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College of Natural Science and Mathematics

Courses: Mathematics (MATH)

6300: Cardinal and Ordinal Numbers
Cr. 3 per semester. (3-0). Prerequisite: graduate standing or consent of instructor. Ordinal and cardinal number theory; transfinite induction, equivalence of the axiom of choice, well-ordering principle, Zorn's lemma, and Hausdorff maximality principle; uses of generalized continuum hypothesis.

6302:6303 Modern Algebra
Cr. 3 per semester. (3-0). Prerequisite: MATH 4333 or MATH 4378, or consent of instructor. Topics from the theory of groups, rings, fields, and modules with special emphasis on universal constructions.

6304: Theory of Matrices
Cr. 3. (3-0). Prerequisite: consent of instructor. Emphasis on canonical forms and finite dimensional spectral theory.

6306:6307: Graph Theory and Combinatorics
Cr. 3 per semester. (3-0). Prerequisite: graduate standing or consent of instructor. Basic graph-theoretical and combinatorial methods and algorithms with their applications.

6315:7315: Master's Tutorial
Cr. 3 per semester. Prerequisite: consent of instructor. May be taken concurrently. Open only to those choosing the non-thesis option for the M.S. degree. Special topics selected by student and instructor to be no less demanding than writing a thesis.

6320:6321: Theory of Functions of a Real Variable
Cr. 3 per semester. (3-0). Prerequisite: MATH 4332 or consent of instructor. Lebesque measure and integration, differentiation of real functions, functions of bounded variation, absolute continuity, the classical Lp spaces, general measure theory, and elementary topics in functional analysis.

6322:6323: Theory of Functions of a Complex Variable
Cr. 3 per semester. (3-0). Prerequisite: MATH 4331 or consent of instructor. Geometry of the complex plane, mappings of the complex plane, integration, singularities, spaces of analytic functions, special functions, analytic continuation, and Riemann surfaces.

6324:6325: Differential Equations
Cr. 3 per semester. (3-0). Prerequisite: MATH 4331. General theories, topics in ordinary and partial differential equations, and boundary value problems.

6326:6327: Partial Differential Equations
Cr. 3 per semester. (3-0). Prerequisite: MATH 4331 or consent of instructor. Existence and uniqueness theory in partial differential equations; generalized solutions and convergence of approximate solutions to partial differential systems.

6340:6341: Algebraic Topology
Cr. 3 per semester. (3-0). Prerequisites: MATH 3330 and MATH 4337, or consent of instructor. An introduction to homology, cohomology, and homotopy groups of spaces, including the homology of simplicial complexes.

6342:6343: Point Set Topology
Cr. 3 per semester. (3-0). Prerequisites: MATH 4331 and MATH 4337; or consent of instructor. An axiomatic development of point set topology; connectivity, separability, cartesian products, topological equivalence, compactness, metrizability, well-ordering, and inner limiting sets.

6344: Topological Semigroups
Cr. 3. (3-0). Prerequisites: MATH 3330 and MATH 4337, or consent of instructor. Elementary properties, Green's relations, the minimal ideal of a compact semigroup, general properties of compact connected semigroups with identity, semilattices, and semigroups of matrices.

6346: Topological Groups
Cr. 3. (3-0). Prerequisites: MATH 3330 and MATH 4337, or consent of instructor. Structure of topological groups. Topics include separation axioms, metrization, quotients, direct products, Haar integration, duality, Lie groups, and transformation groups.

6360:6361: Applicable Analysis
Cr. 3 per semester. (3-0). Prerequisite: graduate standing or consent of instructor. Solvability of finite dimensional, integral, differential, and operator equations, contraction mapping principle, theory of integration, Hilbert and Banach spaces, and calculus of variations.

6366:6367: Optimization and Variational Methods
Cr. 3 per semester. (3-0). Prerequisites: MATH 4331 and MATH 4377, or consent of instructor. Constrained and unconstrained finite dimensional nonlinear programming, optimization and Euler-Lagrange equations, duality, and numerical methods. Optimization in Hilbert spaces and variational problems. Euler-Lagrange equations and theory of the second variation. Application to integral and differential equations.

6370:6371: Numerical Analysis
Cr. 3 per semester. (3-0). Prerequisite: graduate standing in mathematics or consent of instructor. Ability to do computer assignments. Topics selected from numerical linear algebra, nonlinear equations and optimization, interpolation and approximation, numerical differentiation and integration, numerical solution of ordinary and partial differential equations.

6372: Numerical Ordinary Differential Equations
Cr. 3. (3-0). Prerequisite: MATH 3431 and MATH 4331, or consent of instructor. Single- and multistep methods for initial value problems, special methods for stiff systems, stability theory and error analysis, shooting and other methods for two-point boundary value problems.

6374: Numerical Partial Differential Equations
Cr. 3. (3-0). Prerequisite: MATH 6371 or consent of instructor. Finite difference, finite element, collocation and spectral methods for solving linear and nonlinear elliptic, parabolic, and hyperbolic equations and systems with applications to specific problems.

6375: Methods of Approximating Partial Differential Equations
Cr. 3. (3-0). Prerequisite: MATH 4336 or consent of instructor. Theoretical error statements for finite difference and finite element methods in partial differential equations, superconvergence, local error estimates in mesh refinements, domain decompositions, and multi-level schemes for numerical solution of discretized equations.

6376: Numerical Linear Algebra
Cr. 3. (3-0). Prerequisite: MATH 6371 or consent of instructor. Advanced techniques for the direct and iterative solution of linear systems, especially sparse systems, and for the solution of eigenvalue problems.

6378: Parallel Scientific Computing
Cr. 3. (3-0). Prerequisites: a programming course, computing experience, and consent of instructor. Multiple processors solving scientific problems. Issues related to programming languages and algorithms for scientific computation; includes a parallel programming project.

6380:6381: Mathematical Probability
Cr. 3 per semester. (3-0). Prerequisite: MATH 6320 or consent of instructor. Random variables, conditional expectation, weak and strong laws of large numbers, central limit theorem, Kolmogorov extension theorem, martingales, separable processes, and Brownian motion.

6382:6383: Probability Models and Mathematical Statistics
Cr. 3 per semester. (3-0). Prerequisites: MATH 3334, MATH 3338 and MATH 4378, or consent of instructor. A survey of probability theory, probability models, and statistical inference. Includes basic probability theory, stochastic processes, parametric and nonparametric methods of statistics.

6387: Nonparametric Statistics
Cr. 3. (3-0). Prerequisites: MATH 4332 and either MATH 4386 or MATH 6383. General theory of nonparametric statistical inference on location and scale, linear rank statistics, M-estimators, Hodges-Lehmann estimators.

6388:6389: Statistical Inference and Multivariate Analysis
Cr. 3 per semester. (3-0). Prerequisites: MATH 4332 and either MATH 4386 or MATH 6383. General theory of parameter estimation and hypothesis testing, multivariate normal distribution and associated sampling distributions and tests for mean vectors and covariance hypotheses, discriminant analysis, covariance models and time series models.

6394: Selected Topics in Algebra
Cr. 3. (3-0). Prerequisite: consent of instructor. May be repeated with approval of chair.

6395: Selected Topics in Analysis
Cr. 3. (3-0). Prerequisite: consent of instructor. May be repeated with approval of chair.

6396: Selected Topics in Topology
Cr. 3. (3-0). Prerequisite: consent of instructor. May be repeated with approval of chair.

6397: Selected Topics in Mathematics
Cr. 3. (3-0). Prerequisite: consent of instructor. May be repeated with approval of chair.

6399:7399: Master's Thesis
Cr. 3 per semester.

7304: Theory of Groups
Cr. 3. (3-0). Prerequisite: MATH 6302 or consent of instructor. Sylow theorems, the Remak-Krull-Schmidt theorem, solvable and nilpotent groups, free groups and free products, extensions, infinite abelian groups, and homological algebra for Z-modules.

7306:7307: Structure of Rings and Modules
Cr. 3 per semester. (3-0). Prerequisite: MATH 6302 or consent of instructor. Study of structure of rings and modules.

7308: Boolean Algebras with Applications
Cr. 3. (3-0). Prerequisites: Graduate standing or consent of instructor. The lattice and ring theoretic foundations of boolean algebras and Stone's topological duality theory. Particular algebras from topology, analysis, logic, and computer science.

7320:7321: Functional Analysis
Cr. 3 per semester. (3-0). Prerequisite: MATH 6320 or consent of instructor. Linear topological spaces, Banach and Hilbert spaces, duality, and spectral analysis.

7324:7325: Bifurcation Theory
Cr. 3. (3-0). Prerequisites: MATH 3334, MATH 4362 and MATH 4378, or consent of instructor. Course material includes singularity theory, imperfect bifurcation, normal forms, classification by codimension, Liapunov Schmidt reduction, and Hopf bifurcation. The first semester concentrates on singularity theory in systems without symmetry and the second on systems with symmetry. Elements of representation theory, pattern formation and applications will be discussed.

7342: Advanced Point Set Topology
Cr. 3. (3-0). Prerequisite: MATH 6343 or consent of instructor. Upper semicontinuous collections, indecomposable continua, covering theorems, and metrization problems.

7344: Dimension Theory
Cr. 3. (3-0). Prerequisite: MATH 6343 or consent of instructor. The topological study of dimension in metric spaces.

7350;7351: Geometry of Manifolds
Cr. 3 per semester. (3-0). Prerequisites: MATH 3431 and MATH 3333, or consent of instructor. Manifolds and tangent bundles, submanifolds and imbeddings, integral manifolds, triangulation of manifolds, connections and holonomy; Riemannian geometry, surface theory, Morse theory, and G-structures.

7374: Math Theory of Finite Element Methods
Cr. 3. (3-0). Prerequisite: MATH 6320 or consent of instructor. Basic theory of existence, uniqueness, and convergence for finite element methods for approximating solutions of partial differential equations. Introduction to Sobolev spaces, variational formulation of boundary value problems, construction of finite elements, and polynomial approximation in Sobolev spaces.

7394: Selected Topics in Applied Mathematics
Cr. 3. (3-0). Prerequisite: consent of instructor. May be repeated with approval of chair.

7396: Selected Topics in Numerical Analysis
Cr. 3. (3-0). Prerequisite: consent of instructor. May be repeated with approval of chair.

7397: Selected Topics in Probability
Cr. 3. (3-0). Prerequisite: consent of instructor. May be repeated with approval of chair.

8198:8298:8398:8498:8598:8698: Doctoral Research
Cr. 1-6 per semester. Prerequisites: consent of instructor and approval of chair.

8399:8699:8999: Doctoral Dissertation
Cr. 3, 6, or 9 per semester.