# Mathematics (MAT) Course Descriptions

**MAT 5003 Modern Mathematics for Teachers**

(3-0) 3 hours credit.

A practical orientation concerned with the classroom uses of mathematics for teachers of K-12. This course may not be applied toward the Master of Science degree in Mathematics with a concentration in Mathematics.

**MAT 5013 Computers for Mathematics Teachers**

(3-0) 3 hours credit.

A course for mathematics teachers on integrating the computer into the mathematics curriculum, with a focus on mathematical problem solving through the use of mathematical software packages. This course may not be applied to the Master of Science degree in Mathematics with a concentration in Mathematics.

**MAT 5023 Problem-Solving Seminar**

(3-0) 3 hours credit.

Students will have the opportunity to engage in extensive experience and practice in solving mathematical problems.

**MAT 5033 Foundations and Fundamental Concepts of Mathematics**

(3-0) 3 hours credit.

Topics include the study of mathematics in antiquity as an empirical science, the shift from inductive reasoning to axiomatic structures, the development of geometry in the plane and 3-space, the discovery of analysis, the emergence of axiomatic systems, and the focus on algebraic structures. This course may not be applied to the Master of Science degree in Mathematics with a concentration in Mathematics without approval of the Graduate Advisor of Record and the Graduate Review Committee.

**MAT 5043 Euclidean and Non-Euclidean Geometry**

(3-0) 3 hours credit.

Topics will be selected from advanced Euclidean and non-Euclidean geometry, solid analytic geometry, and differential geometry.

**MAT 5103 Introduction to Mathematical Analysis**

(3-0) 3 hours credit. Prerequisite: MAT 4213 or consent of instructor.

Axiomatic construction of the reals, metric spaces, continuous functions, differentiation and integration, partial derivatives, and multiple integration. This course may not be applied to the Master of Science degree in Mathematics with a concentration in Mathematics. (Credit cannot be earned for both MAT 5103 and MAT 5203.)

**MAT 5123 Introduction to Cryptography**

(3-0) 3 hours credit. Prerequisite: MAT 4213.

Congruences and residue class rings, Fermat’s Little Theorem, the Euler phi-function, the Chinese Remainder Theorem, complexity, symmetric-key cryptosystems, cyclic groups, primitive roots, discrete logarithms, one-way functions, public-key cryptosystems, digital signatures, finite fields, and elliptic curves.

**MAT 5173 Algebra I**

(3-0) 3 hours credit. Prerequisite: MAT 4233 or consent of instructor.

The opportunity for development of basic theory of algebraic structures. Areas of study include finite groups, isomorphism, direct sums, polynomial rings, algebraic numbers, number fields, unique factorization domain, prime ideals, and Galois groups.

**MAT 5203 Theory of Functions of a Real Variable I**

(3-0) 3 hours credit. Prerequisite: MAT 4213 or consent of instructor.

Measure and integration theory. (Credit cannot be earned for both MAT 5203 and MAT 5103.)

**MAT 5213 Theory of Functions of a Real Variable II**

(3-0) 3 hours credit. Prerequisite: MAT 5203.

Further development of measure and integration theory, metric space topology, and elementary Banach space theory.

**MAT 5223 Theory of Functions of a Complex Variable I**

(3-0) 3 hours credit. Prerequisite: MAT 3213 or MAT 4213.

Complex integration, Cauchy’s theorem, calculus of residues, and power series.

**MAT 5233 Theory of Functions of a Complex Variable II**

(3-0) 3 hours credit. Prerequisite: MAT 5223.

Infinite products, entire functions, Picard’s theorem, Riemann mapping theorem, and functions of several complex variables.

**MAT 5243 General Topology I**

(3-0) 3 hours credit. Prerequisite: MAT 4273 or consent of instructor.

Topological spaces, metric spaces, continua, and plane topology.

**MAT 5253 General Topology II**

(3-0) 3 hours credit. Prerequisite: MAT 5243.

Topics may include: Metrizable topological spaces, function spaces, covering spaces, homotopy theory and fundamental groups, classification of surfaces, and others.

**MAT 5263 Algebraic Topology**

(3-0) 3 hours credit. Prerequisite: MAT 4273 or MAT 5243.

Fundamental ideas of algebraic topology, homotopy and simplicial complexes, fundamental group, covering spaces, and duality theorems.

**MAT 5283 Linear Algebra and Matrix Theory**

(3-0) 3 hours credit. Prerequisite: MAT 2233 or an equivalent.

A study of linear algebraic structures and algebraic properties of matrices.

**MAT 5293 Numerical Linear Algebra**

(3-0) 3 hours credit. Prerequisite: MAT 2233 or an equivalent.

Direct and iterative methods for solving general linear systems, the algebraic eigenvalue problem, least squares problems, and solutions of sparse systems arising from partial differential equations. (Same as CS 5293. Credit cannot be earned for both MAT 5293 and CS 5293.)

**MAT 5313 Algebra II**

(3-0) 3 hours credit. Prerequisite: MAT 5173.

Areas of study include: groups, rings, fields, Galois theory, ideal theory and representations of groups, module theory, and homological algebra.

**MAT 5323 Mathematical Modeling**

(3-0) 3 hours credit. Prerequisite: MAT 3633 or equivalent.

Techniques of mathematical modeling for applications, including ordinary and partial differential equations, stochastic models, discrete models and optimization, modeling error and uncertainty quantification.

**MAT 5333 Wavelet Analysis**

(3-0) 3 hours credit. Prerequisite: MAT 5213, MAT 5283, or consent of instructor.

Inner products and Hilbert spaces, time-frequency analysis, the integral wavelet transform, multiresolutional analysis, dyadic wavelets, classification of wavelets, wavelet decompositions and reconstructions, wavelet packets, multivariate wavelets, and curvelets.

**MAT 5343 Differential Geometry**

(3-0) 3 hours credit. Prerequisite: MAT 5283 or equivalent.

Multilinear algebra, differentiable manifolds, exterior differential forms, affine connections, Riemannian geometry, and curvature equations.

**MAT 5353 Mathematics of Image Processing**

(3-0) 3 hours credit. Prerequisite: MAT 5213, MAT 5283, or consent of instructor.

Topics include image acquisition, denoising and enhancement, transformations, linear and nonlinear filters, image compression, segmentation and edge detection, morphology, and pattern recognition.

**MAT 5403 Functional Analysis I**

(3-0) 3 hours credit. Prerequisites: MAT 2233, MAT 4273, and MAT 5203, or their equivalents.

Topological vector spaces, inner product spaces, normed spaces, Hilbert spaces and Banach spaces, dual spaces, Hahn-Banach theorem, and bounded linear operators.

**MAT 5413 Functional Analysis II**

(3-0) 3 hours credit. Prerequisite: MAT 5403.

Riesz representation theorem, spectral theory, Banach algebras, and C*-algebras.

**MAT 5553 Harmonic Analysis**

(3-0) 3 hours credit. Prerequisites: MAT 3223 and MAT 4223, or consent of instructor.

Theory of the Fourier, Laplace, and Hilbert transforms. Elements of the distribution theory. Harmonic functions. Function spaces: Lp-spaces, Hardy spaces, Sobolev spaces.

**MAT 5603 Numerical Analysis**

(3-0) 3 hours credit. Prerequisite: MAT 3633 or consent of instructor.

Emphasis on the mathematical analysis of numerical methods. Areas of study include solution of nonlinear equations and function optimization, approximation theory and numerical quadrature. (Same as CS 5603. Credit cannot be earned for both MAT 5603 and CS 5603.)

**MAT 5613 Numerical Solutions of Differential Equations**

(3-0) 3 hours credit. Prerequisite: MAT 5603 or an equivalent.

Emphasis on the mathematical analysis of numerical methods. Areas of study include the analysis of single and multistep methods of ordinary differential equations. Analysis of finite difference and finite element methods for partial differential equations.

**MAT 5653 Differential Equations I**

(3-0) 3 hours credit. Prerequisites: MAT 3613 and MAT 4213, or consent of instructor.

Solution of initial-value problems, linear systems with constant coefficients, exponentials of operators, canonical forms and generic properties of operators, and contractions.

**MAT 5663 Differential Equations II**

(3-0) 3 hours credit. Prerequisite: MAT 5653.

Dynamic systems, the fundamental existence and uniqueness theorem, stability, the Poincare-Bendixson theorem, introduction to perturbation, and bifurcation theory.

**MAT 5673 Partial Differential Equations I**

(3-0) 3 hours credit. Prerequisite: MAT 3623, MAT 5663, or consent of instructor.

Classical theory of initial value and boundary value problems for partial differential equations.

**MAT 5683 Partial Differential Equations II**

(3-0) 3 hours credit. Prerequisite: MAT 5673.

Modern topics in partial differential equations.

**MAT 5833 Perturbation Theory in Applied Mathematics**

(3-0) 3 hours credit. Prerequisite: MAT 3613, MAT 5653, or consent of instructor.

Perturbation theory, asymptotic analysis, and boundary layer expansions.

**MAT 5973 Directed Research**

3 hours credit. Prerequisites: Graduate standing and permission in writing (form available) of the instructor and the student’s Graduate Advisor of Record.

The directed research course may involve either a laboratory or a theoretical problem. May be repeated for credit, but not more than 6 hours, regardless of discipline, will apply to the Master’s degree.

**MAT 6603 Optimization Techniques in Operations Research**

(3-0) 3 hours credit. Prerequisite: MAT 2214, MAT 2233, or consent of instructor.

Analysis and application of optimization techniques in operations research. Emphasis on linear programming, nonlinear programming, and integer programming.

**MAT 6901 Teaching Seminar**

(1-0) 1 hour credit. Prerequisite: Designation as a teaching assistant in the Department of Mathematics.

Designed to improve the instructional effectiveness of graduate students’ teaching at the college level. Topics include boardwork, clear speech, teacher-student interaction, professional responsibilities, course content and pace, grading policy, test writing, sensitivity to student needs, information and technical support and guest lectures on special topics. This course may not be applied as credit toward a Master of Science degree in Mathematics.

**MAT 6953 Independent Study**

3 hours credit. Prerequisites: Graduate standing and permission in writing (form available) of the instructor and the student’s Graduate Advisor of Record.

Independent reading, research, discussion, and/or writing under the direction of a faculty member. For students needing specialized work not normally or not often available as part of the regular course offerings. May be repeated for credit, but not more than 6 hours, regardless of discipline, will apply to the Master’s degree.

**MAT 6961 Comprehensive Examination**

1 hour credit. Prerequisite: Approval of the appropriate graduate program committee to take the Comprehensive Examination.

Independent study course for the purpose of taking the Comprehensive Examination. May be repeated as many times as approved by the Graduate Program Committee. Enrollment is required each term in which the Comprehensive Examination is taken if no other courses are being taken that term. The grade report for the course is either “CR” (satisfactory performance on the Comprehensive Examination) or “NC” (unsatisfactory performance on the Comprehensive Examination).

**MAT 6963 Topics in Mathematics Education**

(3-0) 3 hours credit. Prerequisite: Consent of instructor.

An organized course offering the opportunity for specialized study not normally or not often available as part of the regular course offerings. This course may be repeated for credit when topics vary but not more than 9 hours may be applied toward the Master’s degree. This course may not be applied toward the Master of Science degree in Mathematics with a concentration in Mathematics.

**MAT 6973 Special Problems**

(3-0) 3 hours credit. Prerequisite: Consent of instructor.

An organized course offering the opportunity for specialized study not normally or not often available as part of the regular course offerings. Special Problems courses may be repeated for credit when topics vary, but not more than 6 hours, regardless of discipline, will apply to the Master’s degree.

**MAT 6983 Master’s Thesis**

3 hours credit. Prerequisites: Permission of the Graduate Advisor of Record and thesis director.

Thesis research and preparation. May be repeated for credit, but not more than 6 hours will apply to the Master’s degree. Credit will be awarded upon completion of the thesis. Enrollment is required each term in which the thesis is in progress.