(September 12) P. M. Fitzpatrick: Introduction to the academic year - Chairman Mike Fitzpatrick will introduce new faculty and visitors and take the occasion to discuss plans for the department.

(October 3) Professor Richard Wentworth: Yang-Mills flows on Kaehler manifolds - The Yang-Mills equations have played an important role in mathematics and physics for over thirty years. The moduli spaces of solutions to these equations on manifolds give information about the underlying topology. In the presence of a complex structure on the manifold, the existence of solutions is equivalent to algebraic conditions on an associated holomorphic vector bundle. In the early eighties, Atiyah and Bott discovered a relationship between the Morse theory of the Yang-Mills functional and an algebraic stratification of the space of holomorphic vector bundles on a Riemann surface. This approach gives strong information about the topology of the moduli space. In this talk we will survey some of this theory and indicate recent results in higher dimensions.

(October 10) Professor Vitali Milman: Phenomena of Large Dimension - We will discuss some common properties of finite but high-dimensional normed spaces (or, equivalently, convex bodies). Surprisingly, such objects reveal much less diversity than appriory one would expect. After a short introduction to the subject, we will concentrate on some very new results on symmetrizations and geometric probability. All results have very elementary formulations and will be understood by graduate students.

(October 24) Dr. N. Sri Namachchivaya: Stability and Nonstandard Reduction of Noisy Mechanical Systems - We will discuss both stability of certain class of noisy nonlinear mechanical systems and some dimensional reduction techniques which consists of a sequence of averaging procedures that are uniquely adapted to study these systems.

(October 31) Professor Joel Spruck: Locally convex hypersurfaces of constant curvature with boundary - Given a disjoint collection $\Gamma=\{\Gamma_1, \ldots, \Gamma_m\}$ of closed smooth $(n-1)$ dimensional submanifolds in $R^{n+1}$, does there exist an immersed hypersurface M of constant curvature $f(\kappa(M))=K_0$ (elliptic Weingarten hypersurface) spanning $\Gamma$ for some positive constant $K_0$? Important examples are of this problem include the Plateau problem for minimal or constant mean curvature surfaces. In this talk, we will survey the progress made on this problem for Gauss curvature ($f=S_n(\kappa)$) and other curvature functions defined in the positive cone $\Gamma_n^+$ (so that we insist that M is locally convex).

(November 7) Professor Stephen Brush: Is Mathematics the Key to the Universe? Variations on a Theme of Eugene Wigner - In a famous 1960 paper, Wigner discussed "The Unreasonable Effectiveness of Mathematics in the Natural Sciences." I suggest that the effectiveness of mathematics in producing successful new theories and surprising discoveries is even more unreasonable than Wigner claimed -- unless, of course, you already believe in the Pythagorean-Platonic view of the universe. But watch out -- some persons who believed that mathematics is the key to the universe tried to unlock the wrong door, or when they opened the door found behind it an answer they didn't like.

(November 14) Professor Tom Hou: Multiscale Modeling and Computation of Flow in Heterogeneous Media - Many problems of fundamental and practical importance contain multiple scale solutions. Composite materials, flow and transport in porous media, and turbulent flow are examples of this type. Direct numerical simulations of these multiscale problems are extremely difficult due to the range of length scales in the underlying physical problems. In this talk, I will give an overview of the multiscale finite element method and describe some of its applications, including composite materials, wave propagation in random media, convection enhanced diffusion, flow and transport in heterogeneous porous media. It is important to point out that the multiscale finite element method is designed for problems with many or continuous spectrum of scales without scale separation. Further, we introduce a new multiscale analysis for convection dominated 3-D incompressible flow with multiscale solutions. The main idea is to construct semi-analytic multiscale solutions locally in space and time, and use them to construct the coarse grid approximation to the global multiscale solution. Our multiscale analysis provides an important guideline in designing a systematic multiscale method for computing incompressible flow with multiscale solutions.

(December 5) Professor Albert Marden: On the Mathematical Legacy of Leon Greenberg - From a contemporary perspective I will give an exposition of Leon's work with isometries of the hyperbolic plane and space, and automorphisms of Riemann surfaces. I will bring out the highlights, and Leon's prescience in choice of topics for research.

(February 6) Professor Georgios Pappas: Geometric Galois Modules - Suppose X/Y is a Galois cover of algebraic varieties defined over the integers with finite Galois group G. The various cohomology groups of X are then naturally G-modules. We will discuss work towards the determination of such ``geometric Galois modules". There are interesting relations with ideal class groups of cyclotomic fields and the theory of Artin-Hasse-Weil L-functions.

(February 13) Professor Eugenia Kalnay: Chaos and Atmospheric Predictability - Applications of chaos theory to exploit predictability of the ocean and the atmosphere.

(February 20) Charles Epstein: Adventures in Magnetic Resonance - I will introduce a simple mathematical description of nuclear magnet resonance and explain how this phenomenon is used in medical imaging. important aspect of MRI is selective excitation. I will show that selective excitation is a classical inverse scattering problem and explain how this problem is solved in practice.

(February 27) Professor Michael Fisher: The stochastic dynamics of molecular motors - Molecular motors are protein molecules that drive much active biological motion. Recently, striking experiments have observed single motor protein molecules in vitro pulling loads along linear molecular tracks. Thus a kinesin molecule takes hundreds of discrete steps of 8.2 nm along a microtubule, while consuming one 'fuel molecule' of ATP per step, and may reach an average speed of nearly a micron per second.

How "mechanical" are such motors? And what forces do they exert? How 'should' their motion be described theorectically? Exact results derived for random walks in random environments, and subsequent developments, yield effective tools.

(March 12) Professor Joan Feigenbaum: Incentives and Internet Computation - Traditional treatments of distributed computation typically assume nodes to be either cooperative (i.e., they execute the prescribed algorithm) or Byzantine (i.e., they can act in an arbitrary fashion). However, to properly model the Internet, in which distributed computation and autonomous agents prevail, one must also consider selfish agents that maximize their own utility. To cope with selfish agents, system designers must develop mechanisms in which cooperation is in each agent's self-interest; we call such mechanisms "incentive-compatible." This talk will first introduce the audience to the economic theory that forms the basis for the design and analysis of incentive-compatible Internet algorithms. It will then review previous results in this area, including those on interdomain routing and multicast cost sharing. Finally, it will present several promising research directions and pose some specific open problems.

(March 19) Professor Anthony Quas: Monotonicity in Voting Systems - Arrow's Impossibility Theorem states that in an election with 3 or more candidates, there is no voting satisfying a small number of basic fairness requirements. In spite of this, many voting systems are used with a wide variety of properties. Here, we focus on the requirement of monotonicity: that the more votes you get, the more likely you are to win. Surprisingly, a fairly popular voting system does not have this property. We will discuss the probability that unfairness of this type arises in the single transferable vote system.

(April 9) Professor Henri Darmon: Diophantine equations and modular forms. - The question of solving diophantine equations efficiently has preoccupied number theorists since the dawn of the last millenium, with two classes of equations: Pell's equation, and elliptic curves, being the focus of some of the most intensive investigations. I will discuss how the modularity of elliptic curves, established by Wiles almost ten years ago, and used by him to prove Fermat's Last Theorem, also leads to the best known results to date on solving elliptic curve equations.

(April 16) Professor Herbert Hauptman: The crystallographic phase problem - See here.

(April 23) Professor C. R. Rao: Anti-eigen and anti-singular values of a matrix and applications to problems in statistics - See here.

(May 7) Professor Benoît Perthame: Mathematical models for cell motion - Several transport-diffusion systems arise as simple models in chemotaxis (motion of bacterias or amebia interacting through a chemical signal) and in angiogenesis (development of capillary blood vessels from an exhogeneous chemoattractive signal by solid tumors). These systems describe the evolution of a density (of cells or blood vessels) coupled with the evolution equation for a chemical substance, through a nonlinear transport term depending on the gradient of the chemoattracting substance. Such systems are successful in recovering various qualitative behavior (chemotactic collapse, ring dynamics). Endothelial (i.e. cells forming blood vessels) have a tendency to form different patterns, initiating the vessels shape. Then hyperbolic models seem better adapted to describe this kind of network formation.

We will present these models, their main mathematical properties (quantitative and qualitative), numerical simulations and, for bacteria E. Coli, we will give a microscopic picture based on a kinetic modelling of the interaction (nonlinear scattering equation). We show that such models can have global solutions that converge in finite time to the Keller-Segel model, as a scaling parameter vanishes. This point of view has also the advantage of unifying all the models.