**Graduate Courses**

The official catalog of mathematics courses can be accessed here.

**MATH 501. GENERAL TOPOLOGY I.** Metric spaces, continuity, separation axioms, connectedness, compactness, and other related topics. Prerequisites: Math 305 with a minimum grade of C. (3)

**MATH 502. GENERAL TOPOLOGY II.** Algebraic invariants in topology. Prerequisites: Math 501 with a minimum grade of C. (3)

**MATH 513. THEORY OF NUMBERS I.** Divisibility; properties of prime numbers; congruences and modular arithmetic; quadratic reciprocity; and representation of integers as sums of squares. Prerequisites: Math 305. (3)

**MATH 514. THEORY OF NUMBERS II.** Arithmetic functions and their distribution; distribution of prime numbers; Dirichlet characters and primes in arithmetic progression; and partitions. Prerequisites: Math 513 and Math 555. (3)

**MATH 519. MATRICES.** Basic matrix theory, eigenvalues, eigenvectors, normal and Hermitian matrices, similarity, Sylvester’s Law of Inertia, normal forms, functions of matrices. Prerequisites: Math 319 with a minimum grade of C. (3)

**MATH 520. LINEAR ALGEBRA.** An introduction to vector spaces and linear transformations; eigenvalues, and the spectral theorem. (3)

**MATH 525. MODERN ALGEBRA I.** General theory of groups. Prerequisites: Math 305 with a minimum grade of C. (3)

**MATH 526. MODERN ALGEBRA II.** General theory of rings and fields. Prerequisites: Math 525 with a minimum grade of C. (3)

**MATH 533. TOPICS IN EUCLIDEAN GEOMETRY.** A study of incidence geometry; distance and congruence; separation; angular measure, congruences between triangles; inequalities; parallel postulate; similarities between triangles; circles area. Prerequisites: Math 305 with a minimum grade of C. (3)

**MATH 537. NON-EUCLIDEAN GEOMETRY.** Brief review of the foundation of Euclidean plane geometry with special emphasis given the Fifth Postulate; hyperbolic plane geometry; elliptic plane geometry. (3)

**MATH 540. HISTORY OF MATHEMATICS.** Development of mathematics, especially algebra, geometry, and analysis; lives and works of Euclid, Pythagoras, Cardan, Descartes, Newton, Fuler, and Gauss. Prerequisite requirements for this course may also be satisfied by consent of instructor. Prerequisites: Junior standing (60 hr), Math 305 with minimum grade of C. (3)

**MATH 545. SELECTED TOPICS IN MATHEMATICS FOR SECONDARY SCHOOL TEACHERS.** High-school subjects from an advanced point of view and their relation to the more advanced subjects. May be repeated once for credit. (3)

**MATH 555. ADVANCED CALCULUS I.** Suprema and infima on the real line; limits, liminf, and limsup of a sequence of reals; convergent sequences; Cauchy sequences and series, absolute and conditional convergence of series. Prerequisites: Junior standing (60 hr), Math 305 with minimum grade of C. (3)

**MATH 556. ADVANCED CALCULUS II.** Limits, continuity, power series, partial differentiation; multiple, definite, improper, and line integrals; applications. Prerequisites: Junior standing (60 hr), Math 555 with minimum grade of C. (3)

**MATH 564. INTRODUCTION TO DYNAMICAL SYSTEMS I.** This course is an introduction to the theory of dynamical systems. The course will cover linear maps and differential equations, nonlinear systems, conservative dynamics, one-dimensional dynamics and connections with ergodic theory and number theory. Prerequisites: Math 353 with a minimum grade of C or Graduate Standing. (3)

**MATH 565. INTRODUCTION TO DYNAMICAL SYSTEMS II.** This course is the second semester of an introduction to the theory of dynamical systems. The course will cover some aspects of linear maps and differential equations, nonlinear systems, conservative dynamics, one-dimensional dynamics, and connections with ergodic theory and number theory. Prerequisites: Junior standing (60 hr), MATH 564. (3)

**MATH 567. INTRODUCTION TO FUNCTIONAL ANALYSIS I.** Hilbert spaces, Banach spaces, Hahn-Banach Theorem, Banach Steinhaus Theorem, Open Mapping Theorem, weak topologies, Banach-Alaoglu Theorem, and Classical Banach spaces. Prerequisites: Math 556 with a minimum grade of C. (3)

**MATH 568. INTRODUCTION TO FUNCTIONAL ANALYSIS II.** Topics in Banach space theory. Prerequisites: Math 567 with a minimum grade of C. (3)

**MATH 572. INTRODUCTION TO PROBABILITY AND STATISTICS.** Emphasis on standard statistical methods and the application of probability to statistical problems. Prerequisites: Math 261, Math 262, Math 263, and Math 264, all with a minimum grade of C. (3)

**MATH 573. APPLIED PROBABILITY.** Emphasis on understanding the theory of probability and knowing how to apply it. Proofs are given only when they are simple and illuminating. Among topics covered are joint, marginal, and conditional distributions, conditional and unconditional moments, independence, the weak law of large numbers, Tchebycheff’s inequality, Central Limit Theorem. Prerequisites: Math 261, Math 262, Math 263, and Math 264, all with a minimum grade of C. (3)

**MATH 574. PROBABILITY.** Topics introduced in Math 573 will be covered at a more sophisticated mathematical level. Additional topics will include the Borel-Cantelli Lemma, the Strong Law of Large Numbers, characteristic functions, Fourier transforms. Prerequisites: Math 573 with a minimum grade of C. (3)

**MATH 575. MATHEMATICAL STATISTICS I.** Mathematical treatment of statistical and moment characteristics; probability models; random variables; distribution theory; correlation; central limit theorem; and multiparameter models. Prerequisites: Junior standing (60 hr), Math 262 with a grade of C. (3)

**MATH 576. MATHEMATICAL STATISTICS II.** Mathematical treatment of statistical inference; maximum likelihood estimation and maximum likelihood ratio test; minimum variance unbiased estimators; most powerful tests; asymptotic normality and efficiency; and Baysian statistics. Prerequisites: Math 575 with a minimum grade of C. (3)

**MATH 577. APPLIED STOCHASTIC PROCESSES.** Emphasis on the application of the theory of stochastic processes to problems in engineering, physics, and economics. Discrete and continuous time Markov processes, Brownian Motion, Ergodic theory for stationary processes. Prerequisite requirements for this course may also be satisfied by consent of instructor. Prerequisites: Math 573 with a minimum grade of C. (3)

**MATH 578. STOCHASTIC PROCESSES.** Topics will include general diffusions, Martingales, and Stochastic differential equations. (3)

**MATH 590. TECHNIQUES IN TEACHING COLLEGE MATHEMATICS.** Directed studies of methods in the presentation of college mathematics topics, teaching and testing techniques. This course is required of all teaching assistants, each semester, and may not be used for credit toward a degree. Prerequisites: Junior standing (60 hr). (1-3)

**MATH 597. SPECIAL PROBLEMS I.** Prerequisites: Junior standing (60 hr). (1-3)

**MATH 598. SPECIAL PROBLEMS II.** Prerequisites: Junior standing (60 hr). (1-3)

**MATH 599. SPECIAL PROBLEMS III.** Prerequisites: Junior standing (60 hr). (1-3)

**MATH 625. MODERN ALGEBRA I.** Advanced group theory. Prerequisites: Math 525. (3)

**MATH 626. MODERN ALGEBRA II.** Advanced theory of rings and fields, including Galois theory and modules. Prerequisites: Math 625. (3)

**MATH 631. FOUNDATIONS OF GEOMETRY.** Development of Euclidean geometry in two and three dimensions using the axiomatic method; introduction to high dimensional Euclidean geometry and to non-Euclidean geometrics. (3)

**MATH 639. PROJECTIVE GEOMETRY.** Fundamental propositions of projective geometry from synthetic and analytic point of view; principle of duality; poles and polars; cross ratios; theorems of Desargues, Pascal, Brianchon; involutions. (3)

**MATH 647. TOPICS IN MODERN MATHEMATICS.** Survey of the more recent developments in pure and applied mathematics. (3)

**MATH 649. CONTINUED FRACTIONS.** Arithmetic theory; analytic theory; applications to Lyapunov theory. (3)

**MATH 655. THEORY OF FUNCTIONS OF COMPLEX VARIABLES I.** Complex numbers, analytic functions, complex integration, Cauchy’s theorem and integral formula, Liouvilles’s theorem, maximum modulus principle, Schwarz’s lemma, sequences and series of analytic functions, isolated singularities, the residue theorem. (3)

**MATH 656. THEORY OF FUNCTIONS OF COMPLEX VARIABLES II.** This course covers conformal mappings, harmonic functions, and infinite products. Prerequisites: Math 655. (3)

**MATH 661. NUMERICAL ANALYSIS I.** Numerical linear algebra, error analysis, computation of eigenvalues, and eigenvectors, finite differences. (3)

**MATH 662. NUMERICAL ANALYSIS II.** This course covers techniques for ordinary and partial differential equations, and stability and convergence analysis. Prerequisites: Math 661. (3)

**MATH 663. SPECIAL FUNCTIONS.** Advanced study of gamma functions; hypergeometric functions; generating function; theory and application of cylinder functions and spherical harmonics. (3)

**MATH 664. TOPICS IN DYNAMICAL SYSTEMS.** The course covers some important topics in dynamical systems. In particular, low-dimensional dynamics, circle dynamics, Hamiltonian dynamics. It will also contain an introduction to Kolmogorov-Arnold-Moser theory and renormalization methods in dynamics. Prerequisite: Math 564 with a minimum grade of C. (3)

**MATH 667. FUNCTIONAL ANALYSIS I.** Topological vector spaces (tvs), complete tvs, product and quotient tvs, separation theorems for convex sets, locally convex spaces, Krein-Milman theorem, linear operators, dual pairs and Mackey-Arens theorem, Alaoglu-Bourbaki thorem, bornological and barreled spaces. (3)

**MATH 668. FUNCTIONAL ANALYSIS II.** This course covers topics in applied functional analysis. Prerequisite: Math 667. (3)

**MATH 669. PARTIAL DIFFERENTIAL EQUATIONS I.** Classical theories of wave and heat equations. (3)

**MATH 670. PARTIAL DIFFERENTIAL EQUATIONS II.** Hilbert space methods for boundary value problems. Prerequisites: Math 669 with a minimum grade of C. (3)

**Math 671. STATISTICAL METHODS I.** This course and its sequel, Math 672, cover linear statistical models for regression, analysis of variance, and experimental design. The courses seek to blend theory and application. Topics in this course include simple and multiple linear regressions, model diagnostics, model selection and validation, generalized linear models, nonlinear regression, and neural networks. SAS or R will be used to apply these methods with real data. Prerequisites: Math 576. (3)

**Math 672. STATISTICAL METHODS II.** This course is a continuation of Math 671. Topics covered are one-way and multi-way analysis of variance (ANOVA), balanced and unbalanced designs with fixed effects, random effects and mixed effects, model diagnostics, nested designs, repeated measures designs, fractional designs, Latin squares. SAS or R will be used to apply these methods with real data. Prerequisites: Math 671. (3)

**MATH 673. ADVANCED PROBABILITY I.** Topics in probability at an advanced level; measure theoretic foundations; infinitely divisible laws; and stable laws. Corequisite: Math 754. (3)

**MATH 674. ADVANCED PROBABILITY II.** Multidimensional central limit theorem; strong laws; and law of the integrated logarithm. Prerequisites: Math 673 with a minimum grade of C. (3)

**MATH 677. ADVANCED STOCHASTIC PROCESSES I.** Topics in the theory of stochastic processes; separability; Martingales; stochastic integrals; and the Wiener process. Prerequisites: Math 674 with a minimum grade of C. (3)

**MATH 678. ADVANCED STOCHASTIC PROCESSES II.** Gaussian process; random walk; Ornstein-Uhlenbeck process; and semi- group theory for diffusions. Prerequisites: Math 677 with a minimum grade of C. (3)

**MATH 679. STATISTICAL BIOINFORMATICS.** Introduction to bioinformatics—an interdisciplinary study that combines techniques and knowledge in mathematical, statistical, computational, and life sciences in order to understand the biological significance of genetic sequence data. Prerequisites: Math 575 with a minimum grade of C. (3)

**MATH 681. GRAPH THEORY I.** Primarily topics in Matroid Theory including duality, minors, connectivity, graphic matroids, representable matroids, and matroid structure. Connections between the class of matroids with the classes of graphs and projective geometries are also studied. (3)

**MATH 697. THESIS.** (1-12)

**MATH 700. SEMINAR IN TOPOLOGY.** Prerequisite: consent of instructor. (May be repeated for credit.) (3)

**MATH 710. SEMINAR IN ALGEBRA.** Prerequisite: consent of instructor. (May be repeated for credit.)

(3).

**MATH 750. SEMINAR IN ANALYSIS.** Prerequisite: consent of instructor. (May be repeated for credit.) (3)

**MATH 753. THEORY OF FUNCTIONS OF REAL VARIABLES I.** Lebesgue measure and integration; differentiation; bounded variation and absolute continuity of functions. (3)

**MATH 754. THEORY OF FUNCTIONS OF REAL VARIABLES II.** This course covers general measure theory. Prerequisites: Math 753. (3)

**MATH 775. ADVANCED MATHEMATICAL STATISTICS I.** Math 775 and Math 776 serve as Topics in Statistics courses, and may be repeated once for credit as topics vary. Topics will be selected from but not limited to Bayesian statistics, nonparametric statistics, time series analysis, survival analysis, financial statistics, statistical learning and data mining, robust statistics, and multivariate analysis. Course topics rotate depending on interests of students and faculty. Prerequisites: Math 576 with a minimum grade of C.(3)

**MATH 776. ADVANCED MATHEMATICAL STATISTICS II.** Math 775 and Math 776 serve as Topics in Statistics courses, and may be repeated once for credit as topics vary. Topics will be selected from but not limited to Bayesian statistics, nonparametric statistics, time series analysis, survival analysis, financial statistics, statistical learning and data mining, robust statistics, and multivariate analysis. Course topics rotate depending on interests of students and faculty. Prerequisites: Math 576 with a minimum grade of C. (3)

**MATH 777. SEMINAR IN STATISTICS.** (May be repeated for credit up to a maximum of 9 hours). (3)

**MATH 780. SEMINAR IN GRAPH THEORY.** Prerequisite: consent of instructor. (May be repeated for credit up to a maximum of 9 hours.) (3)

**MATH 782. GRAPH THEORY II.** Topics in Graph Theory including trees, connectivity, matchings, paths, cycles, coverings, planarity, graph colorings, networks and directed graphs. Extremal graph structure, applications, and algorithms will also be studied. (3)

**MATH 797. DISSERTATION.** (1-18)