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**MTH 602: History of Mathematics, 3 credits.**

*Fall Semester*

The development of mathematics from its earliest beginnings through the first half of the twentieth century. Numeral systems, geometry, algebra, analysis and set theory.

*Prerequisite: Two courses in mathematics at the 200 level or above.*

**MTH 604: Foundations of Geometry, 3 credits.**

*Fall Semester*

Axiom systems and models of Euclidean and Non-Euclidean geometry.

*Prerequisite: MTH 230 or 309.*

**MTH 605: Theory of Numbers, 3 credits.**

*Offered By Announcement Only*

Divisibility, primes; congruences; quadratic resides and reciprocity; Diophantine equations. Applications to cryptography.

*Prerequisite: MTH 210 or 504 or 604.*

**MTH 606: Mathematical Logic, 3 credits.**

*Offered By Announcement Only*

Logics, truth, proof, logical consequence, model theory, formalization, and computation. Meta-theory of first-order logic, computability theory, and Gödel's incompleteness theorems. Related results by Church, Turing, and Tarski. Discussion of their philosophical significance.

*Prerequisite: MTH 230.*

**MTH 610: Linear Algebra, 3 credits.**

*Spring Semester*

Abstract vector spaces, bases and dimensions, linear maps, eigen values and eigenvectors, inner product spaces, operators, spectral theorems, canonical forms.

*Prerequisite: MTH 210; transition course in logical reasoning such as MTH 230 or 309 recommended but not required.*

**MTH 612: Elementary Complex Analysis, 3 credits.**

*Spring Semester*

Complex variables; conformal mapping, contour integration.

*Prerequisite: MTH 211 or 310.*

**MTH 613: Partial Differential Equations I, 3 credits.**

*Fall Semester*

Derivation, well posedness, and qualitative properties of initial value and boundary value problems for the heat, wave and Laplace equations. Energy methods, causality, maximum principles, heat kernels, Fourier series, and potential theory.

*Prerequisite: MTH 210, 310, and 311.*

**MTH 614: Partial Differential Equations II, 3 credits.**

*Spring Semester*

Continuation of MTH 613. Approximations of solutions, distributions and integral transform methods, spectral theory and scattering. Applications to physical problems. Nonlinear equations and phenomena.

*Prerequisite: MTH 513 or 613 or permission of the instructor.*

**MTH 615: Ordinary Differential Equations, 3 credits.**

*Fall Semester*

Linear systems, equilibria and periodic solutions, stability analysis, bifurcation, phase plane analysis, boundary value problems, applications to engineering and physics.

*Prerequisites: MTH 311 and either MTH 211 or 310.*

**MTH 616: Dynamics and Bifurcations, 3 credits.**

*Spring Semester*

Bifurcation of equilibria and periodic solutions, global theory of planar systems, planar maps, nonlinear vibrations, forced oscillations, chaotic solutions, Hamiltonian systems, applications to engineering and physics.

*Prerequisites: MTH 515 or 615 or permission of the instructor.*

**MTH 617: Data Structures and Algorithm Analysis, 3 credits.**

*Offered By Announcement Only*

Basic algorithmic analysis. Algorithmic strategies. Fundamental computing algorithms. Distributed algorithms. Cryptographic algorithms. Geometric algorithms.

*Prerequisite: MTH 309 (or 230).*

**MTH 620: Numerical Analysis I, 3 credits.**

*Fall Semester*

Topics from numerical linear algebra including solving systems of equations, LU, QR, and SVD factorizations, eigenvalues and eigenvectors, iterative methods, and applications.

*Prerequisite: MTH 320 or permission of department chair.*

**MTH 621: Numerical Analysis II, 3 credits.**

*Offered By Announcement Only*

Numerical solution of ordinary and partial differential equations.

*Prerequisite: MTH 320 or 520 or 620 or permission of department chairman.*

**MTH 624: Introduction to Probability Theory, 3 credits.**

*Fall Semester*

Probability spaces, random variables, expectation, limit theorems.

*Prerequisite: MTH 224 and 310.*

**MTH 625: Introduction to Mathematical Statistics, 3 credits.**

*Spring Semester*

Probability distributions, theory of sampling and hypothesis testing.

*Prerequisite: MTH 524 or 624.*

**MTH 627: Theory of Computing, 3 credits.**

*Offered By Announcement Only*

Sets, relations, and languages. Automata theory. Basic computability theory. Turing machines. The complexity classes P and NP.

*Prerequisite: MTH 309 or 461.*

**MTH 631: Topology I, 3 credits.**

*Fall Semester*

Set theory, topological spaces, compactness, connectedness, separation properties, quotient spaces, Tychonoff Theorem, compactification, Urysohn Lemma and Tietze Extension Theorem, function spaces.

*Prerequisite: Permission of department chairman.*

**MTH 632: Topology II, 3 credits.**

*Spring Semester*

Differential and topological manifolds, classical groups and associated manifolds, tangent and tensor bundles, vector fields, differential forms, transversality, Sard's theorem, Stokes' Theorem.

*Prerequisite: MTH 210 and 531 or 631.*

**MTH 633: Introduction to Real Analysis I, 3 credits.**

*Fall Semester*

Numerical sequences and series; continuity; differentiation; integration; sequences and series of functions; Fourier series; functions of several variables; implicit and inverse function theorems.

*Prerequisite: MTH 211 (or 310) and 230.*

**MTH 634: Introduction to Real Analysis II, 3 credits.**

*Spring Semester*

Continuation of MTH 633.

*Prerequisite: MTH 533 or 633.*

**MTH 640: Algorithm Design and Analysis, 3 credits.**

*Offered By Announcement Only*

Design techniques include divide-and-conquer, greedy method, dynamic programming, backtracking. Time and space complexity. Sorting, searching, combinatorial and graph algorithms.

*Prerequisite: MTH 517 or 617.*

**MTH 642: Statistical Analysis, 3 credits.**

*Fall Semester*

Statistical inference about one or two populations from interval, ordinal, and categorical data; analysis of variance; simple and multiple linear regression; designing research studies.

*Prerequisite: MTH 210, 224.*

**MTH 647: Introduction to Mathematical Finance, 3 credits.**

*Offered By Announcement Only*

Models of financial markets. Derivative securities: European and American options. Tools of mathematical finance: binomial trees, martingales, stopping times. Concepts of arbitrage and hedging. Risk-neutral valuation of financial derivatives; the Black-Scholes formula and its applications.

*Prerequisite: MTH 224.*

**MTH 648: Stochastic Calculus with Applications to Finance, 3 credits.**

An introduction to Stochastic Calculus developing the basic probabilistic techniques necessary to study analytic models of financial markets. Brownian motion and the stochastic integral, stochastic differential equations, the Black-Scholes formula, Girsanovs theorem and applications to option pricing.

**MTH 651: Introduction to Differential Geometry, 3 credits.**

*Offered By Announcement Only*

Geometry of curves and surfaces in Euclidean space. Local space curve theory, intrinsic and extrinsic curvature of surfaces, geodesics, parallelism, and differential forms.

*Prerequisite: MTH 210 and either MTH 211 or 310, or permission of instructor.*

**MTH 661: Abstract Algebra I, 3 credits.**

*Fall Semester*

Groups, rings; linear algebra; modules.

*Prerequisite: MTH 210 and permission of department chairman.*

**MTH 662: Abstract Algebra II, 3 credits.**

*Spring Semester*

Continuation of MTH 661.

*Prerequisite: MTH 561 or 661.*

**MTH 691-694: Topics in Mathematics, 1-3 credits each.**

*Offered By Announcement Only*

**MTH 709: Data Security and Cryptography, 3 credits.**

Access, information flow, and inference controls. Network security and management. Encryption algorithms. Cryptographic techniques.

*Prerequisite: MTH 517 or 617 or MTH 527 or 627 .*

**MTH 721: Mathematical Probability, 3 credits.**

Development of the measure-theoretic approach probability. Random variables, central limit theory, laws of large numbers, martingales.

*Prerequisite: Permission of department chairman.*

**MTH 722: Stochastic Processes, 3 credits.**

*Prerequisite: MTH 734.*

**MTH 725: Multivariate Analysis, 3 credits.**

Sampling theory for multivariate normal populations. Component and factor analysis. Stochastic difference equations.

*Prerequisite: MTH 525 or 625.*

**MTH 733, 734: Real Variables, 3 credits each.**

Lebesgue measure and the Lebesgue Integral for R^{1}, Banach Spaces. General measure theory, topological groups and Haar Measure.

*Prerequisite: MTH 532 or 632.*

**MTH 735, 736: Complex Variables, 3 credits each.**

Complex numbers line or transformation, analytic function, conformality. Cauchy's Theorem, representation theorems, harmonic functions.

*Prerequisite: MTH 531 or 631.*

**MTH 741, 742: Algebraic Topology, 3 credits.**

Homotopy, covering space, Eilenberg-Steenrod axioms for (co) homology theories, Mayer-Vietoris sequences, Universal Coefficient theorem, Kunneth formula, computations and applications.

*Prerequisite: MTH 532 or 632.*

**MTH 747: Computational Geometry, 3 credits.**

Algorithms for solving geometric problems arising from application domains including raphics, robotics, and GIS.

*Prerequisite: MTH 517 or 617 or permission of instructor.*

**MTH 751, 752: Differential Geometry, 3 credits each.**

**MTH 757: Lie Groups, 3 credits.**

**MTH 761, 762: Abstract Algebra, 3 credits each.**

*Prerequisite: MTH 562 or 662.*

**MTH 770: Directed Reading of Research, 2-4 credits.**

**MTH 780, 781: Topics in Analysis, 3 credits each.**

**MTH 782, 783: Topics in Topology, 3 credits each.**

**MTH 785: Topics in Algebra, 3 credits.**

**MTH 786, 787: Topics in Mathematics, 3 credits each.**

**MTH 792: Seminar, 1-2 credits.**

**MTH 820: Research in Residence, 0 credit.**

*Fall & Spring Semester & First & Second Summer Session*

To establish a residence for non-thesis master's students who are preparing for major examinations. Credit not granted. Regarded as full-time residence.

**MTH 830: Pre-Candidacy Doctoral Dissertation, 1-12 credits.**

*Fall & Spring Semester & First & Second Summer Session*

Credits earned in this course apply towards the 12 credit hour dissertation research requirement of the graduate school. The student will enroll for credit as determined by his/her dissertation advisor. Up to 12 hours may be taken in a regular semester, but not more than six in a summer session.

**MTH 840: Post-Candidacy Doctoral Dissertation, 1-12 credits.**

*Fall & Spring Semester & First & Second Summer Session*

Credits earned in this course apply towards the 12 credit hour dissertation research requirement of the graduate school. The student will enroll for credit as determined by his/her dissertation advisor. Up to 12 hours may be taken in a regular semester, but not more than six in a summer session.

**MTH 850: Research in Residence, 0 credit.**

*Fall & Spring Semester & First & Second Summer Session*

Used to establish research in residence for the Ph.D. after the student has been enrolled for the permissible cumulative total in appropriate doctoral research. Credit not granted. May be regarded as full-time residence as determined by the Dean of the Graduate School.