Advanced Courses Overview

Analysis

In analysis, we focus on harmonic analysis and its many ramifications and connections with complex function theory, real function theory, operator theory, Toeplitz and Hankel operators, probability theory, and partial differential equations. Functions of one and several complex variables, partial differential equations, and function algebras are treated. Pseudodifferential operators, Fourier integral operators, and calculi of integral operators designed for the study of special partial differential equations and boundary value problems are a specialty. Harmonic analysis is studied as part of classical real and complex analysis in the Euclidean setting as well as in settings associated with Lie groups, Riemannian symmetric spaces, pseudoconvex domains in several complex variables, function algebras, spaces of analytic functions, partial differential equations, local fields, martingales, ergodic theory, Markov processes, diffusion semigroups, and with random walks on graphs. Much work is done on characterizations of Hardy spaces in all these settings, as well as in operator theory and interpolation of Banach spaces. Singular integral operators and boundary behavior are common themes. Problems in all these areas are, of course, also studied in their own right. Wavelets are also extensively studied.

Geometry

The geometry curriculum focuses on differential geometry and associated problems in nonlinear partial differential equations, differential topology, foliation theory, low dimensional topology, harmonic maps, minimal surfaces, Kahler geometry, Finsler geometry, and propagation of singularities. Algebraic aspects of geometry are also studied. These include algebraic geometry using methods of differential geometry, commutative algebra, algebraic K-theory, groups which arise in geometry as global and local symmetries, and spectral theory of manifolds.

Algebra

Our offerings in algebra concentrate on algebraic geometry, algebraic K-theory, commutative algebra, group cohomology, quadratic forms, group rings, and combinatorics. Our algebra group and our geometry group have many interests in common.

Probability and Statistics

In probability, we focus on Markov processes, diffusion semigroups, martingales, and ergodic theory. In statistics, our major efforts are in constructing theoretical models, analyzing population-biology data, and applying statistics in psychiatry and other medical disciplines. Washington University is in the forefront of magnetic resonance imaging. Many of the research projects in this area are joint efforts by members of the Department of Mathematics and faculty at the Medical School.

While only highlights, the above descriptions provide some idea of our mathematical focus. These areas should not be considered disjoint, for many of the Washington University faculty use techniques from several of these areas. At Washington University mathematics is viewed in the broad sense, and that is manifested in the curriculum, the teaching, and in the research that is undertaken.