(September 11) Jeffrey Adams: Character Theory - Abstract: For a finite group, a representation pi is determined by its character, i.e. the function g->Trace(pi(g)). This is a conjugation invariant function on the group. The characteristic function of a conjugacy class is also such a function, and this establishes a beautiful duality between representations and conjugacy classes.

Properly interpreted, a representation of a semisimple Lie group is also determined by its character. Much of modern representation theory is concerned with generalizing the representation/conjugacy class duality of finite groups.

This philosophy suggests that if there is a natural relationship between the conjugacy classes of two groups, there should be an analogous relationship between their representations. This is seen in terms of characters, while the correspondence of representations may be quite subtle and reveal deep underlying structure. This idea is one of the key points in the Langlands program, and it also plays a role in Howe's theory of dual pairs.

I will give a survey of the subject, focusing on applications to representation theory of non-linear groups.

(September 18) Tom Illmanen: The inverse mean curvature flow and the Riemannian Penrose inequality - The Penrose conjecture of general relativilty, in its purely Riemannian case, states the following:

In an asymptotically flat 3-manifold of nonnegative scalar curvature, the ADM mass is bounded below by the area of each outermost minimal surface.

Huisken and I succeeded in proving this using the so-called inverse mean curvature flow, even though the evolving surfaces jump around in the manifold. A corollary is the positive mass theorem.

(November 20) Mark Levi: A Lagrangian approach to KAM - I will give a brief survey of the KAM theory and will outline a very short recent proof, due to Moser, of Moser's invariant curve theorem, based on the variational approach. The two key steps of the proof are: first, the reformulation of the problem as a second order non-linear difference equation, and second, a way to deal with non-linearity by utilizing the self-adjointness of the problem.

(February 26) Madhav Nori: Motives - The cohomology groups of algebraic varieties have several enhanced structures: Hodge structures, and Galois actions, for instance. It is believed that there is an Abelian category called "Motives" which contains all these structures. The purpose of this talk is to demonstrate the existence of this Abelian category, when the ground field is contained in the field of complex numbers. In topology, we have a functor that associates to any space its singular chain complex. In analogy, there is a functor from varieties to chain complexes of motives.

(March 5) David Hoff: Some mathematical questions in compressible fluid flow - I will give an overview of the current state of affairs concerning the existence and regularity of solutions of the Navier-Stokes equations of compressible fluid flow in several variables, with passing references to other, related models in fluid mechanics. This talk should be accessible to a very general audience, including to graduate students. At the same time, I will attempt to explain some of the important mathematical ideas which occur, such as the roles of asymptotic decay rates, weak solutions, and weak convergence theory.

(March 12) Steven G. Krantz: Recent results on the automorphism groups of domains - We describe some recent work of the author---joint with several co-authors---concerning the group of biholomorphic self-maps of a domain in complex space. We will relate the dimension of the automorphism group and the geometry of an orbit accumulation point with the global geometry of the domain in question.

The talk should be accessible to anyone acquainted with basic complex variable theory.

(April 9) Charles Peskin: The immersed boundary method for bloodflow in the heart (and related problems in biofluid dynamics). - Humans, like all other life forms, are mostly fiber-reinforced fluid. The immersed boundary method is a mathematical formulation of the fiber-fluid problem, together with a numerical method for the computer simulation of such systems. The immersed boundary method will be explained, and its application to the beating heart will be presented in the form of a computer generated video animation.