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Mathematics
Faculty Research Summaries | Chairman's Introduction
Robert AlmgrenI am interested in various areas of applied and computational mathematics. In fluid dynamics and materials science, I study the motion of free boundaries, such as the solid/liquid interface in a solidifying metal. I also am interested in the applications of mathematics to financial problems, for example, optimal portfolio liquidation, and statistics of real asset prices. Jonathan AlperinRepresentation theory of finite groups emphasizing homological and local methods. Gregory AroneArone is interested in the applications of the Good willie-Weiss calculus
of functors to topology. His work László BabaiSee Department of Computer Science Walter L. Baily, Jr.Investigation of arithmetic and moduli problems connected with symmetric
tube domains, including the exceptional tube domain connected with one
of the real exceptional Lie groups E7. Types of problems to be considered:
(1) Finite generation of algebras of modular form over Z, (2) Reciprocity
laws for special values of arithmetic modular functions on the exception
domain, (3) Problems on the moduli of algebraic varieties connected
with arithmetic quotients of symmetric domains, with emphasis on heretofore
untreated questions about such Jonathan BeckQuantum groups, representation theory, and symmetric functions. Alexander BeilinsonArithmetic algebraic geometry, geometric Langlands program. Patrick BillingsleySpencer BlochAlgebraic geometry, K-theory, and number theory. Cedric BonnafeCharacters of finite reductive groups, character-sheaves, complex reflection groups. Peter ConstantinPartial Differential Equations, Dynamical Systems; Applications to Fluids Mechanics, Nonlinear and Statistical Physics. Kevin CorletteMy research interests lie in the area of differential 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 CowanMy main work is to try to understand brain mechanisms. I use nonlinear stability theory to investigate how neural circuits can generate stable patterns of activity. This approach leads to systems of nonlinear differential equations, the properties of which 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. 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. Vladimir DrinfeldI am interested in algebraic geometry, especially in the Geometric Langlands Program Todd DupontSee Department of Computer Science Alex EskinMy recent research interest has been ergodic theory and discrete groups, most particularly the connections to number theory. Benson FarbI study the relationship between algebraic and geometric properties of groups which arise in geometry and topology. This includes studying 3-dimensional manifolds, mapping class groups, discrete subgroups of Lie groups, and groups acting on trees, among other things. Robert A. FeffermanI 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. Alex FurmanMy research interests are ergodic theory, Lie groups and discrete groups. I am especially interested in various rigidity properties of groups and group actions. Victor GinzburgMy research concerns the representation theory of Lie groups and their generalizations, most particularly the connections of this theory with differential geometry. George GlaubermanI am working mainly on finite groups, especially variations on Thompson's J-subgroup in p-groups, and properties of sporadic simple groups, particularly the Monster group. Brendan HassettMy research is in the field of algebraic geometry, i.e., the geometry of solutions to algebraic equations. I am interested in families of plane curves, especially as they degenerate and acquire singularities of various types. These degenerate curves can be replaced by less singular curves, which are called stable curves or limiting plane curves. I would like to understand which curves arise as limiting plane curves. This may be studied through a careful analysis of algebraic surfaces, especially certain singular surfaces arising as limits of smooth surfaces. I also study rationality questions, i.e., whether the solutions to certain equations admit good parametrizations. One particularly rich example is the four-dimensional cubic hypersurface, which is known to be rational in many instances. Recently, I have become interested in the geometry of projective varieties with algebraic groups actions, especially the group of n-dimensional vectors. This topic has many beautiful connections to other branches of mathematics. Po HuMy research interests are in algebraic topology and its applications to algebraic varieties. Leo KadanoffCarlos E. KenigApplications of harmonic analysis to partial differential equations, especially on boundary value problems in non-smooth domains. Alexander KiselevMy broad research interests are partial differential equations and harmonic analysis. Currently, I am working on reaction-diffusion-convection equations modeling combustion processes, Schrodinger equations, and fluid mechanics. Robert KottwitzI am interested in automorphic forms from a number-theoretic point of view as well as the representation theory of reductive groups over local fields. Sandor J. KovacsHigher dimensional birational geometry, especially the study of singularities that arise in Minimal Model Program (Mori's Program). Vanishing theorems. Relations to singularities arising in characteristic p commutative algebra. Norman LebovitzOne of my research interests is in fluid dynamics. A particular direction that I have explored is that of hydrodynamic-stability theory, both linear and nonlinear. Some of the mathematical problems that I have studied are motivated by issues in the fluid dynamics of planets, stars and galaxies. The nonlinear problems are related to finite-dimensional dynamical systems and treated by methods of nonlinear analysis developed for that subject. A related interest in this theory is that of singular perturbation problems, which are frequently employed as approximations in natural settings like those of geophysics and astrophysics. I am interested both in the applications of singular-perturbation techniques in these areas and in the mathematical justification of the techniques. Arunas L. LiuleviciusThe isotropy structure of linear actions of Lie groups on spheres and other homogeneous spaces is a continuing interest. Equivariant K-theory methods have been used to prove homotopy rigidity of actions of compact Lie groups. Recent work on the multiplicative and exponential structure of similarity classes of self-maps of finite sets may give the right setting for permutation representations and associated characteristic classes. Saunders Mac Lane1. Search for new algebraic descriptions of homotopy types of topological spaces, extending 30-year old results of Mac Lane-Whitehead (three type = crossed module = group in category) 2. Use of elementary topos theory to provide an alternative foundation for mathematics (well pointed topos) and to clarify forcing in set theory. 3. Recent book, with I. Moerdijk. Sheaves in Geometry and Logic, a First Introduction to Topos Theory, Springer 1992, 627pp. 4. Studies of the philosophy of mathematics, beginning with a book, Mathematics: Form and Function, Springer, 1985. 5. Study of the objectives and difficulties of research universities, and the influence of government funding. 6. Connection between coherence theory, category theory, and quantum field theory. 7. Study of n-categories, high n J. Peter MayMay and his students and collaborators have developed new foundations for stable homotopy theory that allow the wholesale importation of techniques of commutative algebra into that subject. They have also systemized this area by proving that this approach to stable homotopy theory, which is based on coordinate-free spectra, is equivalent to a more recent alternative approach based on diagram spectra. The first approach applies equivariantly, and work is in progress to give an equivariant theory of diagram spectra. Many new constructions and theorems come out of this foundational work: new universal coefficient and Kunneth spectral sequences in generalized homology, new and more highly structured constructions of spectra originally obtained via the Baas-Sullivan theory of manifolds with singularities, new constructions of Quillen's algebraic K-theory of rings and Waldhausen's algebraic K-theory of spaces, new constructions of Bokstedt's topological Hochschild homology and of topological cyclic homology, new information about Bousfield localizations of spectra, new completion theorems in equivariant cobordism and related homology and cohomology theories, etc. There is a parallel algebraic theory that has led Mandell (a student of May) to an algebraization of p-adic homotopy theory analogous to the Quillen-Sullivan algebraization of rational homotopy theory. Matam P. MurthyStudy of structure of projective modules and obstruction theory for projective modules over affine algebras. Determining the number of generators for ideals in affine algebras. These problems relate to Chow groups and K-theory of affine varieties. Embedding questions for affine varieties are also of interest. Nikolai NadirashviliPartial differential equations; applications to the problems of geometry and mathematical physics. Raghavan NarasimhanThere are two directions I am pursuing now: 1. Aspects of the old conjecture on immersions of parallelizable Stein manifolds, in both the analytic and algebraic contexts. Recent developments connected with Gromov's h-principle make it likely that some unpublished work on the complement of plan cubics can be extended significantly. 2. Continuation of the work done with C. Fefferman using geometric methods to study analytic questions. The study of Bernstein type "doubling inequalities" is turning out to be more complicated, but also richer than expected. Andre NiesMy research focuses on structures which arise naturally in computability theory. Examples of such structures are in the Turing degrees of computably enumerable (c.e.) sets and the many-one degrees of c.e. and or arithmetical sets. I study global properties of such structures, like definability, automorphisms and whether they form prime models of their first-order theories. Not only the questions, but also the methods used have a model theoretic flavor. For instance, as an important tool, I use coding with first-order formulas. Examples of recent results (2/99) are: the arithmetical degrees form a prime model of their theory, and each definable subset D of the c.e. many-one degrees which contain not only the least and greatest element is an automorphism base (i.e., an automorphism is already determined by its action on D). In a further line of research, I study the model theoretic properties of free groups. M. V. NoriEisentein cohomology classes, Algebraic cycles on abelian varieties, Properties of the fundamental group of smooth projective varieties. Niels NygaardMy 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. Andrei OkounkovMy research is mainly focused on interconnection between combinatorics (such as multivariate orthogonal polynomials) and asymptotic problems arising in the representation theory especially in connection with groups like the infinite symmetric and unitary groups). Vachtang PutkaradzeI am interested in the application of mathematical methods to fluid dynamical problems and the theory of turbulence. Melvin RothenbergUnderstanding covering spaces of compact manifolds. Leonid RyzhikWave propagation, especially waves in random media, radiative transport, non-linear parabolic equations. Paul J. Sally, Jr.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. Matthias SchwartzMy research interests lie in the field of symplectic topology and Hamiltonian dynamics. I am particularly interested in the use of Morse-theoretical methods known as Floer homology to relate symplectic geometric invariants with phenomena in Hamiltonian dynamical systems. This involves the study of pseudoholomorphic curves in symplectic manifolds. Ridgway ScottI study the basic behavior of discrete methods for solving partial differential equations, such as finite element methods. Often these have a much richer structure than that of the corresponding differential equations. Recently, we have made substantial progress in the study of a model for non-Newtonian fluids called the grade-two model, which required estimates for the solution of a simple transport equation in very weak spaces. This study has also demonstrated existence of solutions to flow under much more reasonable restrictions on the data than was previously possible. The key technique was special interpolation methods for functions that are not very smooth, and these will likely be useful in the development of error estimators for the discrete approximations. Brooke ShipleyMy research interests are in algebraic topology and homological algebra. Robert I. SoareSee Department of Computer Science Shankar VenkataramaniI study Partial Differential equations, Dynamical systems and their applications to problems in Physics. I am interested in questions relating to the formation of small scales and singularities in the solutions to Partial Differential equations, especially in equations that arise in the description of fluid flows and in the crumpling of elastic manifolds. I'm also interested in applying ideas from Dynamical system theory to problems in Nonlinear and Statistical Physics, including fluid flows, flows in granular media and other condensed matter systems. Sidney M. WebsterMy work centers on real submanifolds of complex space, as they relate to various problems of multivariable complex analysis. These include problems about differential invariants, boundary regularity of holomorphic maps, Bergman Kernel, reflection principle, and existence of embeddings of CR structures. Brendan WeickertI study holomorphic dynamical systems, and recently have begun investigating the dynamics of unitary maps on Hilbert spaces which arise as quantizations of simple classical systems. Shmuel WeinbergerMost of my research has involved geometry and analysis on manifolds and spaces with singularities which arise naturally when studying geometric structures on manifolds (like compactifications, and quotients). The natural tools here are surgery theory, algebraic topology, algebraic K-theory and operator theory in combination. In recent years (somewhat motivated by some physical problems) I have been studying the moduli space of Riemannian metrics on a given manifold and have discovered it possesses a great deal of geometry invisible from its topology. In some sense, this space is a first example of a "large scale fractal" (spaces of multidimensional knots are others), and I hope that this new type of structure will have impact on the study of variational problems. The techniques combine those of topology and differential geometry with logic and theoretical computer science. Robert J. ZimmerMy work concerns Lie groups and their discrete subgroups, in particular the realization of these groups as diffeomorphism groups of manifolds, as well as related questions of analysis, geometry, and topology, and group theory.
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