- Home
- About the Department
- Research
- People
- Undergraduate Program
- Introduction
- Course Information
- Concentration in Mathematics
- First-year Information
- Paris Math Program
- Opportunities and Extracurricular Activities
- Junior Tutors, VCAs, and Readers
- Directed Reading Program
- Urban Teacher Education Program
- Association for Women in Mathematics
- Employment Resources
- Contact/Help

- Graduate Program
- Outreach
- Resources

## Research Interests

### Algebra, algebraic geometry, number theory

**Jonathan Alperin:**
Representation theory of finite groups emphasizing homological and local methods.

**Laszlo Babai: ** see here.

**Alexander Beilinson:**
Arithmetic algebraic geometry, geometric Langlands program.

**Spencer Bloch****: **
My interests are algebraic geometry, K-theory, and number theory, with focus on the
theory of motives. Recent work concerns motives associated to Feynman
graphs in physics, and Euclidean limits of motives. Recent talks and
publications are posted on my website
here.

**Frank Calegari:**
My research is in the area of algebraic number theory. I am
particularly interested in the Langlands programme, especially, the notion
of reciprocity linking Galois representations and motives to automorphic
forms. I am also very interested in the cohomology of arithmetic groups,
especially in torsion classes.

**Vladimir Drinfeld:** I am mostly interested in the Geometric Langlands
program, which is a part of Geometric Representation theory.
Here you can find
Victor Ginzburg's description of the subject of geometric representation
theory and the literature that he recommends.

Jointly with Mitya Boyarchenko (a student of mine) I am trying to develop the theory of character sheaves for unipotent groups. A unipotent group is a subgroup of the group of strictly triangular matrices defined by algebraic equations. Let G be a unipotent group over a finite field k. For each positive integer n the points of G in the degree n extension of k form a finite group. Let X(n) be the set of its irreducible characters. Our goal is to understand X(n) for all n simultaneously in terms of certain perverse sheaves on G, which are called character sheaves. Such a theory was developed by Lusztig for reductive groups G. Inspired by a remarkable and short e-print by Lusztig, Mitya and I are trying to do this in the quite opposite case of unipotent groups.

**Alex Eskin:** see here.

**Victor Ginzburg:**
I work mostly in geometric representation theory and in noncommutative geometry.

Geometric representation theory tries to apply the methods of algebraic geometry for studying representations of various algebras important from the representation theoretic perspective. Typical examples include:

- Classification of irreducible representations
of Hecke algebras (Deligne-Langlands-Lusztig conjecture)
in terms of K-theory and perverse sheaves;

- Applications of D-modules and perverse sheaves to representations
of complex or real reductive groups and to semisimple Lie algebras
(Kazhdan-Lusztig conjecture);

- The study of integrable representations
of quantum groups using the geometry of quiver varieties (Nakajima);

- Geometric Langlands program.

To get more details I suggest to look at the Intro in our book:
Chriss-Ginzburg, *Representation Theory and Complex Geometry*
(Birkhauser Boston, 1997), or at my survey article
*Geometric Methods in Representation Theory of Hecke Algebras and Quantum Groups*.

During the last 5-10 years, I've also got interested in what may be called noncommutative geometry. Some of the inspiration comes from the theory of quivers (I teach a course on quivers quite frequently). Another source of inspiration comes from Mirror symmetry (e.g., Calaby-Yau categories) and, more generally, from the mathematics appearing in string theory. To get a rough idea of what I mean, you may want to look at the following papers:

- V. Ginzburg,
*Lectures on Noncommutative Geometry* - V. Ginzburg,
*Non-commutative Symplectic Geometry, Quiver varieties, and Operads*, Math. Res. Lett. 8 (2001), no. 3, 377-400 - P. Etingof, V. Ginzburg,
*Symplectic reflection algebras, Calogero-Moser space, and deformed Harish-Chandra homomorphism*, Invent. Math. 147 (2002), no. 2, 243-348

I have 7 graduate students at the moment; all of them choose their own favorite topic for research, not necessarily directly related to what I'm doing myself. However, I do have joint projects with some of my students.

**George Glauberman:**
My main research is in the area of finite p-groups. I have made
extensive use of techniques from algebraic group theory, and have
shown that every finite p-group has "large" characteristic
subgroups that share some of the properties of algebraic groups.
Recently, I have found some applications to local analysis and
representation theory. I am trying to clarify which subgroups
most closely resemble algebraic groups.

**Robert Kottwitz:**
I am interested in automorphic forms from a number-theoretic point of view as well as the representation theory of reductive groups over local fields.

**Madhav Nori:**
I am interested in Algebraic Cycles, K-theory, Hodge theory, Galois theory, and their interactions.

**Niels Nygaard:**
My research is mainly concerned with the interplay between the geometry and arithmetic of modular varieties. These are varieties which are parameter spaces of families of algebraic varieties of certain types with various structures. In particular I have been interested in Siegel modular three folds which parametrize abelian surfaces. Conjecturally the cohomology of these three folds is intimately related to Siegel modular forms and one of the goals of my research is to make this relation explicit. This has been achieved in a number of interesting examples, which has given significant information of what one can expect in general.

### Analysis, probability theory

**Robert Fefferman:**
I am interested in Harmonic Analysis and Partial Differential Equations. Of particular interest are the topics of maximal functions and differentiation of integrals, multi-parameter problems in Harmonic Analysis, and PDE with minimal smoothness assumptions on either the coefficients or domain of definition.

**Carlos Kenig:**
I work in the fields of harmonic analysis and partial differential
equations. I recent years I have been interested in the study of free
boundary problems, particularly regularity questions and the connections
with potential theory and geometric measure theory. I have also been
studying various aspects of unique continuation, and have given
applications of it to problems in mathematical physics, like Anderson
localization and to inverse problems, such as the inverse conductivity
problem. I continue to have an active interest in the theory of elliptic
boundary value problems under minimal regularity assumptions. Finally, a
large chunk of my research is devoted to the study of nonlinear
dispersive equations. I continue to study the issue of well-posedness for
various models, in low regularity spaces of data. I have also been
studying, recently, issues related to the global behavior in time,
asymptotics, scattering and blow-up.

**Gregory Lawler:**
I am a probabilist who specializes in random walks
and continuous analogues (Brownian motion and diffusions)
with a special emphasis
on processes arising in statistical physics such as
self-avoiding random walk, loop-erased random walk
(uniform spanning trees), and percolation. In the
last ten years, I have been involved in the rigorous
study of continuum limits of two-dimensional models.
In these cases, conformal invariance becomes an important
aspect of the systems. A key tool is the
Schramm-Loewner evolution (SLE) although much also can
be done by considering measures derived from Brownian
motion (limits of simple random walks). One of
the big challenges is finding means to
describe the evolution of a random fractal curve
which interacts with its past.

**Paul Sally:**
I am currently working on the representation theory and harmonic analysis on reductive p-adic groups. I am particularly interested in applications to the theory of automorphic forms.

**Wilhelm Schlag:**
In a wide sense, I work on problems in mathematical physics that can be
approached
by means of methods from harmonic analysis and analysis in general.

Topics here include nonlinear PDE such as the nonlinear Schroedinger equation and the nonlinear wave equation, further spectral theory of both self-adjoint and non-selfadjoint operators; as far as the former is concerned, I mainly consider Schroedinger operators with potentials defined by an ergodic process (such as the evaluation of a function on the torus along an orbit of the shift or skew-shift). Such "random potentials" are most famously connected with the Anderson conjecture about existence of extended states for the three-dimensional random Schroedinger operator (first conjectured in 1957). As far as nonlinear equations are concerned, I have studied stability questions of nonlinear boundstates of both the Schroedinger and wave equations. More specifically, I have addressed the existence of center-stable manifolds for orbitally unstable bound states.

Papers and lecture notes can be found here.

**Panagiotis Souganidis:** I work in the field of partial differential
equations. I am interested in the study of qualitative properties of
solutions to elliptic/parabolic and hyperbolic (conservation laws and
Hamilton-Jacobi equations) pde, their applications to, among others,
phase transitions and mathematical biology, and interactions with
stochastic analysis. Among my most recent research programs are the
development of the theory of stochastic viscosity solutions for fully nonlinear
stochastic pde with multiplicative noise, the study of
homogenization in random environments, and the analysis of
biological models for motor effects.

**Sid Webster:** I work on the holomorphic geometry of smooth bounded
domains in the complex space **C**^{n}. It is conjectured, and known in many
cases, that biholomorphic maps of such extend smoothly to the boundary. In
the Levi non-degenerate case, the induced CR structure on the boundary has
a complete system of invariants, manifested in a normal form (Chern-Moser
theory). Some general problems are: 1) Determine Fefferman's asymptotic
expansion of the Bergman and Szego kernels more precisely in terms of these
and related invariants. 2) The holomorphic embedding problem (local
existence
and regularity) for formally integrable CR structures. 3) Geometry of CR
singularities, especially for real n-manifolds in **C**^{n}, normal forms, hulls
of holomorphy, etc.

Recently, my former graduate student, Prof. X. Gong, and I have obtained solutions to the local CR-embedding problem (2), and to the integrability problem for CR vector bundles, which have sharp regularity. This has led to my discovery of new invariants, both local and global, of a fundamental solution to the Cauchy-Riemann equations in several complex variables.

### Applied mathematics

**Jack Cowan:**
My main work is to try to understand the circuitry of the visual cortex and how it mediates visual perception. I use a combination of linear and nonlinear dynamics, symmetry groups and bifurcation theory to investigate how neural circuits can generate stable patterns of activity. The results are relevant to a wide range of observations in neurobiology and in cognitive psychology.

Another interest of mine is the mathematics of the stock market and the throy of option pricing. I am interested in the non-Gaussian aspects of price fluctuations and their origin. I use random graph theory and self-organized criticality to investigate such problems.

**Todd Dupont:**
The main thrust of my research is the construction, analysis, and evaluation of numerical methods for partial differential equations (PDE's), but I also have had interests in related areas such as the construction of mathematical models for physical and biological systems.
Approximate solution of PDE's is frequently computationally expensive, even for problems that are conceptually simple. I have been studying ways of using adaptivity to make some of these calculations more efficient and robust. For time-dependent problems the use of meshes that move smoothly with time can be of significant value in producing high quality solutions to difficult problems. Although a general solution to the question of how to use such meshes has not yet been found, there are many situations that I have looked at with my students in which such procedures can be both effective and simple.
The computation of free surface flows is important in several of the projects that I am working on at the moment. These involve the formation of drops under various conditions, modeling of the flow of a fluid over a solid surface, and two fluid flows.

**Norman Lebovitz**: My
current research is directed towards mathematical questions in
turbulent flows of fluids. In particular, in the onset of turbulence
in shear flows, there is a significant parameter regime in which the
flow appears to become turbulent but subsequently returns to an
ordered state. This raises questions about the geometric structure of
invariant manifolds of the (infinite-dimensional) dynamical systems
describing these flows. I have been investigating this by first
understanding the geometry in a sequence of finite-dimensional analogs
of these systems and identifying features that appear to be
independent of dimension.

**Ridgway Scott:** See my
homepage.

### Geometry, topology

**Danny Calegari**: My
interests include geometry, topology and dynamics in low dimensions;
foliations/laminations; and extremal problems in topology and
geometric group theory. For a fuller description, see http://math.uchicago.edu/~dannyc/advisor.html

**Kevin Corlette:**
My research interests lie in differential and algebraic geometry. I am particularly interested in Kahler geometry and locally symmetric spaces, as well as systems of partial differential equations with geometric meaning, such as the harmonic map and Yang-Mills equations.

**Alex Eskin:**
My recent research interest has been ergodic theory and discrete groups, most particularly the connections to number theory.

**Benson Farb****:**
My interests lie at the juncture of geometry, topology and discrete groups. More recently, the interaction of topology, representation theory, algebraic geometry and arithmetic. One common theme is how complicated objects
are sometimes determined by very simple data. Here is a great example: Let M and N be two closed, locally symmetric (with no local torus
factors), nonpositively curved manifolds of dimension at least 3. The Mostow Rigidity Theorem states that if M and N have isomorphic
fundamental groups, then M must be isometric to N. In particular, invariants such as volume are actually topological invariants!

Examples I study include discrete subgroups of Lie groups, 3-dimensional manifolds, groups acting on nonpositively curved singular spaces (e.g. trees, affine buildings, etc.), and groups of diffeomorphisms. I enjoy working in a number of different directions. These have included: geometric and combinatorial group theory; trying to understand the relationship between volume and degree (a topic combining differential geometry and ideas from dynamics); actions of infinite groups on manifolds; classifying manifolds with (sometimes hidden) symmetry.

In the last few years I have been focussing on studying mapping class groups and the moduli space of Riemann surfaces. Of special interest is the Torelli group, a classically studied but poorly understood group which is in many ways the mysterious part of the mapping class group. The interplay of combinatorial topology, 3-manifold theory, algebraic geometry and symplectic representation theory that one sees in this topic is especially fascinating.

**Peter May****:**
I am interested in a variety of topics in and around algebraic topology.
The calculational parts of the subject tend to focus on stable homotopy
theory, which includes all of homology and cohomology theory, and that
area has changed drastically in the past decade with the introduction of
categories of spectra (``stable spaces'') in which one ``can do algebra''.
Much of the new foundational theory was developed here. There are many
active related areas. For example, there are far-reaching interactions
with algebraic geometry, including applications of algebraic topology to
algebraic geometry (Voevodsky et al), applications of algebraic geometry
to algebraic topology (Hopkins et al), and the development of an algebraic
geometry of ``brave new rings'' in stable homotopy theory (Lurie, Toen and
Vessozi). There is also a new Galois theory of brave new rings (Rognes et
al). Increasingly, ``homotopy theory'' has come to have both a narrow sense
(the homotopy theory of topological spaces and spectra) and a broad sense
(homotopical algebra, including derived categories, DG-categories, etc).

I've also become interested in a variety of topics in and around category theory. There is a new subject of higher category theory that is just beginning to be understood. Here at Chicago, the development of parametrized homotopy theory, which allows one to do stable homotopy theory while keeping track of such basic unstable structure as fundamental groups, has led unexpectedly to a new duality theory in symmetric bicategories which turns out to give exactly the right framework for fixed point theory in algebraic topology and Morita theory in algebra. The unfocused focus of the algebraic topology and category theory group is on such interactions between different areas of mathematics.

**Leonid Polterovich****:**
I am mainly working in symplectic and contact topology with
applications to Hamiltonian dynamics. Some of my favorite projects
include:

- function theory on symplectic manifolds, including rigidity of the Poisson bracket and theory of symplectic quasi-states (with Misha Entov);
- algebraic properties of groups of Hamiltonian diffeomorphisms such as existence of quasi-morphisms coming from Floer theory (with Misha Entov) and restrictions on finitely generated subgroups in the spirit of the Zimmer program;
- invariant geometric structures on diffeomorphism groups, including partial order on groups of contact transformations (with Yasha Eliashberg) and Hofer's geometry on groups of Hamiltonian diffeomorphisms.

In a different direction, I am interested in eigenfunctions of the Laplace-Beltrami operator, in particular in geometry of the nodal sets (with Misha Sodin, Fedor Nazarov and Iosif Polterovich).

**Sid Webster:** see here.

**Shmuel Weinberger****:**
Most of my research is concerned with understanding things
geometrically or understanding geometric things. The main directions are:

- Topology of (mainly high dimensional) manifolds.
- Global analysis (e.g. L^2 cohomology and index theory) on noncompact manifolds and its coarse nature.

These topics are somewhat related to the Novikov conjecture (although that is only one important aspect). Jonathan Rosenberg maintains a web page of developments related to this problem (and to the Borel and Baum-Connes conjectures). In general, one often connects the fundamental group to invariants of manifolds with that fundamental group.

- Singularities, e.g. orbifolds, but much more serious as well. The main reference is probably to my book "The topological classification of stratified spaces" (but that is somewhat out of date.)
- Applications of logical and computer scientific ideas to variational problems and the large scale geometry of certain moduli spaces. Again I have written a book on this ("Computers, Rigidity and Moduli: The large Scale Fractal Geometry of Riemannian Moduli Space"): this is mainly joint work with Alex Nabutovsky.
- Quantitative topology: i.e. studying the precise nature of solutions to problems that are produced existentially by algebraic topology. This is a vague theme that includes a number of points of contact with the previous topics. But, you can do well to look at the papers in Gromov's bibliography that allegedly refer to this to capture a good deal of the scope.

I have also been involved recently in applications of algebraic topology to the analysis of large data sets, to robotics, and to economics. Some of this is the focus of a program at MSRI in Fall 2006, and a conference in Zurich (July 2006). See this article about the last one.

### Logic, theoretical computer science and combinatorics

**Laszlo Babai:**
I work in the fields of theoretical computer science and discrete mathematics;
more specifically in computational complexity theory, algorithms, combinatorics, and finite groups, with an emphasis on the interactions between these fields. Asymptotic questions and probabilistic methods are common features in my work in each of these areas. The introduction of Las Vegas algorithms, interactive proofs, holographic proofs (proofs verifiable by spotchecks) are among the conceptual highlights. A recent example: methods of the complexity theories of Boolean circuits and branching programs have been brought to bear on the analysis of a popular random sampling technique in computational group theory.

**Denis Hirschfeldt:**
I work in computability theory, and am particularly interested in
applying methods from computability theory to computable model theory,
computable algebra,
algorithmic randomness, and the computability theoretic and
reverse mathematical analysis of combinatorial principles.

The articles in the *Handbook of Recursive Mathematics*
(Ershov, Goncharov, Nerode, and Remmel, eds., Stud. Logic
Found. Math. 138 - 139,
Elsevier, Amsterdam, 1998) are a good introduction to computable model
theory and related areas. See also *Computability-Theoretic
Complexity of Countable Structures* by V. S.
Harizanov.

My article *Calibrating
Randomness*
with R. Downey, A. Nies, and S. A. Terwijn is a
survey of work in algorithmic randomness. A more recent and detailed
survey
is my book *Algorithmic
Randomness and Complexity* with Downey.

My article *Slicing the
Truth* is an introduction to the
computability theoretic and reverse mathematical analysis of
combinatorial principles.

My papers can be found on my website.

**Alexander Razborov****:**
During my career I have been working on various topics in logic, TCS and
combinatorics, including combinatorial group theory, circuit complexity,
quantum computing and communication complexity. These are the two projects that are
currently the most active.

1. Continuous Combinatorics (flag algebras, graph limits etc.) It is a relatively new area striving to study traditional combinatorial structures via their infinite abstractions, usually on a measure space. This study involves methods from and make connections to algebra, analysis, measure theory, mathematical logic, probability theory etc.

2. Propositional Proof Complexity. This area draws inspiration both from
mathematical logic (what true statements possess *efficient* proofs?)
and from Computer Science (e.g., what are the general mathematical principles
underlying practical SAT solvers?)

More information, including a few ad hoc problems, can be found here.

**Robert Soare****:**
I maintain a list of research interests and other information on my homepage,
here.