Graduate Courses: College of Natural Sciences and Mathematics

Mathematics

 

Courses: Mathematics (MATH)

Math 5310. History of Mathematics
Cr. 3 (3-0).  Prerequisites: Graduate standing or consent of instructor.  Mathematics of the ancient world, classical Greek mathematics, the development of calculus, notable mathematicians and their accomplishments.

5315: Graph Theory with Applications
Cr. 3. (3-0). Prerequisites: Graduate standing or consent of instructor. Introduction to fundamental concepts of graph theory. Does not apply towards the Master of Science in Mathematics or Applied Mathematics.

5330: Abstract Algebra
Cr. 3. (3-0). Prerequisites: Graduate standing. Groups, rings and fields; algebra of polynomials, Euclidean rings and principal ideal domains. Does not apply toward the Master of Science in Mathematics or Applied Mathematics.

5331: Linear Algebra With Applications
Cr. 3. (3-0). Prerequisites: Graduate standing and consent of instructor. Systems of linear equations, matrices, vector spaces, linear independence and linear dependence, determinants, eigenvalues; applications of the linear algebra concepts will be illustrated by a variety of projects.

5332: Differential Equations
Cr. 3. (3-0). Prerequisites: Math 5331 or consent of instructor. Linear and nonlinear systems of ordinary differential equations; existence, uniqueness and stability of solutions; initial value problems; higher dimensional systems; Laplace transforms. Theory and applications illustrated by computer assignments and projects. Applies toward the Master of Arts in Mathematics degree; does not apply toward the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees.

5333: Analysis
Cr. 3. (3-0). Prerequisites: Two semesters of calculus or consent of instructor. A survey of the concepts of limit, continuity, differentiation and integration for functions of one variable and functions of several variables; selected applications. Applies toward the Master of Arts in Mathematics degree; does not apply towards the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees.

5336: Discrete Mathematics
Cr. 3. (3-0). Prerequisites: Graduate standing or consent of instructor. Logic and proof, sets and relations; elementary set theory; the axiom of choice. Does not apply toward the Master of Science in Mathematics or Applied Mathematics.

5350: Linear Algebra With Applications
Cr. 3. (3-0). Prerequisite: MATH 2433, or consent of instructor. Curves, arc-length, curvature, Frenet formula, surfaces, first and second fundamental forms, Guass' theorem egregium, geodesics, minimal surfaces. Does not apply toward the Master of Science in Mathematics or Applied Mathematics.

5382: Probability
Cr. 3. (3-0). Prerequisites: Two semesters of calculus and one semester of linear algebra or consent of instructor. Sample spaces, events and axioms of probability; basic discrete and continuous distributions and their relationships; Markov chains, Poisson processes and renewal processes; applications. Applies toward the Master of Arts in Mathematics degree; does not apply toward Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees.

5383: Number Theory
Cr. 3. (3-0). Prerequisites: Graduate standing and consent of instructor. Divisibility and factorization, linear Diophantine equations, congruences and applications, solving linear congruences, primes of special forms, the Chinese remainder theorem, multiplicative orders, the Euler function, primitive roots, quadratic congruences, representation problems and continued fractions.

5385: Statistics
Cr. 3. (3-0). Prerequisites: Graduate standing and consent of instructor. Data collection and types of data, descriptive statistics, probability, estimation, model assessment, regression, analysis of categorical data, analysis of variance. Computing assignments using a prescribed software package (e.g., EXCEL, Minitab) will be given.

5386: Regression and Linear Models
Cr. 3. (3-0). Prerequisites: Two semesters of calculus, one semester of linear algebra, and Math 5385, or consent of instructor. Simple and multiple linear regression, linear models, inferences from the normal error model, regression diagnostics and robust regression, computing assignments with appropriate software. Applies toward Master of Arts in Mathematics degree; does not apply toward the Master of Science in Mathematics or the Master of Science in Applied Mathematics degrees.

5397: Selected Topics In Mathematics
Cr. 3. (3-0). Prerequisites: Graduate standing and consent of instructor. May be repeated with approval of chair when topics vary.

6198:6298:6398:6498:6598: Special Problems
Cr. 1-5 per semester. Prerequisites: consent of instructor and approval of chair.

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.

6308: Advanced Linear Algebra I
Cr. 3 per semester. (3-0). Prerequisite: graduate standing, MATH 2331 and a minimum of 3 semester hours of 3000-level mathematics. Linear systems of equations, matrices, determinants, vector spaces and linear transformations, eigenvalues and eigenvectors. An expository paper or talk on a subject related to the course content is required.

6309: Advanced Linear Algebra II
Cr. 3 per semester. (3-0). Prerequisite: graduate standing, and MATH 6308. Similarity of matrices, diagonalization, hermitian and positive definite matrices, canonical forms, normal matrices, applications. An expository paper or talk on a subject related to the course content is required.

6312: Introduction to Real Analysis
Cr. 3 per semester. (3-0). Prerequisite: graduate standing and MATH 3334 or consent of instructor. Properties of continuous functions, partial differentiation, line integrals, improper integrals, infinite series, and Stieltjes integrals. An expository paper or talk on a subject related to the course content is required.

6313: Introduction to Real Analysis
Cr. 3 per semester. (3-0). Prerequisite: graduate standing and MATH 6312 or consent of instructor. Properties of continuous functions, partial differentiation, line integrals, improper integrals, infinite series, and Stieltjes integrals. An expository paper or talk on a subject related to the course content is required.

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: Topology
Cr. 3 per semester. (3-0). Prerequisites: MATH 4331 and MATH 4337 or consent of instructor. Point-set topology: compactness, connectedness, quotient spaces, separation properties, Tychonoff's theorem, the Urysohn lemma, Tietze's theorem, and the characterization of separable metric spaces.

6343: Topology II
Cr. 3 per semester. (3-0). Prerequisites: MATH 4331 and MATH 4337 or consent of instructor. Algebraic topology: the fundamental group covering spaces, surfaces, basic homology and applications.

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 Eigen value problems.

6377: Basic Tools for the Applied Mathematician
Cr. 3. (3-0). Prerequisite: MATH 1431, MATH 1432, MATH 2433 and either MATH 2431 or MATH 3321 and graduate standing or consent of instructor. Finite dimensional vector spaces, linear operators, inner products, eigen values, metric spaces and norm, continuity, differentiation, intergration of continuous functions, sequences and limits, compactness, fixed point theorems, applications to initial value problems.

6378: Basic Scientific Computing
Cr. 3. (3-0). Prerequisites: MATH 4364 and MATH 4365 or equivalent, and either COSC 1304 or COSC 2101 or equivalents, or consent of instructor. A project-oriented course in fundamental techniques for high performance scientific computation. Hardware architecture and floating point performance, code design, data structures and storage techniques related to scientific computing, parallel programming techniques, applications to the numerical solution of problems such as algebraic systems, differential equations and optimization. Data visualization.

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.

6384: Discrete - Time Models in Finance
Cr. 3 per semester. (3-0). Prerequisites: MATH 6382 or consent of instructor. Single-period securities markets, arbitrage, risk-neutral probabilities, complete and incomplete markets, consumption investment problems, mean-variance portfolio analysis, equilibrium models, valuation of options, futures and other derivatives on equities, currencies, commodities and fixed-income securities.

6385: Continuous-Time Models in Finance
Cr. 3 per semester. (3-0). Prerequisites: MATH 6384 or consent of instructor. Stochastic calculus, Brownian motion, change of measures, Martingale representation theorem, pricing financial derivatives whose underlying assets are equities, foreign exchanges, and fixed income securities, single-factor and multi-factor HJM models, and models involving jump diffusion and mean reversion.

6386: Computational Statistics
Cr. 3. (3-0). Prerequisites: MATH 4377 and MATH 3334 or consent of instructor. Common descriptive, graphical, and inferential procedures in statistics and their implementation in standard software packages, simulation, linear and generalized linear models, smoothing, classification and clustering, time series.

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). 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.

7374:7375: Finite Element Methods
Cr. 3. (3-0). Prerequisites: MATH 6326; MATH 6327 or consent of the instructor. Introduction to variational formulations of boundary value operators, construction of finite element spaces, existence and convergence of finite element solutions, mixed and hybrid finite element methods, algebraic formulation of finite element equations, iterative methods for large scale finite element systems, applications in fluid mechanics and electromagnetics.

7380: Stochastic Differential Equations
Cr. 3. (3-0).  Prerequisites: Math 6380 or 6382 or consent of instructor.  Brownian motion and its properties, martingales, the Ito integral, solutions of stochastic differential equations, numerical schemes, diffusion processes.  Applications to mathematical finance (arbitrage and option pricing) and connections to PDE's.

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.

Catalog Publish Date: January 14, 2013
This Page Last Updated: November 29, 2012