### MATH 501. Topology (3)

Prerequisite: MATH 350. Metric spaces, topological spaces, compactness, completeness and connectedness. Introduction to function spaces, with emphasis on the uniform topology.

### MATH 510A/B. Algebra and Number Theory (3-3)

Prerequisite: Admission to the Graduate Program. A 2-course sequence on integers and prime numbers, rational and complex algebraic numbers, symmetry and group theory, rings of polynomials and algebraic integers, basic algebraic geometry and algebraic extensions, elementary Galois Theory and the theory of equations. MATH 510A is the prerequisite for MATH 510B. These courses cannot be taken for credit toward the MasterÕs Degree in Options I and II.

### MATH 511A/B. Linear Algebra and Geometry (3-3)

Prerequisite: Admission to the Graduate Program. A 2-course sequence on modern applications of mathematics that involve matrices, basic properties of vectors of R2 and R3, dot product, orthonormal basis, cross product, linear transformations of Euclidean 2- and 3-Space and the classification of its rigid motions, symmetric bilinear forms, conics and quadrics, basic topology of Rn, spherical geometry, PoincarŽÕs models of the hyperbolic plane and their isometries. MATH 511A is the prerequisite for MATH 511B. These courses cannot be taken for credit toward the MasterÕs Degree in Options I and II.

### MATH 512A/B. Concepts of Analysis (3-3)

Prerequisite: Admission to the Graduate Program. A 2-course sequence on the real number system, countable and uncountable sets, cardinal numbers, Cantor diagonal argument, well-ordered sets, ordinal numbers, numerical sequences and numerical series of real numbers, continuity, differentiability and integration of functions of one variable, sequences and series of functions, uniform convergence and ordinary differential equations. MATH 512A is the prerequisite for MATH 512B. These courses cannot be taken for credit toward the MasterÕs Degree in Options I and II.

### MATH 513A/B. Discrete Mathematics (3-3)

Prerequisite: Admission to the Graduate Program. A 2-course sequence on permutations, combinations, multinomial coefficients and Pascal triangles, pigeon hole principle, inclusion-exclusion principle, Ramsey numbers, vharacteristic functions and algorithms, generating functions, finite probabilities, recurrence relations, vonnected graphs, graph volorings, planar graphs, trees, adjacency matrices, Eulerian paths, Hamiltonian paths, tournaments, matching and covering, networks, information transmission, coding and decoding, and error correcting codes. MATH 513A is the prerequisite for MATH 513 B. These courses cannot be taken for credit toward the MasterÕs Degree in Options I and II.

### MATH 514A/B. Probability and Statistics (3-3)

Prerequisite: Admission to the Graduate Program. A 2-course sequence on probability rules, discrete and continuous random variables and their distributions, central limit theorem, and on elementary topics in statistics from the advanced point of view, including exploratory analysis, graphical display, random phenomena, probability distributions, simulation, correlation and regression, survey sampling and experimental design, sampling distributions, confidence intervals and significance tests for proportions and means, and chi-square tests. MATH 514A is the prerequisite for MATH 514B. These courses cannot be taken for credit toward the MasterÕs Degree in Options I and II.

### MATH 540. Regression Analysis (3)

Prerequisite: MATH 440A. General linear model in matrix form, simple and multiple regression analysis, transformations, variable selection, multicollinearity, analysis of variance, robust regression, logistic regression, principal components and factor analysis. Statistical software utilized.

### MATH 542A-D. Probability and Statistics (3-3-3-3)

Prerequisite: MATH 340 or 440A. This course will cover topics in probability and statistics not covered elsewhere in the program. Part A is usually devoted to multivariate statistics, Part B to stochastic processes, and Part C to probability theory. Part D is left to a topic chosen by the individual instructor.

### MATH 550. Calculus On Manifolds (3)

Prerequisite: MATH 450. Integration of functions of several variables. Differential forms and differential manifolds, Line integrals, integration on manifolds, StokesÕ Theorem and PoincarŽÕs Lemma.

### MATH 552. Real Analysis (3)

Prerequisite: MATH 501. Introduction to measure theory and Lebesgue integration, and their application to probability theory. Monotone and dominated convergence theorems, FubiniÕs theorem, Fourier analysis and Banach spaces.

### MATH 560. Abstract Algebra III (3)

Prerequisite: MATH 460. Graduate course in abstract algebra. Group theory, Galois theory and other topics.

### MATH 570. Differential Geometry (3)

Prerequisite: MATH 450. The local theory of regular curves in R3 and Frenet formulas. Regular surfaces in R3, the first and second fundamental forms, Gaussian and mean curvatures, and the Egregium Gauss theorem. Geodesics and the Gauss-Bonnet theorem.

### MATH 581. Numerical Methods for Linear Systems (3)

Prerequisite: MATH 462. Methods for solving large linear problems and eigenvalue problems are presented at an advanced level. Direct methods such as LU factorization, Cholesky factorization and the Least Squares method, and Iterative methods, such as the Jacobi, Gauss-Seidel, SOR and conjugate Gradient methods, are discussed in detail. Eigenvalue problems are solved via power iteration, the QR method and the Jacobi method.

### MATH 582 A-D. Topics in Numerical Analysis (3-3-3-3)

Prerequisite: MATH 581 or consent of instructor. The course will cover topics in numerical analysis which are important in many applications and which are not covered elsewhere in the program. Part A usually covers numerical methods in optimization, Part B covers numerical methods for ordinary differential equations ,and Part C covers numerical solution of partial differential equations. Part D covers a subject chosen by the instructor.

### MATH 589. Seminar in Mathematics (1)

Prerequisite: Senior or graduate standing in the Mathematics Department. Students will read about advanced topics in the recent literature in mMathematics and report on them in a lecture. This course may be taken up to two times with the consent of the advisor. (Credit/No Credit only)

### MATH 592A-D. Topics in Applied Mathematics (3-3-3-3)

Prerequisites: MATH 552 or consent of instructor. This course is devoted to a variety of important topics in applied Mmathematics that are not covered elsewhere in the Program. In particular, Part A will cover the mathematical theory of partial differential equations, Part B covers mathematical optimization and operations research, and Part C covers mathematical biology. The topic of Part D is left to the individual instructor.

### MATH 595A-Z. Experimental Topics (1-3)

Prerequisite: Consent of instructor. Specialized topics from a concentrated field of current interest presented at an advanced level.

### MATH 625. Advanced Mathematical Modeling (3)

Selected problems in ecology, biology, economics, finance, social sciences, life sciences, physical sciences and engineering are used to develop advanced techniques of mathematical modeling.

### MATH 651 ABC. Advanced Topics in Analysis, Geometry and Topology (3-3-3)

Prerequisite: Consent of instructor. Advanced topics not covered in the previous classes on the subject. Part A covers topics in analysis, Part B covers topics in geometry, and Part C covers topics in topology. May be repeated with the consent of the advisor.

### MATH 655. Complex Analysis (3)

Prerequisites: MATH 501, 455. Topics covered incluyde the general Cauchy theorem, power series and analytic continuation, series and product expansions, conformal mapping and the Dirichlet problem.

### MATH 661 ABC. Advanced Topics in Algebra, Number Theory and Discrete Mathematics (3-3-3)

Prerequisite: Consent of instructor. Advanced topics not covered in the previous classes on the subject. Part A covers topics in algebra, Part B covers topics in number theory, and Part C covers other topics in discrete mathematics. May be repeated with the consent of the advisor.

### MATH 680A/B. Applied Functional Analysis (3-3)

Prerequisites: MATH 501, 552. This 2-semester sequence gives an introduction to Banach and Hilbert spaces and their applications. Fixed Point Theorems and their applications to differential and integral equations and variational principles. Adjoint and self-adjoint operators and spectral theory of linear operators. MATH 680A is a prerequisite for MATH 680B.

### MATH 697A-C. Directed Comprehensive Studies (1-3)

No course description.

### MATH 698A-C. Thesis or Graduate Project (1-3)

No course description.

### MATH 699A-F. Independent Study (1-6)

See Independent Study under courses of study.