(September 8) C. David Levermore: Fluid Dynamical Limits for the Boltzmann Equation - The endeavor to understand how fluid dynamical equations can be derived from kinetic theory goes back to the founding works of Maxwell and Boltzmann. Most of these derivations have been well understood at several formal levels for some time, and yet their full mathematical justifications are still missing. This talk will introduce this general problem and describe recent works in which the acoustic and incompressible Stokes limits are now globally establish for the classical Boltzmann equation considered over any periodic spatial domain of dimension two or more. Recent partial results regarding incompressible Navier-Stokes and incompressible Euler limits will also be discussed.

(September 22) Sheldon Newhouse: The structure of surface diffeomorphisms - Recently there has been considerable progress toward an understanding of the dynamics of typical diffeomorphisms of surfaces. We review some of that progress discussing results obtained by various workers over the past fifty years including some surprising connections between number theory and bifurcation theory.

(September 27) Jack Morava: Topology and gravity in dimensions two and four - Formulating geometric variational problems to accomodate topology change is a familiar and longstanding problem, but in recent years ideas from conformal field theory have led to some progress in dimension two, and it may be that such methods can be stretched to accomodate some form of Donaldson or Seiberg-Witten theory in dimension four.

(October 20) Henk van der Vorst: Eigenvalue Approximations from Subspaces - The computation of eigenvalues of high dimensional matrices, say of order 10,000 or more, is a very expensive task in terms of computer resources. An alternative technique is to restrict the matrix to convenient low-dimensional subspaces and to compute the eigenvalues of the restricted matrices. Krylov subspaces have become very popular for this purpose. The resulting eigenvalue approximations are known as the Ritz values.

We will give some background of this approach and we will discuss known and less well known aspects of the behavior of the eigenvalue approximations. There exists nice theory for the errors in the approximations for exact arithmetic. The behavior in finite precision arithmetic may lead to strange effects, as we will see.

In the 1990s much attention has been given to the so-called harmonic Ritz approximations. The harmonic Ritz pairs can be interpreted as to correspond with the inverse of A, restricted to A times the Krylov subspace. It turns out that one can generate a variety of such harmonic Ritz pairs, depending on a shift. The choice of the shift may be very critical for the detection of proper approximations. We will show some experimental results and surprising behavior.

(October 27) H. Blaine Lawson, Jr.: Poincare-Pontrjagin Duality for Differential Characters - In 1973 Jeff Cheeger and Jim Simons introduced the theory of differential characters into geometry. This theory, which included the basic Chern-Simons invariant, had a wealth of applications. It provided new invariants for flat vector bundles. It gave a unified derivation of the Bott-Godbillon-Vey classes for foliations. It was used to prove new non-conformal-immersability theorems in riemannian geometry. A basic feature of the theory is that it provides an important refinement of the classical Chern-Weil Theory of characteristic classes.

In recent years differential characters have appeared in Deligne cohomology and also in mirror symmetry (as ``gerbes with connection''). A holomorphic analogue of differential characters occurs in Arakelov theory -- in the Arithmetic Riemann-Roch Theorem of Bismut-Gillet-Soulé.

In my lecture I will sketch a derivation of the basic theory from a de Rham-Federer point of view. I will show how the rings of differential characters can be defined in terms of differential forms and currents, in the spirit of the classical theorems of de Rham and Federer for cohomology.

(November 06) Sergei Yakovenko: Bezout-type theorems for solutions of polynomial differential equations. - The classical Bezout theorem implies an explicit upper bound on the number of isolated zeros (either real or complex) of algebraic functions and their polynomial combinations in terms of the degrees involved. In general, one cannot count so easily zeros of arbitrary analytic functions. However, for functions satisfying polynomial ordinary differential equations, a "restricted quasialgebraicity" holds: zeros of such functions and their polynomial combination can be counted in bounded subdomains of the real line or the complex plane. Ultimately, for solutions of Fuchsian (linear ordinary differential) equations having only simple poles of the coefficients, the "global quasialgebraicity" can be proved and bounds for zeros on the whole Riemann sphere obtained in terms of norms of the residue matrices. If time permits, we discuss connections between these results and the tangential Hilbert 16th problem on limit cycles appearing by perturbations of planar Hamiltonian polynomial systems, that reduces to bounding the number of zeros of complete Abelian integrals.

Auxiliary reading: ftp://ftp.wisdom.weizmann.ac.il/pub/yakov/fields.ps

(November 17) Jeremy Teitelbaum : Locally analytic p-adic representation theory - Peter Schneider and I have been working on an approach to a theory of continuous representations of p-adic groups in locally convex vector spaces over p-adic fields. Such representations arise naturally as spaces of global sections of vector bundles on p-adic symmetric spaces. I will outline some of the features of this theory, some of our progress, and describe some of the open problems.

(December 01) G. Lawler: Universality, conformal invariance, and the dimension of the Brownian frontier - I will present recent work with Oded Schramm and Wendelin Werner calculating (rigorously) the exact values of the intersection exponents for planar Brownian motion. As a corollary the exact Hausdorff dimension of exceptional sets of Brownian motion is derived: the dimension of the frontier or outer boundary is 4/3 (this establishes a conjecture of Mandelbrot), the dimension of the set of cut points is 3/4 (a conjecture of Duplantier and Kwon), and the dimension of the set of pioneer points is 7/4. Conformal invariance and universality are essential components of the proof as well as a new process, Stochastic Loewner Evolution, introduced recently by Schramm.

(December 08) John W. Cahn: Mathematical Formulations of Some Time-Dependent Problems in Materials Science - The science of materials has led to a variety of mathematical formulations about the equilibria of assemblies of atoms and molecules, and how such assemblies evolve in time, if taken out of equilibrium. Many of these formulations are based on gradient systems on various energy functions or functionals. Of particular interest are cases in which formulations leads to ill-posed PDE, such as backward diffusion equations, or equations resulting from non-differentiable or non-convex energy functionals.

(February 09) Marcelo Viana: Dynamics: Beyond uniform hyperbolicity - It is fair to say that Dynamical Systems was born as a field of Mathematics around the end of the XIX century, with Poincaré's bringing of Topology into the realm of Celestial Mechanics. Progress, in such a short period, has been spectacular.

After the contribution of Birkhoff in the thirties, two path-breaking developments would take place in the mid XX century. One was the Kolmogorov-Arnold-Moser theory of invariant tori, that was further developed by Herman, Mather, and several other mathematicians. Another, the theory of hyperbolic systems, initiated by Smale and developed by his students and collaborators, as well as by Anosov, Sinai, and Ruelle, among others.

The advent of personal computers provided a powerful tool, and was partially responsible for the attention the field has attracted in the last decades. Models of systems in Nature have taught us that their evolution can be highly complex and, to some extent, unpredictable, raising some deep mathematical problems that we are now beginning to understand. The turn of the millenium found the field in a particularly exciting stage, as it addresses, yet and again, a major challenge: Can we build a general theory of complex dynamical behavior ?

(March 2) Robert J. McCann: Exact Solutions to the Transportation Problem on the Line - This lecture concerns a classical optimization problem formulated by Monge in 1781. Motivated by economics, the problem is described as follows: Given a distribution $f$ of iron mines throughout the countryside, and a distribution $g$ of factories which require iron ore, decide which mines should supply ore to each factory in order to minimize the total transportation costs. Taking the mines and factories to be distributed continuously throughout Euclidean space --- or a Riemannian manifold --- and the cost per ton of ore transported from the mine at $x$ to the factory at $y$ to be specified as a function of the distance, yields a problem with deep connections to geometry and non-linear PDE.

For costs $c(x,y) = h(d(x,y))$ given by strictly convex or strictly concave increasing functions $h\geq 0$ of the distance, the solution takes the form of a measure-preserving map between the measures $f$ and $g$. It is unique, and uniquely characterized by its geometry. Even on the line this mapping may be intricate. This talk explores some unexpected features of the solution for concave costs, including the emergence of a local / global hierarchy which seems as fascinating from the economic as the mathematical point of view. In one-dimension, this structure may be exploited to provide an algorithm for obtaining exact solutions to the infinite-dimensional problem by a combinatorial sequence of finite-dimensional optimizations involving convex, separable network flows. Future research directions are sketched.

W. Gangbo and R.J. McCann, The geometry of optimal transportation, Acta Math. 177 (1996) 113-161.

R.J. McCann, Exact Solutions to the Transportation Problem on the Line, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 455 (1999) 1341--1380.

(March 02) Yuri Grabovsky: The general theory of exact relations for effective moduli of polycrystals - The elastic (or dielectric) behavior of polycrystalline materials usually depends strongly on the microstructure: relative position and orientation of crystalline grains. The tensors describing these properties will therefore occupy a set in the space of all tensors, as we vary the microstructure over all conceivable configurations. This set, called the G-closure set, is of fundamental importance in the mathematical theory of composites. Usually, this set has a non-empty interior. However, there are known special cases when some combinations of material moduli (components of the tensor describing the material properties) do not depend on the microstructure. This leads to the G-closure set degenerating into a surface. When this happens we say that there is an exact relation for effective moduli of polycrystals. In this talk I will describe a general method for finding all exact relations for effective moduli of polycrystals. The method is applicable to a variety of physical settings such as elasticity, thermo-electricity and piezo-electricity. The method reduces the problem of finding exact relations to an algebraic problem of characterizing the rotationally invariant families of Jordan algebras. The use of representation theory for the rotation group SO(3) permitted us to find all exact relations in relatively low dimensional cases. The higher dimensional cases corresponding to several coupled problems are still open. The new progress can be made only if the Jordan algebra structure is used in a deeper way than it has been used before.

(March 09) Paul Baum: Statement and proof of the Atiyah-Singer Index Theorem - TThis is an expository talk intended for a general mathematical audience. The talk begins with some low-dimensional examples of the Atiyah-Singer theorem. Then a proof of the theorem is outlined. The aim is to show precisely how the Atiyah-Singer formula is implied by Bott periodicity.

(March 30) Daniel Stroock: The differentiable structure for the space of probability measures and the realization of Markov processes as integral curves - In one of his first articles, Itô suggested that Kolmogorov's forward (a.k.a. the Fokker-Planck) equation can be thought of as describing a vector field on the space of probability measures. In this talk I will develop Itô's suggestion and examine the picture which it gives. In particular, I will say how Itô's theory of stochastic differential equations is a natural outgrowth of these considerations.

(April 06) Fanghua Lin: Recent progress on high dimensional Ginzburg-Landau equations - The complex Ginzburg-Landau equations in 2-D can be used to describe vortices in superconductors and superfluids.It has been extensively studied in recent years.The similar issues in higher dimensions are much more difficult and various questions concerneded with associated flows remains to be challenging.In this talk,I shall present some works concerning minimizing and stationary solutions.The analysis also leads to some results of C.Taubes on Sieberg-Witten equations.I shall also discuss some geometric flow problems which arise naturally in the study of flows of the Ginzburg-Landau equations.

(April 20) Tai-Ping Liu: Shock wave theory  - We will present some of the recent progress on shock wave theory. The stability of Navier-Stokes shocks and the positivity of Boltzmann shocks are among the topics to be discussed. The talk will start with basics on hyperbolic conservation laws and then proceed with the recent stability analysis for dissipative physical systems.

(April 23) Franco Brezzi: Recent approaches in the treatment of subgrid scales - In a certain number of applications, one has to deal with phenomena that take place on a scale that is smaller than the smallest scale affordable in a finite element discretization. These subgrid phenomena cannot however be neglected, as their effect on the bigger (computable) scales can be quite relevant. In these cases, the use of parallel computers allows the implementation of suitable strategies, in which a preprocessing is constructed (and executed in parallel, element-by-element) to simulate the effect of the subgrid scales on the computable ones. The lecture shall present a rather general framework in which these pre-processing strategies can be studied.