Our department is highly active in research. Our research specialties are in algebra, analysis, topology, geometry, combinatorics, probability, number theory, and mathematical biology.

## Faculty Research Interests

*Algebraic Geometry*

I work in algebraic geometry, mainly using derived categories of coherent sheaves. My interests include compact hyperkähler manifolds, rationality questions, and classical algebraic geometry.

*Quantum Groups, Representation Theory, Algebraic Combinatorics*

My research interests include Representation Theory of Lie Algebras, Quantum Groups, Coxeter Groups, Hopf Algebras, Algebraic Combinatorics, Cluster Algebras, Noncommutative Algebra, and related aspects.

*Differential Topology, Positive Scalar Curvature, Morse Theory*

I study algebraic topology and differential geometry, with a focus on conformal geometry and the space of metrics of positive scalar curvature.

*Harmonic Analysis, Wavelets, Approximation Theory*

I work in the area of harmonic analysis and wavelets. More specifically my research areas include construction of wavelet bases with good time-frequency localization for large classes of dilations; limitations on the existence of such wavelet bases; and anisotropic function spaces and their study through wavelet bases, general L^2 theory of wavelets, frame wavelets, and generalized multiresolution analysis.

*Algebraic Groups, Combinatorial Representation Theory, Lie Superalgebras*

I study representation theory and combinatorics arising from semisimple Lie algebras and algebraic groups, like the Lie algebra gl_n(C) of all n by n matrices over C and the group GL_n(C) of invertible such matrices.

*Homotopy Theory, K-theory, Homological Algebra*

Most of my work is in motivic and equivariant homotopy theory, these days largely focused on the latter. I study equivariant cohomology theories that are graded by representations rather than by integers.

*Number Theory*

I work primarily in number theory, especially algebraic number theory. My research is largely driven by questions concerning automorphic forms (a class of functions that includes modular forms) and L-functions (a class of functions that includes the Riemann zeta function). I use a variety of techniques, including algebraic, p-adic, arithmetic geometric, and analytic.

*Representation Theory, Categorification*

I study categorical representation theory, a relatively new field that takes representation theory to the next level. Categorical representation theory has an astounding amount of structure, making it an interesting topic of study in its own right, but it also can be used to study ordinary representation theory, especially in finite characteristic.

*Geometry, Gauge Theory*

My research program centers around the asymptotic geometry of the Hitchin moduli space and K3 surfaces. One main goal is to address the beautiful conjectural description of the hyperkähler metric on the Hitchin moduli space by the physicists Gaiotto, Moore, and Neitzke. I approach these intricate conjectures from physics with tools coming out of geometric analysis.

*Differential Geometry and Partial Differential Equations*

I study complex geometry and Kahler geometry, extremal metrics and Calabi flow; geometric evolution equations and mean curvature flow; and nonlinear partial differential equations.

*Algebraic and Topological Combinatorics*

I work in algebraic and topological combinatorics. The topics that interest me include posets, stratified spaces, and the development and refinement of combinatorial-topological techniques such as shellability and discrete Morse theory, total positivity theory, Coxeter groups, and Bruhat order.

*Representation Theory, Lie Theory, Group Theory*

I study representation theory of Lie algebras, algebraic groups and related objects, such as symmetric groups, Hecke algebras, etc.

*Markov Chains and Random Walks, Multiparameter Processes, Potential Theory*

My research is in probability theory, including: random walks, Markov chains, multiparameter processes, jump processes, and related potential theory. Recently, I am interested in quantatitive estimates on the time for ergodic Markov chains to equilibriate.

*Functional Analysis, C*-algebras, Dynamical Systems*

I am currently interested in the structure of C*-algebras and applications of C*-algebra theory in classical topological dynamical systems and non-commutative dynamical systems.

*Low-dimensional and Symplectic Topology*

I use techniques from symplectic geometry and, to a lesser extent, algebraic topology and abstract algebra, to study questions about knots and 3- and 4-dimensional manifolds. Most of this falls under the category of Floer homology and pseudo-holomorphic curves.

*Geometric Analysis, Ricci Flow, Complex Geometry*

My research is in geometric analysis. Currently I am working on Ricci flow, a heat type equation which evolves Riemannian metrics by its Ricci curvature. More precisely I am interested in the ancient solutions and the singularity analysis of Ricci flow.

*Theoretical Neuroscience*

This highly collaborative research group aims to understand how the collective activity of large networks of neurons leads to the emergence of cognitive function and behavior, how information processing in the brain arises through learning and plasticity, and how it is modulated by context and behavioral states.

*Harmonic analysis, partial differential equations, and inverse problems*

My research focuses largely on the long-time behavior of solutions to nonlinear dispersive partial differential equations. This includes problems related to decay and scattering of solutions, as well as stability properties of special solutions known as solitons. In addition, I study inverse problems related to scattering, which ask whether the scattering behavior of solutions is sufficient to recover information about an unknown nonlinearity.

*Theoretical Neuroscience*

Our group’s research involves developing mathematical models of brain computations, drawing from dynamical systems theory, stochastic processes, and reinforcement learning. We also draw on recent developments in artificial intelligence, using artificial neural networks such as the ones used in AI to emulate the complex information processing performed by the brain. Finally, our group works together with experimental collaborators, where our role is to use modeling and data analysis to interpret data and to contribute to experimental design by generating predictions that can be tested in future experiments.

*Geometric Lie Theory, Tensor Categories, Hopf Algebras*

I am currently interested in the categorification of ring theory, that is study of tensor categories and module categories over them, and in geometric representation theory, which means study of representation theoretic questions using tools from algebraic geometry (perverse sheaves and D-modules).

*C*-algebras, Functional Analysis, Noncommutative Geometry*

C*-algebras are special algebraic structures which arise in analysis. The combination of strong extra structure and usefulness in applications has made C*-algebras a broad and very active branch of mathematics. Most of my current research concerns group actions on C*-algebras (often ones of the form C(X)), with emphasis on but not limited to the structure and classification of crossed products.

I also work on operator algebras on L^p spaces for p different from 2. This is a very new area, with less structure but still quite promising, and there are many problems which nobody has looked at yet. The methods are less algebraic than for C*-algebras.

*Algebraic Geometry, Noncommutative Geometry*

My general area of research is algebraic geometry. More specifically, recently I work with problems involving derived categories of coherent sheaves on algebraic varieties, noncommutative geometry and higher homotopy structures (such as A-infinity algebras) appearing in algebraic geometry.

*Combinatorics and Algebraic Geometry*

My work is somewhere in between algebraic geometry, combinatorics, representation theory, and algebraic topology. I work mostly with algebraic varieties that are built using the data of a hyperplane arrangement. I then study the relationship between the algebraic invariants of these spaces and the combinatorial invariants of the input data. Sometimes I use combinatorics to compute objects of intrinsic geometric interest (categories of sheaves, cohomology rings, etc.) and sometimes I use geometric machinery to prove purely combinatorial theorems.

*Mathematical Biology, Evolution, Statistics*

I work on probability and statistics as applied to understanding ecology and evolution, in particular developing new stochastic models of biological evolution and using statistical inference and visualization methods to find out what genomes can tell us about biology.

*Stable Homotopy Theory, Homological Algebra, Complex Cobordism*

I work primarily in stable homotopy theory which is the part of algebraic topology concerned with properties of maps which are preserved after “suspending” (cross X with the unit interval, and identify X x 0 to a point and X x 1 to a point). At the moment the questions I’m working on concern understanding how to compute generalized homology theories (functors from spaces to graded groups which obey most of the axioms of homology theory) on certain types of spaces arising from limit constructions, and their applications.

*Algebraic Geometry and Mathematical Physics*

I am interested in algebraic geometry and mathematical physics. My recent work focus on curve counting theories, such as Gromov-Witten theory and Fan-Jarvis-Ruan-Witten theory, and the mirror symmetry beyond them. These theories have deep connections to complex geometry, number theory, and representation theory.

*Random Matrix Theory, Heights of Polynomials*

I am interested in the statistics of eigenvalues of random matrices and the roots of random polynomials. I am also interested in measures of complexity of polynomials (heights) and the distribution of roots of polynomials with low height.

*Algebraic and Geometric Topology*

I like to see the geometry which underlies homotopical structures. My interests are broad, with my most recent projects being in cohomology operations on manifolds, cohomology of groups, in group theory related to rational homotopy theory, in knot theory, and in neural networks. Configuration spaces are a part of much of what I do, and I like to study topology, geometry, algebra and combinatorics related to them.

*Algebraic Geometry, Knot Theory, Mathematical Physics, Lie Theory*

I study algebra and geometry motivated by physics. My current interests involve algebraic geometry, in particular orbifolds. My past interests have included knot theory and representation theory.

*Geometric Analysis, Partial Differential Equations*

I study Geometric Analysis and Geometric PDE. My most recent work has involved fourth order elliptic minimal surface equations, and other recent work has been on constructions of Ricci curvature with applications to machine learning. In general I study fully nonlinear elliptic PDE such as the Monge-Ampère (sometimes involving optimal transportation) and special Lagrangian equations.

*Approximation Theory, Harmonic Analysis, Orthogonal Polynomials, Numerical Analysis*

I work in several directions in analysis, applied mathematics. My main research is in approximation theory, Fourier analysis, orthogonal polynomials and special functions, which are really all connected. I’m also interested in constructing minimal numerical integration formulas. Most of my work focuses on multidimensional problems.

*Algebraic and Enumerative Combinatorics, Perfect Matchings*

I work in enumerative, bijective and algebraic combinatorics. Most of what I am working on at the moment is related to the dimer model, or to Schubert calculus and the combinatorics of reduced words. Lately I’ve also been spending a lot of time thinking about Kazhdan-Lusztig polynomials for matroids. I use computers heavily in my work.