Topic: An algebra of integral operators and correlation functions February 10

of interacting electron gas

Presenter: Vladimir Korepin, SUNY, Stony Brook

Topic: Asymptotic Aspects in Geometric Functional Analysis February 10

Presenter: Nicole Tomczak-Jaegermann, University of Alberta

Abstract: We shall present several directions in which close connections can be found between the geometry and linear-metric structure of infinite-dimensional Banach spaces and finite-dimensional phenomena. The geometry of convex bodies in $R^n$ has a fundamental influence on structural phenomena in random quotients of the corresponding normed spaces; and this in turn can be used in infinite-dimensional constructions with various non-local properties.

Topic: Pointwise Theorems for Amenable Groups February 11

Presenter: Elon Lindenstrauss, Hebrew University

Abstract: The classical ergodic theory deals with measure preserving Z actions. Since the 50's, it has become increasingly clear that most of the results of the classical ergodic theory can be extended to actions of amenable groups, i.e. groups for which there exists an invariant mean.

Somewhat surprisingly, two of the basic theorems of ergodic theory, the pointwise ergodic theorem and the Shannon-McMillan Breiman theorem were not known for general amenable groups. I will describe a proof of these two theorems for actions of discrete groups (these techniques should work also in more general contexts).

Topic: Down-up algebras February 11

Presenter: G. Benkart, University of Wisconsin, Madison

Abstract: Down-up algebras originated in the study of differential posets. These algebras exhibit many remarkable features and many interesting connections with enveloping algebras of Lie algebras, Weyl algebras, quantum groups, and Witten's deformations of the enveloping algebra of sl(2). This talk will focus on their combinatorial origins, representations, and structure.

Topic: Foliations with good geometry February 11

Presenter: Sergio Fenley, Princeton University

Topic: Random Matrix Ensembles with General Beta February 12

Presenter: Peter Forrester, Melbourne University

Topic: Functional on Kahler manifolds February 12

Presenter: E. Calabi, University of Pennsylvania

Topic: Elementary Introduction February 12

Presenter: Charles Fefferman, Princeton University

Topic: How Drops Break and Other Singularities February 15

Presenter: Michael P. Brenner, Massachusetts Institute of Technology

Abstract: This talk will give an overview of some physical phenomena whose essence involves singularity formation in a nonlinear partial differential equation. We will describe in detail two different problems where understanding the structure of a singularity provides insight into the underlying physical process. The first problem is droplet breakup: Based on a combination of numerics, asymptotics and experiments we will argue that viscous drops develop long necks when they break, which then spawn a series of smaller necks with ever thinner diameters. The second problem involves singularities occurring during the clumping of material interacting by long ranged, Laplacian interactions, motivated by experiments involving {\it e. Coli} (E. O. Budrene and H. C. Berg, Nature, 376:49 (1995)). The dynamics raises general questions about what happens when a system has several singular solutions in different spatial dimensions competing with each other.

Topic: Wave maps in 2+1 dimensions February 15

Presenter: Daniel Tataru, Northwestern University

Topic: Stability of Matter and the Interacting Relativistic Electron-Positron February 17

Field in Hartree-Fock Approximation

Presenter: Heinz Siedentop, University of Regensburg

Topic: Nonlinear hyperbolic equations February 17

Presenter: Daniel Tataru, Northwestern University

Abstract: We present some recent results concerning the local theory for nonlinear second order hyperbolic equations with nonsmooth initial data. These results are based on some ne strichartz type estimates for hyperbolic operators with nonsmooth coefficients.

Topic: Applications of Homogeneous Dynamics to Diophantine February 18 & 25

Approximation on Manifolds

Presenter: Dmitry Kleinbock, Rutgers University

Topic: The factorial function... and generalizations February 18

Presenter: Manjul Bhargava, Princeton University

Abstract: Though ubiquitous in combinatorics, the factorial function also occurs surprisingly often in number theory. A closer examination of these occurrences leads to a series of intriguing generalizations of the factorial function, which recently have been applied to a variety of number-theoretic, ring-theoretic, and combinatorial problems. In particular, a fundamental question about integer-valued polynomials, put forth by Polya in 1919, is now resolved.

Topic: Periods of Eisenstein series

Presenter: E. Lapid, Institute for Advanced Study

Abstract: The well-known Maass-Selberg relations are formulas for the inner product of two Eisenstein series, suitably truncated to overcome convergence problems. They were proved in full generality by Langlands and Arthur. An analogous formula exists for periods of truncated Eisenstein over certain subgroups, the prototype being $GL(n,Q)<GL(n,Q(i))$. We will discuss the relation with a regularization method developed by Rogawski. The main application is for the relative trace formula.This is a joint work with Jonathan Rogawski.

Topic: TBA February 18

Presenter: Tianjun Li, Yale University

Topic: Orthogonal Geometry and Quantum Error Correction February 24

Presenter: Robert Calderbank, AT&T Research

Abstract: Quantum effects are seldom evident in today's electronic devices since the quantum states of many millions of atoms are averaged together blurring their discreteness. But in quantum computing the foundations of quantum mechanics are finding direct and visible application in information processing. The unreasonable effectiveness of quantum computing is founded on coherent quantum superposition or entanglement which allows a large number of calculations to be performed simultaneously. This coherence is lost as a quantum system interacts with its environment and an important challenge today is to devise means of preserving it.

A quantum error correcting code is a way of encoding quantum states into qubits so that error or decoherence in a small number of individual qubits has little or no effect on the encoded data. This talk will describe a beautiful group theoretic framework that implifies the presentation of known quantum error correcting codes and greatly facilitates the construction of new examples.

Topic: Applications of Homogeneous Dynamics to Diophantine February 18 & 25

Approximation on Manifolds

Presenter: Dmitry Kleinbock, Rutgers University

Topic: The Nash Conjective for Threefolds February 25

Presenter: Janos Kollar, University of Utah

Topic: Lanford's Program March 4

Presenter: Michael Yampolsky, Yale University

Topic: Statistical properties of weak Gibbs measures for certain March 25

nonhyperbolic systems

Presenter: Michiko Yuri, Sapporo University

Abstract (see attached)