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.

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:

  1. Classification of irreducible representations of Hecke algebras (Deligne-Langlands-Lusztig conjecture) in terms of K-theory and perverse sheaves;
  2. Applications of D-modules and perverse sheaves to representations of complex or real reductive groups and to semisimple Lie algebras (Kazhdan-Lusztig conjecture);
  3. The study of integrable representations of quantum groups using the geometry of quiver varieties (Nakajima);
  4. 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. This subject is rather vaguely defined. Some of the inspiration comes from the theory of quivers (I teach a course on quivers quite frequently). For a good survey you may look at the lectures by Crawley-Boevey. Another source of inspiration comes from Mirror symmetry 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:

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: I am working mainly on finite groups, especially properties of the large abelian subgroups that generate Thompson's J-subgroup in p-groups.

Mark Kisin: My current interests are mainly in p-adic Galois representations, and in particular the question of when such a Galois representation arises from a modular form.

Some of the key tools in this area are p-adic Hodge theory and the arithmetic of modular curves. There is also an exciting connection with the p-adic local Langlands correspondence, which is just beginning to be understood. For more details, see here.

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: Exploring the properties of the Motives. Questioning how far Voevodsky's triangulated category is from the derived category of Motives. Giving motivic structure to the higher homotopy groups of the nilpotent tower of a variety. Construction of a category of motivic sheaves.

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.

Paul Sally: see here.

Analysis, probability theory

Peter Constantin: see here.

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.

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.

Sid Webster: I work on the holomorphic geometry of smooth bounded domains in the complex space Cn. 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 Cn, normal forms, hulls of holomorphy, etc.

Applied mathematics

Peter Constantin: I am interested in partial differential equations: regularity and blow up of solutions, long time dynamics, statistical solutions, singular limits. The tools involved come from harmonic analysis, dynamical systems and stochastic analysis. The applications are to problems of nonlinear and statistical physics arising from condensed matter physics, geophysics and astrophysics. I'll single out three areas:

  1. Navier-Stokes equations. For some directions of research look at the first item here and here . Related topics: the QG equation, active scalars, Littlewood-Paley spectra, background field method for dissipation bounds, subgrid models, shell models. In recent years I have been interested in diffusive Lagrangian representations for the Navier-Stokes equation (see my web page and that of my former student G. Iyer here).
  2. Reaction-diffusion systems coupled with fluid equations. In the basics physics group of the Astrophysical Thermonuclear Flashes center in Chicago devoted to exploding stars, we are interested in flame models, and in particular in the effect that hydrodynamic instabilities have on flame propagation. A surprising tool in combustion theory: the Malliavin calculus.
  3. Smoluchowski equations and more general nonlinear Fokker-Planck systems. The applications are to complex fluid-particle systems, in which microscopic inclusions are suspended in fluids. Together with my collaborators we are interested in the transition from disorder to order (by lowering temperature) and in the patterns and dynamical modes of propagation that characterize this transition.

Some papers on all the above can be found on my webpage.

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.

Lenya Ryzhik: My research deals with the mathematical description of systems with random noise and the effect of the noise on the dynamical system. A typical and quite old example is what happens to a particle in a weakly random force field - after a long time it behaves as a Brownian particle. Similar limit theorems can be shown for partial differential equations with oscillatory coefficients (random or periodic) that describe numerous physical phenomena - from the semiclassical limits of the Schroedinger equation to seismic waves in the Earth. The basic machinery is known as homogenization theory.

Another area that I have been interested in recently are the reaction-diffusion equations. This is a very classical area that has its origins in the works of Kolmogorov, Petrovksii, Piskunov and Fisher in the 1930's and has applications ranging from combustion to biology. The main aspect that interests me is the interplay between fluid dynamics and reaction-diffusion and its effect on the qualitative behavior of solutions . This area is full of simple questions that lead to cute answers. For instance, imagine that you want to stir a cup of coffee in the most effective way. The best way to do that is to make sure that your flow does not have smooth eigenfunctions.

My papers can be found on my website.

Good references (even if not most up to date) for the some of the basic issues in random media are:

  • G. Papanicolaou, Mathematical Problems in Geophysical Wave Propagation. In: Proceedings of the International Congress of Mathematicians, `Documenta Mathematica', Extra Volume ICM 98 I, pp. 241-265, Berlin 1998
  • G. Papanicolaou, Diffusion in Random Media. In: Surveys in Applied Mathematics, edited by J.B.Keller, D.McLaughlin and G. Papanicolaou, Plenum Press, 1995; and for reaction-diffusion equations
  • J. Xin, Front Propagation in Heterogeneous Media, SIAM Review, Vol. 42, No. 2, June 2000, pp 161-230
  • H. Berestycki, The influence of advection on the propagation of fronts in reaction-diffusion equations. In: Nonlinear PDE's in Condensed Matter and Reactive Flows, NATO Science series C: Mathematical and physical Sciences, vol. 569 (H. Berestycki and Y. Pomeau, eds.), Kluwer Acad. Publ., 2002, p. 11-48.

Ridgway Scott: See my homepage.

Geometry, topology

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, group theory and dynamical systems. 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.

Sid Webster: see here.

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

  1. Topology of (mainly high dimensional) manifolds.
  2. 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.

  1. 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.)
  2. 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.
  3. 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

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, 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.

The article Calibrating Randomness by R. Downey, D. R. Hirschfeldt, A. Nies, and S. A. Terwijn is a survey of recent work in algorithmic randomness.

Surveys of some results on computability-theoretic and reverse-mathematical aspects of combinatorial principles can be found in the early sections of On the Strength of Ramsey's Theorem for Pairs by P. A. Cholak, C. G. Jockusch, Jr., and T. A. Slaman, and Combinatorial Principles Weaker than Ramsey's Theorem for Pairs by D. R. Hirschfeldt and R. A. Shore.

My papers can be found on my website.

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