Devoted to chomology theory: cohomology groups, the universal coefficient theorem, the Kunneth formula, cup products, the cohomology ring of manifolds, Poincare duality, and other topics if time permits. Prerequisite: MATH 7800.

Examines fiber bundles; induced bundles, principal bundles, classifying spaces, vector bundles, and characteristic classes, and introduces K-theory and Bott periodicity. Prerequisite: MATH 7800.

Definition of homotopy groups, homotopy theory of CW complexes, Huriewich theorem and Whitehead's theorem, Eilenberg-Maclane spaces, fibration and cofibration sequences, Postnikov towers, and obstruction theory. Prerequisite: MATH 7800.

Theory of distributions. Sobolev spaces and their properties (trace and embedding theorems). Theory of elliptic equations. Time-dependent partial differential equations: parabolic and hyperbolic equations. Topics in nonlinear partial differential equations. Prerequisites: MATH 7410 and 7250.

Topics in the theory of operators on a Hilbert space and related areas of function theory.

Topics in the theory of operators on a Hilbert space and related areas of function theory.

This course presents the basic theory of stochastic differential equations and provides examples of its applications. It is an essential topic for students preparing to do research in probability. Topics covered include a review of the relevant stochastic process and martingale theory; stochastic calculus including Ito's formula; existence and uniqueness for stochastic differential equations, strong Markov property; and applications. Prerequisite: MATH 7360 and 7370, or instructor permission.

Discusses fundamental problems and results of the theory of random matrices, and their connections to tools of algebra and combinatorics: Wigner's semicircle law, free probability, Gaussian, circular, and beta ensembles of random matrices, bulk and edge asymptotics and universality, Dyson's Brownian motion, determinantal point processes, and discrete analogues of random matrix models. Prerequisite: MATH 7360 or instructor permission.

Theory of number fields and local fields, ramification theory, further topics as chosen by instructor.

Selected advanced topics in algebraic topology.