Graduate Course Descriptions

4000 Level - Electives

MTH 4314 - Abstract Algebra Prerequisite(s): A grade of C or above in MTH 3312. Fundamentals of group, ring, and field theory. Topics include permutation groups, group and ring homomorphisms, direct products of groups and rings, quotient objects, integral domains, field of quotients, polynomial rings, unique factorization domains, extension fields, and finite fields.

MTH 4322 - Numerical Analysis (Cross-listed as CSI 4322) Prerequisite(s): A grade of C or above in MTH 2321. Numerical evaluation of derivatives and integrals, solution of algebraic and differential equations, and approximation theory.

MTH 4326 - Advanced Calculus I Prerequisite(s): A grade of C or above in MTH 3323. Sequences and series of functions, multiple integrals, improper multiple integrals, functions of several variables, extreme value problems, and implicit function theorems.

MTH 4327 - Advanced Calculus II Prerequisite(s): A grade of C or above in MTH 4326. Line and surface integrals, Green, Gauss, Stokes theorems with applications, Fourier series and integrals, functions defined by integrals, introduction to complex functions.

MTH 4328 - Numerical Linear Algebra (Cross-listed as CSI 4328) Prerequisite(s): A grade of C or above in MTH 2311 and 3324. Numerical methods for solution of linear equations, eigenvalue problems, and least squares problems, including sparse matrix techniques with applications to partial differential equations.

MTH 4329 - Theory of Functions of a Complex Variable Prerequisite(s): A grade of C or above in MTH 2321. Number systems: the complex plane; fractions, powers, and roots; analytic functions; elementary functions; complex integration; power series; mapping by elementary functions; calculus of residues.

5000 Level - Core Courses

MTH 5310 - Advanced Abstract Algebra I Prerequisite(s): MTH 4314 and consent of the instructor. Finite groups, Sylow theorems, nilpotent and solvable groups, principal ideal domains, unique factorization domains, and sub rings to algebraic number fields.

MTH 5311 - Advanced Abstract Algebra II Prerequisite(s): MTH 5310. Field theory, Galois theory, modules, finitely generated modules, principal ideal domains, homological methods, and Wedderburn-Artin theorems.

MTH 5323 - Theory of Functions of Real Variables I Prerequisite(s): MTH 4327. Borel sets, measure and measurable sets, measurable functions, and the Lebesque integral.

MTH 5324 - Theory of Functions of Real Variables II Prerequisite(s): MTH 5323. Function spaces, abstract measure, and differentiation.

MTH 5330 - Topology Prerequisite(s): Graduate standing. Topological spaces, continuous functions, metric spaces, connectedness, compactness, separation axioms, Tychenoff theorem, fundamental group, covering spaces, metrization theorems.

MTH 5331 - Algebraic Topology I Prerequisite(s): MTH 5330. Homology theory, simplicial complexes, topological invariance, relative homology, Eilenberg-Steenrod axioms, singular homology, CW complexes.

MTH 5350 - Complex Analysis Prerequisite(s): MTH 4327. Complex numbers, complex functions, analytic functions, linear fractional transformations, complex integration, Cauchy's formula, residues, harmonic functions, series and product expansions, Gamma function, Riemann mapping theorem, Dirichlet problem, analytic continuation.

MTH 5360 - Applied Mathematics I Prerequisite(s): Graduate standing. Dynamical systems (ODE and PDE, discrete and continuous), linear and nonlinear systems theory, transform methods, control theory and optimization, calculus of variations, stability theory.

MTH 5361 - Applied Mathematics II Prerequisite(s): Graduate standing. Eigenvalue theory, projections for linear equations iterations and multilevel methods, fast Fourier transforms, approximations of differential equations, grid adaption and numerical stability, weak solutions and Sobelov space, wavelets with applications.

5000 Level - Electives

MTH 5316 - Linear Algebra and Matrix Theory Prerequisite(s): MTH 3312. Matrix calculus, eigenvalues and eigenvectors, canonical forms, orthogonal and unitary transformations, and quadratic forms. Applications of these concepts. A course project is required and will be specified by the professor at the beginning of the course.

MTH 5325 - Theory of Differential Equations Prerequisite(s): MTH 3325 and 5323. Initial value problems for ordinary differential equations: existence, uniqueness, continuous dependence, stability analysis, oscillation theory, general linear systems, phase plane analysis, limit cycles and periodic solutions. Topics of current interest in dynamical systems.

MTH 5326 - Theory of Partial Differential Equations Prerequisite(s): MTH 5324 and 5325. Linear and quasilinear first order equations; shocks, characteristics, the Cauchy problem, elliptic, hyperbolic, and parabolic equations, maximum principles, Dirichlet problem, operators, Sobolev spaces, distributions.

MTH 5332 - Algebraic Topology II Prerequisite(s): MTH 5331. Cohomology theory, homology with coefficients, homological algebra, Kunneth theorem, duality in manifolds.

MTH 5340 - Differential Geometry Prerequisite(s): MTH 4327, 5316, and 5330. Differentiable manifolds, submanifolds, vector fields, tensor fields, integration on manifolds, Riemannian geometry.

MTH 4344 - Probability Theory Prerequisite(s): MTH 5323. Topics include the law of large numbers, central limit theorems, random walks, martingales, Markov chains, Egodic theorems, and Brownian motion.

MTH 5345 - Functional Analysis Prerequisite(s); MTH 5324. Banach spaces, Hilbert spaces, linear operators, and spectral theory.

MTH 5351 - Applications of Complex Analysis Prerequisite(s): MTH 5350. Poisson summation, Mellin transformation, zeta function of Riemann, special functions, zeta functions associated with eigenvalue problems, heat kernel, asymptotic expansion of the heat kernel, meromorphic structure of zeta functions, theta functions, elliptic functions.

MTH 5375 - Linear Programming Prerequisite(s): MTH 2311 or consent of instructor. Introduction to the theory and applications of linear programming, including the simplex algorithm, duality, sensitivity analysis, parametric linear programming, integer programming, with applications to transportation and allocation problems and game theory. A course project is required and will be specified by the professor at the beginning of the course.

MTH 5376 - Nonlinear Programming Theory and algorithms for the optimization of unconstrained problems including gradient and Quasi-Newton methods; and constrained problems to include feasible direction methods, Lagrange multipliers, and Karush-Kuhn-Tucker conditions. Students must have a knowledge of linear algebra, third-semester calculus.

MTH 5390 - Special Problems in Mathematics Project course for the project option in the M.S. degree.

MTH 5V91 - Special Topics in Algebra for Graduates 1 to 3 sem. hrs. May be repeated for credit with instructor's consent.

MTH 5V92 - Special Topics in Analysis for Graduates 1 to 3 sem. hrs. May be repeated for credit with instructor's consent.

MTH 5V93 - Special Topics in Mathematics for Education Students 1 to 3 sem. hrs. Prerequisite(s): Consent of departmental chair and the course instructor. May be repeated for credit for a maximum of nine semester hours if under different topics. Maximum 9 sem. hrs.

MTH 5V96 - Special Topics in Graph Theory Prerequisite(s): Consent of instructor. Introduction to graph theory; Euler tours, matching, connectivity, planar graphs, coloring, random graphs. Additional topics may vary by semester.

 

Master's Thesis

MTH 5V99 - Thesis 1 to 6 sem. hrs. Credit to be given for the amount of work done. In no case will less than six semester hours be accepted. Maximum 10 sem. hrs.

6000 Level - Electives

MTH 6310 - Commutative Rings and Modules Prerequisite(s): MTH 5311. Noetherian rings, quotient rings, primary decomposition, integral dependence and valuations, Dedekind domains, and discrete valuation rings, completions, dimension theory.

MTH 6311 - Non-Commutative Rings and Modules Prerequisite(s): MTH 6310. Semi-simple rings and modules, radicals, chain conditions, decomposition of modules, Goldie's theorem, density and Morita theory.

MTH 6312 - Abelian Group Theory Prerequisite(s): MTH 5311. An introduction to the fundamental theory of torsion, torsion-free, and mixed abelian groups.

MTH 6322 - Approximation Theory Prerequisite(s): MTH 4322 and 4328. Approximation of real functions including polynomial and rational interpolation, orthogonal polynomials, Chebyshev approximation, the fast Fourier transform, splines, wavelets, and tensor product interpolation.

MTH 6325 - Numerical Solutions of Partial Differential Equations Prerequisite(s): MTH 4322 and 4328. Finite difference and finite element methods for elliptic, parabolic, and hyperbolic problems in partial differential equations.

MTH 6340 - Compact Lie Groups Prerequisite(s): MTH 5310 and 5340. Compact Lie groups, Lie algebras, representation theory, orthogonality relations, Peter Weyl theorem, structure theory, roots, Weyl character formula.

MTH 6341 - Lie Algebras Prerequisite(s): MTH 5310 and 5316. Lie algebras, semisimple Lie algebras, root systems, conjugecy theorems, classification theorem, representation theory, Chevalley algebras.

MTH 6342 - Semisimple Lie Groups Prerequisite(s): MTH 6340 and 6341. Structure theory for noncompact groups, induced representations, tempered representations, Langland's classification of irreducible admissible representations.

MTH 6350 - Set and Model Theory Prerequisite(s): MTH 5311. Propositional and predicate calculus, Loewenheim-Skolem theorems, properties of ultraproducts, model completeness, Goedel's completeness/incompleteness proofs, infinitary language, axioms of set theory, ordinal and cardinal arithmetic, models of set theory and large cardinals.

MTH 6362 - Fourier Analysis on Euclidean Spaces Prerequisite(s): Graduate standing. Introduction to Fourier Analysis; singular integrals, pseudodifferential operators, Lp estimates, and applications to partial differential equations. Additional topics may vary by semester.

MTH 6363 - Analytic Number Theory Prerequisite(s): Graduate standing. Unique factorization, quadratic reciprocity, arithmetical functions, Dirichlet series, distribution of prime numbers. Additional topics may vary by semester.

MTH 6364 - Algebraic Number Theory Prerequisite(s): Graduate standing. Class field theory, cyclotomic fields, p-adic L functions, and elliptic curves. Additional topics may vary by semester.

MTH 6365 - Topics in Combinatorics Prerequisite(s): Graduate standing. Graphs, Ramsey theory, extremal set theory, generating functions, and partitions. Additional topics may vary by semester.

MTH 6366 - Topic in Noncommutative Analysis Prerequisite(s): Graduate standing. Introduction to Positive definite matrices, Matrices of the trace class and the Schatten-p classes, Lp spaces associated with von Neumann algebras, Markov semigroup of operators, Noncommutative Hardy/BMO spaces, Free Fourier Multipliers, Shannon entropy and Fisher information. Additional topics may vary by semester.

MTH 6367 - Topics in Complex Analysis: Elliptic and Automorphic Functions Prerequisite(s): Graduate standing. Topics which may vary by semester include periodic meromorphic functions, elliptic Weierstrass functions, elliptic Jacobi functions, modular functions, Picard’s theorems, modular group, automorphic functions, applications to completely integrable systems.

MTH 6368 - Topics in Spectral Theory I Prerequisite(s): Graduate standing. Maximal and minimal operators, Weyl-Titchmarsh theory, spectral functions for 2nd order ODE operators, eigenfunction expansions. Topics may vary by semester.

MTH 6369 - Topics in Operator Theory II: Compact Operators Prerequisite(s): Graduate standing. Compact operators, canonical decomposition of compact operators, singular values, l^p-based Schatten-von Neumann trace ideals, (regularized) Fredholm determinants, applications to the spectral theory of differential operators. Topics may vary by semester.

MTH 6V00 - Graduate Research 1 to 10 sem. hrs. Prerequisite(s): Graduate standing. Grants full-time status. For research credit prior to admission to candidacy for an advanced degree. May be repeated for credit through 45 hours.

MTH 6V13 - Advanced Topics in Algebra 1 to 3 sem. hrs. Prerequisite(s): Consent of instructor. May be repeated for credit with instructor's consent if under different topic. Maximum 12 sem. hrs.

MTH 6V23 - Advanced Topics in Analysis 1 to 3 sem. hrs. Prerequisite(s): Consent of instructor. May be repeated for credit with instructor's consent if under different topic. Maximum 12 sem. hrs.

MTH 6V24 - Advanced Topics in Applied Mathematics 1 to 3 sem. hrs. Prerequisite(s): Consent of instructor. May be repeated for credit with instructor's consent if under different topic. Maximum 12 sem. hrs.

MTH 6V28 - Advanced Topics in Numerical Analysis 1 to 3 sem. hrs. Prerequisite(s): Consent of instructor. May be repeated for credit with instructor's consent if under different topic. Maximum 12 sem. hrs.

MTH 6V30 - Advanced Topics in Topology Prerequisite(s): Consent of instructor. Topology is the study of abstract mathematical spaces with the ultimate goal of finding invariants which are preserved under continuous transformation. Along with algebra and analysis, topology is one of the main areas of modern mathematics and as such every doctoral program in mathematics should have a course designed to cover the more advanced aspects of topology. This course would be taken primarily by doctoral candidates with a strong interest in topology.

MTH 6V43 - Advanced Topics in Representation Theory 1 to 3 sem. hrs. Prerequisite(s): Consent of instructor. May be repeated for credit with instructor's consent if under different topic. Maximum 12 sem. hrs.

 

Ph.D. Thesis

6V99 Dissertation 1 to 12 sem. hrs. Grants full time status. Requires passing the Preliminary Exam. Supervised research for the doctoral dissertation.