Graduate Mathematics Courses

MAA 5306 — INTRODUCTION TO REAL ANALYSIS (3)

Differentiation, Riemann-Stieltjes integrals, uniform convergence, Fourier series, special functions.

Prerequisite(s): MAA 4211

Note: This course is taught in the spring semester.

MAA 5307 — REAL ANALYSIS I (3)

Lebesgue measure and integration on the real line, classical Banach spaces.

Prerequisite(s): MAA 5306

Note: This course is taught in the fall semester.

MAA 5405 — APPLIED COMPLEX VARIABLES (3)

Complex numbers, analytic and harmonic functions, series, contour integrals, residue theory, conformal mappings; a survey course emphasizing techniques and applications.

Prerequisite(s): MAP 2302

Note: This course is taught in the spring semester of even years.

MAA 5616 — REAL ANALYSIS II (3)

Banach spaces, measure and integration, Riesz Representation Theorem, Radon-Nikodym Theorem.

Prerequisite(s): MAA 5307

Note: This course is taught in the spring semester.

MAA 5XXX — APPROXIMATION THEORY (3)

Approximation in normed spaces, best approximation, alternation points, existence and unicity, Weierstrass' theorem, Weierstrass-Stone theorem, smoothness spaces, approximation and smoothness, direct and converse inequalities, Jackson's theorem, Stechkin's theorem, polynomial approximation and polynomial inequalities, approximation by operators, rational approximation, spline approximation, wavelets.

Prerequisite(s): MAA 5307

Note: This course is taught in the fall semester of odd years.

MAA 5XXX — HARMONIC ANALYSIS (3)

Fourier series, conjugate series, L² theory, convergence and summability, Fourier transforms, Hilbert transform, singular integrals, H^p spaces, inner functions, atomic decomposition, maximal functions and maximal inequalities, harmonic analysis in Rⁿ.

Prerequisite(s): MAA 5307

Note: This course is taught in the spring semester of even years.

MAA 5XXX — ORTHOGONAL POLYNOMIALS AND SPECIAL FUNCTIONS I (3)

Special functions, orthogonal polynomials and their q-analogues, as well as moment problems and their applications to spectral theory of Schrödinger operators.

Prerequisite(s): MAA 5307

Note: This course is taught in the fall semester of even years.

MAA 5XXX — ORTHOGONAL POLYNOMIALS AND SPECIAL FUNCTIONS II (3)

Functional analytic, function theoretic and potential theoretic techniques and their applications to various problems in the theory of orthogonal polynomials and extremal problems.

Prerequisite(s): MAA 5307

Note: This course is taught in the spring semester of odd years.

MAA 6406 — COMPLEX ANALYSIS I (3)

Linear transformations, analytic functions, conformal mapping, Cauchy's theorem and applications, power series, partial fractions and factorization, elementary Riemann surfaces, Riemann mapping theorem.

Prerequisite(s): MAA 5405 or CI

Note: This course is taught in the fall semester of even years.

MAA 6407 — COMPLEX ANALYSIS II (3)

Conformal mappings, normal families, Picard's theorem, univalent functions, extremal properties, elliptic functions, approximation theory, Riemann surfaces.

Prerequisite(s): MAA 6406 or CI

Note: This course is taught in the spring semester of odd years.

MAA 6506 — FUNCTIONAL ANALYSIS I (3)

Normed linear spaces and topological vector spaces, open mapping, closed graph, and Hahn-Banach Theorem, UB principle, compact operators, dual spaces.

Prerequisite(s): MAA 5307 & MAS 5107 or CI

Note: This course is taught in the fall semester of odd years.

MAA 6507 — FUNCTIONAL ANALYSIS II (3)

Hilbert spaces, spectral theory, and other topics.

Prerequisite(s): MAA 6506

Note: This course is taught in the spring semester of even years.

MAD 5101 — LISP: PROGRAMMING WITH ALGEBRAIC APPLICATIONS (3)

Programming in LISP, functional languages, foundations of the Lambda Calculus, and algebraic applications (theorem proving and game playing).

Prerequisite(s): MAP 2302 or MAS 4301 or CI

Note: This course is taught in the spring semester of odd years.

MAD 5305 — GRAPH THEORY (3)

Brief introduction to classical graph theory (4-color theorem, etc.), directed graphs, connected digraphs, condensations, incidence matrices, Polya's Theorem, networks.

Prerequisite(s): CI

Note: This course is taught in the spring semester of even years.

MAD 5XXX — NUMERICAL ANALYSIS (3)

Interpolation and quadrature, finite differences, numerical solution of algebraic and transcendental equations, numerical solution of differential equations, computer techniques.

Prerequisite(s): MAP 2302 and MAS 3105.

Note: This course is taught in the fall semester of odd years.

MAD 6206 — COMBINATORICS I (3)

Elementary counting principles, distributions, sets, multisets, partitions of sets and integers, generating functions and recurrences, graph theory, probabilistic methods.

Prerequisite(s): MAS 3105 and MAS 4301 or CI

Note: This course is taught in the fall semester of odd years.

MAD 6207 — COMBINATORICS II (3)

Combinatorics of finite sets: posets, hypergraphs and external problems, matroids, block designs, Möbius inversion for partially ordered sets, Polya's enumeration theory.

Prerequisite(s): MAS 5311 and MAD 6206 or CI

Note: This course is taught in the spring semester of even years.

MAD 6510 — ADVANCED THEORY OF COMPUTATION (3)

Advanced topics in theoretical computer science, such as recursion theory, process theory, computational or descriptive complexity theory, abstract computations, or related topics.

Prerequisite(s): CI

Note: This course is taught in the spring semester of even years.

MAD 6616 — AUTOMATA THEORY (3)

Deterministic and non-deterministic finite automata, regular and other languages, Turing machines and related machine models, cellular automata and other network models.

Prerequisite(s): MGF 3301 or MAS 4301 or CI

Note: This course is taught in the spring semester of odd years.

MAD 6617 — TOPICS IN ABSTRACT STRUCTURES (3)

Advanced topics in combinatorial structures, logical structures and models, algebraic structures, set theory and topology, or related topics.

Prerequisite(s): CI

Note: This course is taught in the fall semester of odd years.

MAD 6XXX — TOPICS IN COMBINATORICS (3)

Advanced topics in combinatorics, combinatorial analysis, combinatorial enumeration (including algebraic methods, probabilistic methods, etc.), or related fields.

Prerequisite(s): CI

Note: This course is taught in the fall semester of even years.

MAE 5875 — ABSTRACT ALGEBRA FOR TEACHERS (3)

Groups, fields, vector spaces as they relate to high school algebra and geometry. (No credit for Mathematics majors.)

Prerequisite(s): MAS 3105 and MAS 4301 and Bachelor's degree or CI

Note: This course is not currently taught.

MAE 5877 — MATHEMATICAL ANALYSIS FOR TEACHERS (3)

Limits, continuity, derivatives, differentials. (No credit for Mathematics majors.)

Prerequisite(s): MAC 2313 and Bachelor's degree or CI

Note: This course is not currently taught.

MAP 5316 — NONLINEAR ANALYSIS I (3)

Sobolev Spaces, Degree Theories, Fixed point Theory, Operators of Monotone Type, Applications to Elliptic/Parabolic PDEs.

Prerequisite(s): MAA 4211 or CI

Note: This course is taught in the fall semester of even years.

MAP 6317 — NONLINEAR ANALYSIS II (3)

Convexity, Variational Inequalities, Minimizers, Approximations, Applications to Elliptic/Parabolic PDEs.

Prerequisite(s): MAP 5316 or CI

Note: This course is taught in the spring semester of odd years.

MAP 5345 — APPLIED PARTIAL DIFFERENTIAL EQUATIONS (3)

Separation of variables, the heat equation, wave equation, Laplace's equation, classification, Green's functions with emphasis on applications.

Prerequisite(s): MAP 5407 or CI

Note: This course is taught in the spring semester of odd years.

MAP 5407 — METHODS OF APPLIED MATHEMATICS (3)

Sturm-Liouville theory, Fourier series, Green's functions, matrix methods for linear systems of ordinary differential equations, and topics from calculus of variations, control theory, numerical solutions of differential equations.

Prerequisite(s): MAP 2302 or CI

Note: This course is taught in the fall semester of even years.

MAP 5XXX — DYNAMICAL SYSTEMS I (3)

From differential equations to dynamical systems, introduction to chaos, local bifurcations, hyperbolic sets, averaging method, Melnikov method.

Prerequisite(s): MAA 4211, MAA 5306, MAP 2302, and MAS 3105

Note: This course is taught in the fall semester of even years.

MAP 6205 — CONTROL THEORY AND OPTIMIZATION (3)

Projection theorems and minimum norm problems, convex analysis, duality principle, constrained optimization, finite dimensional linear systems, controllability, optimal control and Pontryagin maximum principle.

Prerequisite(s): MAP 2302, MAS 3105, and MAA 5307 or CI

Note: This course is taught in the spring semester of even years.

MAP 6336 — THEORY OF ORDINARY DIFFERENTIAL EQUATIONS (3)

Existence theorems, linear systems, perturbed linear systems, Floquet theory, stability, boundary value problems, two-dimensional equilibria and limit cycles, Poincaré-Bendixson theory, nonlinear second-order equations, and Runge-Kutta method.

Prerequisite(s): MAP 2302 and MAA 4211 or CI

Note: This course is taught in the fall semester of odd years.

MAP 6356 — PARTIAL DIFFERENTIAL EQUATIONS (3)

Advanced topics from: elliptic boundary value problems, semigroup theory, Sobolev spaces, degree theory, regularity, evolution equations.

Prerequisite(s): MAP 5345 and MAA 5307 or CI

Note: This course is taught in the fall semester of odd years.

MAP 6XXX — DYNAMICAL SYSTEMS II (3)

Nonlinear evolutionary equations, global attractors, inertial manifolds, approximations of global dynamics, and applications to reaction-diffusion equations, nonlinear wave equations, and Navier-Stokes equations.

Prerequisite(s): MAA 5307, MAA 6506, and MAP 5345 or MAP 6356

Note: This course is taught in the spring semester of odd years.

MAS 5107 — ADVANCED LINEAR ALGEBRA (3)

Finite-dimensional vector spaces over arbitrary fields, dual spaces, canonical forms for linear transformations, inner product spaces, orthogonal, unitary, and self-adjoint operators and quadratic forms.

Prerequisite(s): MAS 3105 and MAS 4301; Co-requisite(s): MAS 5311

Note: This course is taught in the fall semester.

MAS 5215 — NUMBER THEORY (3)

Fundamental theorem of arithmetic, modular arithmetic, Chinese remainder theorem, Mersenne primes, perfect numbers, Euler-Fermat theorem, pseudoprimes, primitive roots, laws of quadratic reprocity, factorization and primality testing algorithms.

Prerequisite(s): MAS 3105 and MAS 4301 or CI

Note: This course is taught in the spring semester of odd years.

MAS 5311 — ALGEBRA I (3)

Group theory: Sylow theorems; classification of groups of small order. Ring theory: ideals, quotient rings, polynomial rings, Euclidean domains, principal ideal domains and unique factorization.

Prerequisite(s): MAS 3105 and MAS 4301 or CI

Note: This course is taught in the fall semester.

MAS 5312 — ALGEBRA II (3)

Finitely generated modules over a principal ideal domain, basic field theory, finite fields, Galois theory.

Prerequisite(s): MAS 5311 or CI

Note: This course is taught in the spring semester.

MAT 5932 — SELECTED TOPICS (1-4)

Each course covers a single topic outside the usual curriculum.

Prerequisite(s): CI

MAT 6908 — INDEPENDENT STUDY (1-19 Var.)

Independent study in which student must have a contract with an instructor. Rpt. S/U.

MAT 6911 — DIRECTED RESEARCH (1-19 Var.)

Rpt. S/U

Prerequisite(s): Master's degree

MAT 6932 — SELECTED TOPICS (1-4)

Each course covers a single topic outside the usual curriculum.

Prerequisite(s): CI

MAT 6939 — GRADUATE SEMINAR (1-4)

Direction of this seminar is by a faculty member. Students are required to present research papers from the literature. S/U

MAT 6971 — THESIS: MASTER'S (1-19 Var.)

Rpt. S/U

Prerequisite(s): CI

MAT 7912 — DIRECTED RESEARCH (1-19 Var.)

Rpt. S/U

Prerequisite(s): Ph.D. level

MAT 7980 — DISSERTATION: DOCTORAL (1-19 Var.)

Rpt. S/U

Prerequisite(s): Admission to Candidacy

MHF 5306 — FOUNDATIONS OF MATHEMATICS AND COMPUTING I (3)

Recursion and computability theory, predicate calculus, incompleteness.

Prerequisite(s): MAS 4301 or CI

Note: This course is taught in the fall semester of even years.

MHF 5402 — EARLY HISTORY OF MATHEMATICS (3)

A study of the history and development of mathematics and its cultural impact from the formation of number systems to the Renaissance.

Prerequisite(s): MAC 2312

Note: This course is taught in the fall semester of both even and odd years and in the spring semester of even years.

MHF 5405 — HISTORY OF MODERN MATHEMATICS (3)

Traces the development of mathematical ideas from the Renaissance to the 19th century. Open to non-majors.

Prerequisite(s): MAC 2313

Note: This course is taught in the spring semester of odd years.

MHF 6307 — FOUNDATIONS OF MATHEMATICS AND COMPUTING II (3)

Classical model theory, completeness, set theory.

Prerequisite(s): MAS 4301 or CI

Note: This course is taught in the spring semester of odd years.

MTG 5326 — DIFFERENTIAL GEOMETRY (3)

Exterior calculus, differentiable manifolds, integration of differential forms, surfaces in 3-space, covariant derivative, curvature, matrix groups.

Prerequisite(s): MAA 4211 and MAS 3105

Note: This course is taught in the spring semester of odd years.

MTG 5316 — TOPOLOGY I (3)

Topological spaces, continuity, homeomorphisms, connectedness, compact spaces, separation axioms, product spaces.

Prerequisite(s): MAA 4211

Note: This course is taught in the fall semester.

MTG 5317 — TOPOLOGY II (3)

The fundamental group; elements of homotopy theory and homology theory.

Prerequisite(s): MTG 5316

Note: This course is taught in the spring semester.