**Graduate Courses**

**MATH 501. GENERAL TOPOLOGY I.** Metric spaces, Baire’s theorem, topological spaces, continuity, separation axioms, connectedness, compactness, quotient and product topologies. Prerequisite: Math 305 with minimum grade of C. (3)

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

**MATH 513. THEORY OF NUMBERS I.** Divisibility; properties of prime numbers; congruences and modular arithmetic; quadratic reciprocity; representation of integers as sums of squares. Prerequisite: 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; partitions. Prerequisite: Math 513, Math 555 or approval of the instructor. (3)

**MATH 519. MATRICES.** Basic matrix theory, eigenvalues, eigenvectors, normal and Hermitian matrices, similarity, Sylvester’s Law of Inertia, normal forms, functions of matrices. (3)

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

**MATH 525. INTRODUCTION TO ABSTRACT ALGEBRA I.** General properties of groups. (3)

**MATH 526. INTRODUCTION TO ABSTRACT ALGEBRA II.** General properties of rings and fields. Prerequisite: Math 525. (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; circle area. Prerequisite: Math 305 or graduate standing. (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, Euler, and Gauss. Prerequisite: Math 305 or consent of instructor. (3)

**MATH 545. TOPICS FOR SECONDARY SCHOOL TEACHERS.** High school subjects from an advanced point of view and their relation to the more advanced subjects. (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, series, absolute and conditional convergence of series. Prerequisite requirements for this course may also be satisfied by consent of instructor. Prerequisite: Math 305 with minimum grade of C. (3)

**MATH 556. ADVANCED CALCULUS II.** Limit of a function, metric spaces, limits in metric spaces, complete metric spaces, uniform continuity, pointwise and uniform convergence of sequences and series of functions, power series. Prerequisite: Math 555 with minimum grade of C. (3)

**MATH 564. DYNAMICAL SYSTEMS.** 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. Prerequisite: Math 353 with minimum grade of

C or Graduate Standing. (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, Classical Banach spaces. Prerequisite: Math 556 with minimum grade of C. Prerequisite requirements for this course may also be satisfied by consent of instructor. (3)

**MATH 568. INTRODUCTION TO FUNCTIONAL ANALYSIS II.** Topics in Banach space theory. Prerequisite: Math 567 with 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. Prerequisite: Math 264. (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. Prerequisite: Math 264. (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). Prerequisite: Math 573. (3)

**MATH 575. MATHEMATICAL STATISTICS I.** Mathematical treatment of statistical and moment characteristics; probability models; random variables; distribution theory; correlation; central limit theory; multi-parameter models. Prerequisite: Math 262 with minimum 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; Baysian statistics. Prerequisite: Math 575 with 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: Math 573 or consent of instructor. (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. Z grade. This course is required of all teaching assistants, each semester, and may not be used for credit toward a degree. Prerequisite: departmental consent. (1-3)

**MATH 597. SPECIAL PROBLEMS.** (1-3)

**MATH 598, 599. SPECIAL PROBLEMS.** (1-3)

**MATH 625. MODERN ALGEBRA I.** Advanced group theory. (3)

**MATH 626. MODERN ALGEBRA II.** Advanced theory of rings and fields, including Galois theory and modules. (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. Prerequisite: consent of instructor. (3)

**MATH 649. CONTINUED FRACTIONS.** Arithmetic theory; analytic theory; applications to Lyapunov theory. Prerequisite: consent of instructor. (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.** Conformal mappings, analytic continuation, harmonic functions, infinite products. (3)

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

**MATH 662. NUMERICAL ANALYSIS II.** Techniques for ordinary and partial differential equations, stability and convergence analysis. (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 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.

**MATH 668. FUNCTIONAL ANALYSIS II.** Topics in applied functional analysis.

**MATH 669. PARTIAL DIFFERENTIAL EQUATIONS I.** Classical theories of wave and heat equations. Prerequisite: Math 353 or Math 555. (3)

**MATH 670. PARTIAL DIFFERENTIAL EQUATIONS II.** Hilbert space methods for boundary value problems. Prerequisite: Math 669. (3)

**MATH 673. ADVANCED PROBABILITY I.** Topics in probability are treated at an advanced level. Measure theoretic foundations, infinitely divisible laws, stable laws. Corequisite: Math 654. (3)

**MATH 674. ADVANCED PROBABILITY II.** Multidimensional central limit theorem, strong laws, law of the iterated logarithm. Prerequisite: Math 673 with minimum grade of C. (3)

**MATH 677. ADVANCED STOCHASTIC PROCESSES I.** Topics in the theory of stochastic processes, separability, martingales, stochastic integrals, the Wiener process. Prerequisite: Math 674 with minimum grade of C. (3)

**MATH 678. ADVANCED STOCHASTIC PROCESSES II.** Gaussian processes, random walk, Ornstein-Uhlenbeck process, semi-group theory for diffusions. Prerequisite: Math 677 with 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. Prerequisite: Math 575 with 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.** General measure theory. (3)

**MATH 775. ADVANCED MATHEMATICAL STATISTICS I.** Univariate distribution functions and their characteristics; moment generating functions and semi-invariants; Pearson’s system; Gram-Charlier series; inversion theorems. (3)

**MATH 776. ADVANCED MATHEMATICAL STATISTICS II.** Multivariate distributions and regression systems; multiple and partial correlation; sampling theory; statistical hypotheses; power and efficiency of tests. (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)