# 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:
the *Institute for Applied Mathematics and Computational Science* and the
*Institute 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.

**
Data Science »**

Compressive sensing, data assimilation, deep learning, inverse problems, kernel and multiscale methods, optimal recovery, scientific computation, and topological data 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 L^{2}-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