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.
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.
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.
In probability, our advanced curriculum focuses on stochastic processes in dynamical systems, ergodic theory, and applications in mathematical finance. Within applied probability for mathematical finance, two topics of emphasis are jump diffusion models and Levy processes.
In statistics, our curriculum reflects our diverse faculty research interests, covering many areas of mathematical statistics, statistical methodology and computational statistics. Asymptotic theory and higher-order asymptotic theory are used to provide accurate distributional approximations for statistical inference. Advanced courses on linear models develop the theory and methodology for linear regression and generalized linear models. Time series analysis is used in many application areas for which observations are made at multiple timepoints. Statistical computation concerns tools for approximating integrals and functions, simulating random variables and processes, sampling from distributions, and more. Multivariate statistical analysis is concerned with both classical tools for clustering and grouping, and also modern techniques for statistical learning and prediction with high-dimensional data. A new course on the mathematical foundations of data science studies high-dimensional probability, random matrix theory, and linear dimension reduction from a mathematical and statistical perspective. Courses in Bayesian statistics present approaches to prior specification, Markov chain Monte Carlo techniques, and Bayesian model selection.