Research

Algebra and Combinatorics »
Hopf algebras, coxeter groups, number theory, algebraic combinatorics, algebraic geometry, complexity theory, graded groups, and cohomology of rings. Applications include computer graphics, coding, and cryptography. There are significant overlaps with the research groups in number theory, geometry, and groups and dynamics.

Applied Mathematics and Interdisciplinary Research »
Many of our faculty conduct research with applications to other areas of science and engineering. Applied and interdisciplinary research overlaps with nearly every other research group in the department. These activities include computational materials science, porous media, fluid mechanics, transport equations, inverse problems and imaging, complexity theory, computational algebra, computational geometry, and mathematical biology. The applied mathematics faculty play key roles in two major university interdisciplinary institutes: theInstitute for Applied Mathematics and Computational Scienceand theInstitute of Scientific Computation.

Approximation Theory »
Approximations by orthogonal polynomials, radial basis functions, and wavelets; futher topics of interest include scattered data surface fitting, rates of approximation, constrained approximation, polynomial inequalities, orthogonal polynomials, wavelets, splines, non-linear approximation. There is significant overlapping interests with the groups in partial differential equations and numerical analysis.

Functional Analysis »
Banach spaces, operator spaces, C*-algebras, von Neumann algebras, nonlinear functional analysis; applications include: probability theory, free probability theory, wavelets, mathematical finance, convex geometry

Geometry and Topology »
Algebraic geometry, algebraic topology, differential geometry, geometric analysis, and discrete geometry. Areas of interest include geometry of distributions, exterior differential systems, projective differential geometry, homogeneous varieties, Fano varieties, calculus of variations in the large, minimal surfaces, sub-Riemannian geometry, calibrations, equivariant, motivic and Deligne cohomology, toric and real algebraic geometry, and applications to theoretical computer science, signal processing and control theory.

Groups and Dynamics »
Topics of interest of this research group include: combinatorics, group theory, geometric methods in group theory, asymptotic group theory, amenability, topological groups and invariant means, random walks on groups and graphs, representations, associated C*-algebras and von Neumann algebras, bounded and L²-cohomology, actions on trees, growth, self-similar groups, groups generated by finite automata, groups of homeomorphisms of the real line, the mapping class groups and other groups arising in topology. The topics related to dynamical systems include theory of billiards, geodesic flows on flat surfaces, symbolic dynamics, substitutional dynamical systems, holomorphic dynamics, analysis on graphs and fractals, entropy, ergodic theorems, low-dimensional dynamics, statistical models on groups and graphs.

Number Theory »
Analytic and algebraic number theory, arithmetic geometry, diophantine approximations, transcendental number theory, elliptic curves and modular forms

Numerical Analysis and Scientific Computation »
Numerical methods for computing approximate solutions to partial differential equations, multiscale methods, geometric partial differential equations, fractional diffusion, complex fluid dynamics, adaptive methods, radiative transport, magnetohydrodynamics, porous media, large scale scientific computation with industrial applications. There is significant overlapping interests with the groups in partial differential equations, approximation theory and data science.

Partial Differential Equations and Mathematical Physics »
Analytical, geometric, and computational approaches to partial differential equations; mathematical aspects of quantum theory, relativity, and other physics; spectral theory and harmonic analysis; inverse problems

Probability Theory »
Probability in Banach spaces, limit theorems, empirical processes, U-processes, probability inequalities, convex geometry, ergodic theory, stochastic differential equations, diffusions, and Brownian motion.

Several Complex Variables »
Bergman kernel, boundary regularity for solutions to the Cauchy-Riemann equations, d-bar-Neumann problem, CR manifolds, CR Extension, CR Approximation