# Department of Mathematics

## Faculty Research Interests

### László 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.

### Alexander Beilinson

Arithmetic algebraic geometry, geometric Langlands program.

### 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 my webpage.

### 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.

### 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.

### Jack D. 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 theory 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.

### Marianna Csörnyei

I am mostly interested in the Geometric Langlands program, which is a part of geometric representation theory. Below 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.

### 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.

### Alex Eskin

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

### Benson Farb

My interests include geometric group theory; low-dimensional topology; the interaction of differential geometry, Lie groups and their discrete subgroups; nonpositive curvature; as well as various interactions between topology, representation theory, algebraic geometry and number theory (especially concentrating on various moduli spaces, e.g. of polynomials, rational maps, Riemann surfaces, etc).

### 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.

### 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. 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:

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.

### 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.

### Carlos E. Kenig

I work in the fields of harmonic analysis and partial differential equations. In 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 F. 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.

### 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.

### Bảo Châu Ngô

I am interested in algebraic cycles, K-theory, Hodge theory, Galois theory, and their interactions.

### 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?)

### 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.

### Sidney Webster

I work on the holomorphic geometry of smooth bounded domains in the complex space $$\mathbb{C}^n$$. 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 $$\mathbb{C}^n$$, 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.

### 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 in July 2006. See this article about the last one.

### Amie Wilkinson

My research lies in the area of smooth dynamical systems and is concerned with the interplay between dynamics and other structures in pure mathematics – geometric, statistical, topological and algebraic.

The broad scope of dynamical methods and applications can be traced to its origins. The field of dynamical systems was pioneered around the turn of the 20th Century by Henri Poincare and George Birkhoff, who attempted to attach a rigorous mathematical framework to natural questions arising in physics (in particular, the statistical properties of ideal gases and the stability of planetary motion). Other early contributors to the field were Eberhard Hopf, who used dynamics to describe the statistical behavior of geodesics on negatively curved surfaces, John von Neumann, who laid the foundations of Ergodic Theory, an analytic and measure-theoretic approach to dynamics, and Andrey Kolmogorov, who created the modern field of dynamics as we know it today.

In simple terms, a dynamical system is a space with a set of rules (a transformation) that can be iterated. Each iteration represents the passing of one unit of time, and one asks what phenomena can be observed under long-term iteration. A more modern definition of dynamical system replaces the single transformation by the action of an infinite group or semigroup. In smooth dynamics, the action of this group is by smooth transformations, such as diffeomorphisms or flows given by a smooth vector field. I am interested in the following broad questions in smooth dynamics:

1. What are the mechanisms for chaotic behavior? By mechanism, I mean a coarse geometric and/ or topological property of the system that can be verified in practice and is robust under perturbations of the system. Chaotic can mean several things depending on the context, but a common property of chaotic systems is mi g: arbitrary subsets of the system will evenly intertwine over time. In my own work, I have extensively studied the mechanism of partial hyperbolicity and how it produces stable mixing. The notion of mechanism can be weakened to allow for phenomena that appear for a "typical system," but not robustly. In recent work with Artur Avila and Sylvain Crovisier, I've proved a formulation of the ergodic hypothesis for systems with positive entropy: in the $$C^1$$ topology, ergodicity is generic (i.e. holds for a residual set of conservative diffeomorphisms). The generic mechanism here is nonvanishing Lyapunov exponents, invariants that detect infinitesimal long-term expansion and contraction rates.
2. What geometric properties of a Riemannian manifold produce chaotic distribution of its geodesics? A dynamical system associated to any Riemannian manifold is the geodesic flow, first studied by Morse, Hadamard, Hedlund and E. Hopf. Dynamical properties of the geodesic flow capture the geometric properties of geodesics, including the distribution of closed geodesics and the typical behaviors of infinite geodesics. If the manifold is compact and negatively curved, then the geodesic flow is chaotic, as measured by any reasonable standard (the list of names behind these assertions is huge, going back to the pioneers and including works as recent as the early 2000s). When either negative curvature or compactness is relaxed, the dyanmics can change radically. I have studied the geodesic flow for certain incomplete, negatively curved manifolds carrying the so-called Weil-Petersson metric from Teichmueller theory.
3. Which infinite groups can act smoothly on a manifold, and under what conditions can such actions be deformed? Zimmer initiated an ambitious program to classify actions of "large" infinite groups, such as lattices in semisimple Lie groups. Typically, such groups cannot act, or actions when they do exist are quite rigid (i.e., they cannot be deformed). I've studied actions of groups that are "smaller" in some sense (for example, discrete solvable groups) and can be deformed smoothly, but only in certain prescribed ways. These actions parallel in many ways the properties of partially hyperbolic diffeomorphisms: flexible enough to produce a rich class of examples, but rigid enough to admit some level of classification.
4. What features of a smooth dynamical system are rigid, and how flexible are invariants under perturbation? Continuing with the theme of flexibility and rigidity, I am interested in how much information about a dynamical system is needed to determine the system (up to a reasonable form of equivalence). There are several "soft" invariants, such as entropy, Lyapunov exponents, and dimension of invariant sets, which taken alone can be perturbed freely, but cannot be perturbed independently. Perturbing Lyapunov exponents can produce desirable chaotic behavior, which I have exploited in my work on stable mixing. On the other hand, Lyapunov exponents of certain classes of systems such as geodesic flows cannot be perturbed in certain directions and can completely determine the manifold up to isometry (as has been shown in recent work by my student Clark Butler).

A list of recent papers can be found on my website http://math.uchicago.edu/~wilkinso/.

While my research lies firmly in the area of pure mathematics, I have been exploring recently the connections between smooth dynamics and the physics of particle accelerator design. This has culminated in an interdisciplinary NSF grant with members of the UChicago physical sciences community, commencing in the Fall of 2015. I have funds available to support exploration into this topic on undergraduate through postdoctoral level.