(September 7) Professor Peter Petersen: Distance Estimates in Riemannian Geometry - The talk will be a survey of old and recent work on results about bounding the diameter of manifolds with positive curvature. Simply stated these results say that any space which curves more than a round sphere is smaller than that sphere. We shall also explain how these results can be proved using variational techniques or the maximum principle.

(September 14) Professor Hillel Furstenberg: Boundaries of Groups: A View from the Inside - We will discuss the role of so-called group boundaries in various contexts: random walks on groups, the geometry of "flexible" group actions, representations of groups on convex (rather than linear) spaces, and rigidity theory. A crucial idea will be to regard the boundaries of a group G as G-spaces with certain characteristic geometric properties which also have dynamical implications.

(October 5) Professor Israel Gohberg: Othogonal Systems and Convolution Equations - In this lecture I will discuss the material of a joint book with R.L. Ellis, which is about to be finished. The main topic is concerned with distributions of zeros of orthogonal polynomials and their generalizations in spaces with weighted inner products. The impetus for our research where three Theorems of G.Szego, M.G. Krein, and M.G. Krein and H.Langer. During the last fifteen years these results were generalized and extended by D. Alpay, A. Ben-Artzi, H. Dym, R.L. Ellis, I. Gohberg, H. Landau, D.C. Lay, and L. Lerer. The results of this research will be presented.

(October 19) Professor Mary F. Wheeler: Mathematical and Computational Challenges in Energy and Environmental Modeling - Fluid flows at and below the earth's surface are the cause and cure for problems of water and soil pollution. Petroleum and natural gas production depends on flows in the earth's subsurface. Length scales vary from the microscale to kilometers. Moreover, different physical processes occur simultaneously in different parts of the domain (e.g. single flow within an aquifer, multiphase flow in the vadose zone above the aquifer, and shallow water transport in a river or wetland in contact with the porous media).

In this presentation we discuss both mathematical and computational issues that arise when attempting to model these complex flow problems. We desribe our basic approach for parallel multiphysics/multiscale simulation based on the concept of multiple blocks or domains. In this approach, two levels of domain decomposition are considered: physical and mathematical/computational. First the physical problem is decomposed with appropriate hierarchical models (representing the geometry, geology, and chemistry/physics/biology); then the governing equations must be efficiently decomposed for accurate and efficient computations. We emphasize a multiblock or macro-hybrid approach to decomposition, in which we describe a domain as a union of regions or blocks. This offers great flexibility to accomodate the shape of the external boundary, the presence of internal features such as faults, the need to refine a region of the domain (and thus to treat it as a distinct block), and to accomodate models of multiscale and multiphyscial phenomena.

(October 29) Leon Ehrenpreis: The Radon Ansatz and the Watergate method - The Radon Ansatz is the study of properties of functions in terms of their properties on lower dimentional sets. The most classical example is a theorem of Bernstein and Hartogs which asserts that a function f on Rⁿ is holomorphic if it has a holomorphic extension to certain complex lines in Cⁿ. Extensions of this result to other systems of partial differential equations will be presented, as well as other examples of this type of phenomenon. The Watergate Method is one of the main tools in establishing these results.

(October 30) Leon Ehrenpries: Periods of Eisenstein and Poincaré series - There are classical results of Hecke and Siegel which compute the periods of the Eisenstein series over hyperbolic cycles. The results involve zeta functions with grossencharactere. Calculation of parabolic periods of certain Poincaré series go back to Rademacher in his work on p(n); the results are expressed in terms of Kloosterman sums. We shall present a new method for making these computations which allow us to compute the periods of Poincaré series over hyperbolic cycles; ther results involve a new type of series, which we term "Hyperbolic Kloosterman sums". Extensions to higher dimentions are presented.

(November 2) Professor Peter Constantin: An Eulerian-Lagrangian Approach to the Navier-Stokes Equations - We will review results concerning the regularity problem for the Navier-Stokes equations. We will then describe an approach to the Navier-Stokes equations using diffusive near-identity maps.

(November 9) Professor Sun-Yung Alice Chang: Conformal Covariants, Elliptic Equations and Curvature Pinching - Elliptic equations have always been an important tool in the study of problems in geometry. In the recent decades, non-linear second order elliptic equations with critical exponents have played a special role in the solutions of several important problems in conformal geometry; a common feature of these equations is that they relate naturally to some conformal covariant operators. In this talk, I will describe some recent effort to extend the role played by second order equations to higher order ones. First, I will describe properties of a class of conformal covariant operators-- in particular a 4-th order operator with its leading symbol the bi-Laplace operator, discovered by Paneitz in 1983 --; then I will describe the relations of these operators to the study of the integrand of Chern-Gauss-Bonnet formula and some fully non-linear equations of Monge-Ampere type. As an application, I will describe some sharp "pinching " theorem of 4-sphere with L² curvature bound --a recent joint work of M. Gursky, Paul Yang and myself.

(November 16) Professor Roger Temam: Aziz Lecture: Mathematical Problems in Meteorology and Oceanography - In this lecture we will present some recent and some less recent mathematical results concerning the equations of the atmosphere, the ocean and the coupled atmosphere ocean (the so-called Primitive Equations first considered by Richardson).

(November 30) Professor Mark Friedlin: Distinguished University Professor Lecture Series: Deterministic and Stochastic Perturbations in Dynamical Systems - Two classes of problems will be considered. The first class concerns large deterministic effects caused by small stochastic perturbations. These effects include oscillations of order 1 as noise tends to zero (stochastic resonance) as well as stabilization caused by the small noise. Such effects related to the large deviation theory. The second class of problems concerns small deterministic perturbations of dynamical systems which lead to stochastic behavior of the perturbed system. We will show, in particular, how stochastic perturbations help to regularize the original deterministic problem. This class of problems is related to the averaging principle. Simple examples will be considered, and more general results will be discussed shortly.

(December 7) Professor Stuart Antman: Distinguished University Professor Lecture Series: Mathematical Prospects in Continuum Physics - Continuum physics, which includes the disciplines of continuum mechanics, continuum thermodynamics, continuum electromagnetism, and certain fields of chemistry, furnishes refined mathematical models for the behavior of material bodies that are not invisibly small. The governing equations for theories of continuum physics typically involve partial differential equations and generalizations thereof.

The mathematical foundations of these theories consist of a handful of fundamental principles, whose simplicity and universality gives continuum physics, a science of great technological importance, much of its beauty and elegance. These fundamental principles are supplemented with constitutive equations, which describe the mechanical, thermal, electromagnetic, and other kinds of behavior of specific materials as diverse as steel, terbium, rubber, glass, heart muscle, sand, liquid crystals, air, water, blood, lubricating oil, chewing gum, paint, and sour milk.

The purpose of this talk is to give an impression of how the governing equations of continuum physics are derived from the basic principles, to comment on their mathematical structure, to sketch what is known about their solutions, and primarily to indicate the challenges and opportunities for future research.

(February 1) Professor Yakov Pesin: Fubini's Nightmare in Dynamical Systems - It has been known for quite a while that stable and unstable invariant foliations of a hyperbolic dynamical system possess the crucial property of absolute continuity. In other words, the classical Fibini theorem applies to these foliations. Although examples of foliations without the absolute continuity property were known they were considered rare and insignificant.

Recent advances in smooth ergodic theory have overturned this point of view. Pathological foliations -- the phenomenon known as Fubini's nightmare -- have been discovered in many partially hyperbolic systems and is intimately related to nonzero Lyapunov exponents. In the recently developed stable ergodicity theory this phenomenon is considered typical in some sense.

In the talk I will describe the Fubini's nightmare phenomenon and will explain how it is related to strong stochastic properties of the system.

(February 8) Professor Boris Dubrovin: Integrable Systems, Gromov - Witten Invariants and Riemann - Hilbert problem - Abstract. An approach to the problem of classification of 1+1 integrable evolutionary PDEs will be presented. The explicit construction of these PDEs based on a "quantization" of certain Riemann - Hilbert problem will be given. We will also discuss the conjectural connections of the structure of integrable PDEs to topology of the Deligne - Mumford moduli spaces of stable algebraic curves.

(February 15) Jim Yorke: Distinguished University Professor Lecture Series: Determining the Genome of the Human and other animals. - One of the great challenges is to figure out what the genomes of humans and other species can tell us. One critical but relatively tiny step is to determine the sequence of letters that make up the genome. A genome can be thought of as a sequence of four letters, A C G T, about 3 billion letters for each mammal. Recently much of this sequence has been determined for humans and a couple other species, and significant parts remain unknown. Our goal is to develop better techniques that will make this task simpler and more efficient. The task of making sense of this sequence will take a long time.

(February 22) Professor Yakov Eliashberg: Topology of Lagrangian Embeddings - This will be a survey of old and new results, as well as open problems about the topology of Lagrangian embeddings.

(March 1) Karl H. Hofmann: The Structure of Locally Compact and Pro-Lie Groups - In {\it harmonic analysis}, locally compact groups are at the focus of research where the existence of a left (or right) invariant measure plays the crucial role. The {\it structure theory} of a locally compact group is largely reducible to that of either {\it Lie groups} or {\it compact groups}. This observation provides the strategy for the project of a monograph on the structure theory of locally compact groups by Sidney A. Morris and the speaker, which is to follow their book on ``The Structure of Compact Groups'' (de Gruyter, Berlin, 1998). As a subcategory of the category of all topological groups and continuous group homomorphisms, the category of locally compact groups has certain defects: it fails to be closed under the formation of products (let alone other category theoretical limits). There is a smallest subcategory of the category of topological groups which contains all finite dimensional real Lie groups and is closed the formation of all limits and passing to closed subgroups; its members are called pro-Lie groups and the category has a useful Lie algebra theory, dealing in general, however, with infinite dimensional topological Lie algebras. It will be a part of the authors' strategy to emphasize the the theory of pro-Lie groups as a tool for dealing with the structure of locally compact groups as a special case and utilizing their infinite dimensional Lie theory. The lecture will overview the project of presenting a coherent structure theory for locally compact groups and pro-Lie groups.

(March 15) Professor Thomas C. Hales: Characters, motives, and logic - Several years ago, Kontsevich introduced a new type of integration (called motivic integration). The wonderful feature of this theory of integration is that it works perfectly well for spaces that are not locally compact. The unusual thing about this integral is that it takes values, not in the field of complex numbers, but in a rather mysterious ring constructed from algebraic varieties.

This talk will describe how this new theory of integration gives a powerful new tool in the representation theory of classical groups over local fields.

(March 22) Professor Vitaly Bergelson : Multiple Recurrence and The Properties of Large Sets. - Many familiar theorems in various areas of mathematics have the following common feature: if A is a large set, then the set of its differences (or, sometimes, the set of distances between its elements) is VERY large. For example:

(i) If A is a set of reals having positive Lebesgue measure, then there exists a positive real a, so that A-A contains the interval (-a,a). (ii) If A is a set of natural numbers having positive upper density, then for any polynomial p(n) having integer coefficients and zero constant term, the set A-A contains infinitely many integers of the form p(n). (iii) If F is an infinite algebraic field and G is a subgroup of finite index in the multiplicative group F*, then G-G = F.

In this talk we shall discuss these and other similar results from the perspective of Ergodic Ramsey Theory. This discussion will lead us to new interesting results and conjectures. In particular we will see the foregoing as a special case of the appearance of rather arbitrary finite configurations inside sufficiently large sets.

The talk is intended for a general audience.

(April 5) S. Novikov: Distinguished University Professor Lecture Series: Integrable Systems in Modern Mathematics - Complete Integrability Property in the modern understanding is not simply a continuation of the XVIII Century way to solve analytical problems. It is also much more than any specific qualitative statement about the behavior of solutions. This property is based on the series of deep identities unifying such areas as the scattering theory, nonlinear dynamics, algebraic geometry and representation theory.

(April 12) Professor Vladmir Sverak: Navier-Stokes and Other Super-critical Equations - The Navier-Stokes equation is a nonlinear equation with a well understood linear part, natural energy estimates, and a non-trivial scaling symmetry. In two space dimensions, the energy is invariant under the scaling. In the theory of non-linear PDE such a situation is often called "the critical case". Due to fundamental contributions of many researchers, for parabolic and elliptic equations this situation is quite well-understood. In three space dimensions, the natural Navier-Stokes energy decreases if we scale towards smaller lengths. This is often called "the super-critical case". Questions regarding existence of regular solutions seem to become much harder, and the specifics of the equation come much more into play. Perhaps studying some simpler super-critical equations, and trying to view Navier-Stokes as a member of a suitable family of super-critical equations might be useful. In the lecture I will talk about some results in this direction.

(April 19) Professor Murad S. Taqqu: Connection between self-similar stable mixed moving averages and flows - Self-similarity involves invariance of the probability distribution under scaling and it is characterized by a parameter H. Brownian motion, for example, is self-similar with H=1/2. Fractional Brownian motion is a stochastic process parameterized by H with three characteristics: it is Gaussian, is self-similar and has stationary increments. It is the unique process with these characteristics.

If the Gaussian distribution is replaced by an infinite variance symmetric alpha-stable distribution, then one does not have unicity anymore. There are in fact an infinite number of processes X that are symmetric alpha-stable, self-similar with stationary increments. We want to classify a subclass of them, the so-called "mixed moving average" ones by relating their representations to flows.

We obtain a decomposition of the process X, unique in distribution, into three independent components, which we characterize and associate with flows. The first component is associated with a dissipative flow. Examples include the limit of telecom process, the so-called ``random wavelet expansion'' and Takenaka processes. The second component is associated with a conservative flow. Particular cases include linear fractional stable motions.

This is joint work with Vladas Pipiras.

(April 26) Dr. Jeffrey Lagarias: Trisecting Angles: Alain Connes' proof of Morley's Theorem - Morley's theorem states that, given a triangle, the trisectors of the three interior angles determine three points of intersection inside the triangle, and these intersection points form an equilateral triangle. Since its discovery around 1900, many proofs have been found. In 1998 Alain Connes found an intriguing new proof of Morley's theorem. This talk describes the history of this problem and related problems, and presents Connes' proof. There will be many digressions.

(May 3) Professor Randy Bank: Aziz Lecture: Multigrid: From Fourier to Gauss - Multigrid iterative methods are one of the most significant developments in the numerical solution of partial differential equations in the last twenty years. Multigrid methods have been analyzed from several diverse perspectives, from Fourier-like spectral decompositions to approximate Gaussian Elimination, with each perspective yielding new insight. In this lecture we will introduce the multigrid method, and survey several of the tools used in its analysis.