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. My main technical tool is the derived category of coherent sheaves, but my taste is very much geometric rather than categorical.
Quantum Groups, Representation Theory, Algebraic Combinatorics
My research interests include Representation Theory of Lie algebras and Coxeter Groups, Hopf Algebras, Algebraic Combinatorics 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, frames, and wavelets. My interests span from real-variable harmonic analysis, anisotropic Hardy spaces, frame and wavelet expansions, to more functional analysis topics such as Kadsion-Singer problem and characterization of diagonals of self-adjoint operators.
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 × n matrices over C and the group GL_n(C) of invertible such matrices. Recently, I have been working on various diagrammatic monoidal categories which have a rather combinatorial flavor, take a look for example at https://arxiv.org/abs/2305.05877 to see the spirit of the genre.
Random Matrices, Spin Glasses, Theoretical Machine Learning
My research is in probability. More specifically, I study random matrices and their applications to spin glasses and machine learning. My spin glass research focuses on the spherical Sherrington-Kirkpatrick model and variations of it (particularly fluctuations and phase transitions). My machine learning research focuses on limit theorems for the dynamics of stochastic gradient descent. The problems I study are generally high-dimensional, inspired by questions in physics (where high-dimensional means many random particles) or data science (where high-dimensional means lots of data parameters). While the inspiration for these problems is applied, my techniques and results are more theoretical in nature. Mathematical skills needed for this type of research come from measure theory, probability, complex analysis, and differential equations.
Homotopy Theory, K-theory, Homological Algebra
My current interests are mostly in homological commutative algebra. I maintain some interest in various parts of homotopy theory and K-theory, but for the most part I am not actively working on those topics.
Number Theory
My research lies primarily in algebraic number theory and focuses on mainly on topics concerning automorphic forms, L-functions, and p-adic methods. Number theory is a beautiful subject, partly because it contains many statements that are simple to understand but whose proofs rely on hard, deep (but elegant) mathematics. The simplicity with which some questions can be stated and the ease with which one can prove some number theory results using only undergraduate algebra lead some students to the false conclusion that number theory is an especially accessible area of mathematics. In fact, modern algebraic number theory integrates sophisticated results from many areas of mathematics, and so if you work in algebraic number theory, you should have an open mind and be prepared to keep learning different flavors of mathematics.
Dynamical Systems, Ergodic Theory
My research interests include smooth dynamical systems, ergodic theory, and the interplay between dynamics, geometry, and spectral theory. My work focuses on studying flexibility and rigidity of smooth dynamical systems and producing natural invariant measures that encode dynamical behavior.
Representation Theory, Categorification
I work in the field of categorification, which is the act of taking something you love and replacing it with something far more interesting, and the study of this new and interesting structure. For example, integers are categorified by vector spaces: the number 2 being replaced by a 2-dimensional vector space, addition being replaced by direct sum, and so forth. However, one can discuss linear transformations between vector spaces, while there is no corresponding structure to be studied between integers. Some of the best examples of categorification come from geometric representation theory. This is a pretty high-tech area of math, but my specialty lies in making it very explicit and hands-on, and in doing lots of calculations. I expect students to do lots of examples and gain expertise through exercise.
Geometry, Gauge Theory
I am interested in the moduli spaces of solutions of gauge theoretic equations (PDEs that have a lot of local symmetries that often have origins in physics). The spaces of solutions of these equations tend to be interesting geometric objects in their own right. At one extreme, the space of solutions might be a manifold with a hyperkaehler structure. One example of this is the Hitchin moduli space. At the other extreme, the space of solutions might be a collection of points, and counting them might be an interesting invariant of something.
Techniques in my work involve differential geometry, nonlinear elliptic PDEs, complex geometry and in particular Kaehler and hyperkaehler geometry.
Symplectic Topology, Algebraic Geometry, Mirror Symmetry
I study symplectic topology and algebraic geometry. These two fields are related by homological mirror symmetry, and I am working on understanding this correspondence and its consequences. On its face, homological mirror symmetry is quite surprising at it relates invariants of differential geometric or topological nature (Fukaya categories) to objects of the more rigid world of algebraic geometry. The distinct natures of the two sides make this area particularly intriguing and allow for new viewpoints and applications in both symplectic and algebraic geometry. Thus far, I have largely focused on a thorough understanding of homological mirror symmetry in computable cases with an eye towards applications and more general structural results. This area of research also benefits from frequent interaction with many adjacent areas of mathematics.
Differential Geometry and Partial Differential Equations
I work in differential geometry, geometric analysis and nonlinear elliptic partial differential equations. One of my main interest is the study of complex geometry, including Kahler geometry, complex Monge-Ampere equations, the extremal metrics and Sasaki-Einstein metrics. I also work in geometric evolution equations such as the Calabi flow, mean curvature flow and Sasaki-Ricci flow. To work with me, a student is supposed to take 600 differential geometry course. Depending on your interest, my students can start with a year long reading course including nonlinear elliptic PDE, Riemannian geometry and complex geometry.
Algebraic and Topological Combinatorics
I am interested in topological and algebraic combinatorics, including studying combinatorial-topological structure of partially ordered sets (posets) and of stratified spaces arising e.g. from combinatorial representation theory and total positivity theory. I especially like advising students in poset topology, including working with techniques such as shellability and discrete Morse theory and studying group actions on posets.
Representation Theory, Lie Theory, Group Theory
I study representation theory of Lie algebras, algebraic groups and related objects, such as symmetric groups, Hecke algebras, quantum groups, etc. You can find more details on my web page.
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.
Low-dimensional and Symplectic Topology
I work on applications of symplectic geometry and related tools to smooth 3- and 4-dimensional topology. This involves techniques from algebraic topology, differential geometry, partial differential equations, and homological algebra. Some background and interest in all of those subjects is needed in order to make progress, but which subjects are most important depends on the problem.
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. I am willing to take one student.
Theoretical Neuroscience
I was trained as a theoretical physicist (string theory), with a focus on how macroscopic phenomena emerge from the collective dynamics of strongly coupled degrees of freedom. My current research interests are in theoretical neuroscience. My goal is to understand the neural basis of sensory perception, how it is modulated by expectations leading to behavior. Some of my recent projects address the following questions: Which mechanisms underlie associative learning at the level of synaptic plasticity in local cortical circuits? How does behavior emerge from the temporal dynamics of cortical networks?
To study these questions, I use methods from statistical physics, information theory, machine learning, and dynamical systems. I combine statistical analysis of neurophysiological and imaging data from large populations of neurons in behaving animals with theoretical models based on neural networks. My projects rely heavily on numerical analysis and computer simulations of recurrent networks.
Harmonic analysis, partial differential equations, and inverse problems
My research is mostly in the analysis of nonlinear partial differential equations, with a focus on nonlinear dispersive equations like the nonlinear Schr¨odinger equation. My research uses a lot of tools from different areas of analysis (e.g. harmonic analysis, functional analysis, spectral theory). I am particularly interested in understanding the long-time dynamics of solutions (scattering theory, stability of solitons, and so on). I generally like to study problems that are motivated by some interesting underlying physics (e.g. nonlinear optics). I have also done some work on inverse problems, primarily still in the setting of nonlinear dispersive PDE, and I am currently working to learn more about completely integrable models. I am always looking to broaden my research interests. For motivation, I tend to look to physics and other applications.
Theoretical Neuroscience
My research is in theoretical neuroscience. The scientific questions that my group seeks to address focus on how neurons within and between different brain regions learn things such as selecting actions or how to perform a new motor skill. First, we address these questions by developing mathematical models that draw on random matrix theory, dynamical systems, spin glass theory, stochastic processes, optimal control theory, and reinforcement learning. Second, our group’s research also leverages connections between the fields of neuroscience and artificial intelligence, which works by training artificial neural networks to perform tasks ranging from face recognition to playing video games. Much of our research takes inspiration from recent AI advances, applying them to questions about the architectures of neural circuits in the brain and the algorithms that they use to learn. 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. I would be happy to talk to students interested in mathematical model building and/or deep learning who are also eager to learn about and apply their research to address neuroscience questions.
Geometric Lie Theory, Tensor Categories, Hopf Algebras
My research area is study of tensor categories (which can be considered as a categorification of ring theory). I am also interested in applications of tensor categories to classical representation theory, e.g. to study of decompositions of large tensor powers of representations into indecomposable summands. I am particularly focused on the study of symmetric tensor categories over fields of positive characteristic.
C*-algebras, Functional Analysis, Noncommutative Geometry
I currently have one student, and would be happy to take one or two more students. A new student would be able to also interact with my current student.
I study C*-algebras, which are special algebraic structures which arise in analysis. The easiest examples of C*-algebras are C(X), the algebra of all continuous functions on a compact Hausdorff space X, and L(H), the algebra of all continuous linear operators on a Hilbert space H. The combination of strong extra structure and usefulness in applications has made C*-algebras a broad and very active branch of mathematics. For example, the C*-algebra associated to a locally compact group G encodes the representation theory of G. More generally, the crossed product C*(G, A) is made from an action of G on a C*-algebra A. The study of these algebras connects with dynamical systems.
Much of my current research concerns group actions on C*-algebras, with emphasis on but not limited to the structure and classification of crossed products. Even when the group is Z and the C*-algebra is C(X), or when the group is Z/2Z and the C*-algebra is simple, many questions remain open. Some, but by no means all, of my research uses substantial algebraic topology.
I have also done some work on operator algebras on L^p spaces for p ̸= 2. This is a promising new area, and there are many problems which nobody has looked at yet.
Algebraic Geometry, Noncommutative Geometry
My general area of research is algebraic geometry. I am interested in topics related to moduli spaces of curves, derived categories of coherent sheaves, mirror symmetry, noncommutative geometry and supergeometry. I can take one new student.
Combinatorics and Algebraic Geometry
My interests lie at the interface between algebraic geometry, combinatorics, representation theory, and algebraic topology. In practice, this usually means that I start with some combinatorial data (such as a collection of hyperplanes in a vector space), use it to build an algebraic variety (such as the complement of the hyperplanes, or some compactification thereof), and study algebraic invariants of this variety (such as its cohomology). Here is an example of a problem that I have worked on:
Suppose you are given a set of 53 hyperplanes in R^10 whose common intersection is the origin, and you look at all of the vector spaces of any dimension that can be obtained by intersecting some of these hyperplanes. Then the number of 3-dimensional vector spaces that you get is at least is great as the number of 7-dimensional vector spaces. This was an open conjecture for over 40 years, and was recently proved by studying the cohomology ring of a certain variety.
My mathematical tastes run toward the concrete. I respect abstraction and I use categories a lot in my work, but I’m happiest working on problems in which the abstract tools can be used to tell you something about actual numbers.
Mathematical Biology, Evolution, Statistics
I work at the interface of computational biology, evolutionary theory, stochastic processes, and data analysis. The general goal of my work is to better understand how evolution works - how species adapt to environmental change, how they can maintain useful genetic differences across diverse conditions, and how one species splits into two (or many, or two merge into one) - and how we can learn about evolution by looking at genomes. The basic rules (called “population genetics”) are about as well understood as those of thermodynamics, but there are many major outstanding questions about how evolution works in practice, and to resolve them we need both new mathematical models and new methods for analyzing large-scale genomic data. I am particularly interested in problems related to geography (2D stochastic processes) and inference with ill-posed inverse problems.
Students working with me might try to identify universality classes for stochastic models of genetic change in populations, or might spend a lot of time on the computer analyzing large genomic datasets, or something in between.
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 × 0 to a point and 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. I’m also working on understanding the relationship between operads and a certain kind of homotopy approximation that is analogous to Taylor approximations.
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 work in random matrix theory, mathematical statistical physics and number theory. Specifically I study the statistics of eigenvalues of random matrices and roots of random polynomials especially as applied to electrostatic systems and the distribution of algebraic numbers. The background to approach such problems should include measure theory, linear algebra and probability (and algebra for number theory). If you think you may be interested in working with me, I am always happy to chat about my research and/or develop a plan of study of mutual interest.
Algebraic and Geometric Topology
I work in a range of areas in topology, mostly in algebraic topology but some in geometric topology as well. I like to see the geometry which underlies homotopical structures. My interests are broad, with my most recent projects being in cohomology of alternating and symmetric 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. Some of my research along with other topics of interest in algebraic topology are the subject of a lecture series available at:
Algebraic Geometry, Knot Theory, Mathematical Physics, Lie Theory
I study problems in algebra and geometry motivated by physics. Currently I am interested in quantum invariants in algebraic geometry (Gromov-Witten theory, mirror symmetry, quantum singularity theory) and low-dimensional topology (invariants of knots, links and three manifolds coming from quantum groups and Lie algebras). My other area of interests is theory of supermanifolds and its applications in Poisson geometry and Lie theory.
Geometric Analysis, Partial Differential Equations
I study Geometric Analysis and Geometric PDE. Recently I have been considering problems dealing with optimal transportation, including synthetic constructions of Ricci curvature, fully nonlinear elliptic PDE such as the Monge-Amp`ere and special Lagrangian equations, and also some problems in minimal surface theory. I would be happy to talk to anybody who is interested.
Approximation Theory, Harmonic Analysis, Orthogonal Polynomials, Numerical Analysis
I work in several areas in analysis: approximation theory, Fourier analysis, orthogonal polynomials and special functions, mostly in multi-dimensional setting. My recent work is on approximation by polynomials in Sobolev spaces on regular domains, and orthogonal polynomial on curves and surfaces.
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. To some degree I am also a generalist, solving combinatorial problems from other areas of mathematics. I use computers heavily in my work.
Symplectic Geometry
My research focuses on using tools of symplectic geometry applied to low-dimensional topology. Most of my work uses Heegaard Floer homology, with some work on Khovanov homology and other gauge theoretic invariants. Most of my work applies these tools to study smooth structures on 4-manifolds, surfaces in 4-manifolds, knot concordance, or homology cobordism. Problems in low-dimensional topology can involve a variety of mathematical areas, including homological algebra, algebraic topology, PDEs on manifolds, algebraic geometry, and representation theory. Students will typically focus on one sub-area or tool initially.
