00pm
Ungar Room 402

Abstract: The link of a complex normal surface singularity is an oriented 3-manifold. We will review the definition of the lattice cohomology associated with such a plumbed 3-manifold. It makes connection between low-dimensional topology and singularity theory. E.g., its Euler characteristic is the Seiberg-Witten invariant of the link, while in 'nice' cases it also provides the geometric genus of the analytic structure. We will focus on such connections and interplays, together with a historical overview of the topological characterizations of the geometric genus.