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: Boundaries of symmetric spaces February 18

Presenter: Adrian Banner, Princeton University

Abstract: The hyperbolic plane has an ideal boundary, which we can identify with the circle or the union of the real axis and the point at infinity, depending on whether we use a disk or upper half-plane model. The behavior of certain functions on the hyperbolic plane is essentially determined by their extensions to the boundary. In particular, the Cayley transform which takes the disk model to the upper half-plane model extends to a conformal map of the boundary. We use so-called groups of Heisenberg type to generalize to all symmetric spaces of rank one of noncompact type. Note: this talk will be substantially different from the one I gave at the student seminar in Fall '97. Pizza available.

Topic: On the number of permutations avoiding a given pattern February 18

Presenter: Ehud Friedgut, IAS

Abstract: In this talk we count the number of permutations in S_n that avoid a given pattern on k numbers. (For example, "Sigma avoids 132" means that the sequence sigma(1), sigma(2), ...sigma(n) has no subsequence x y z with y>z>x .) The main conjecture in this area, made by Stanley and Wilf, is that for any pattern tau, out of the n! permutations only an exponential number avoid tau.

We prove this in a few cases, and in general show that the number of avoiding permutations in S_n is bounded by Exp(n G(n)) where G(n) is a function related to the Ackermann hierarchy, and has a VERY slow growth rate. This is joint work with Noga Alon.

Topic: Applications of Homogeneous Dynamics to Diophantine February 18

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: Lefschetz fibrations and symplectic four-manifolds February 18

Presenter: Tianjun Li, Yale University

Topic: Positivity of a conformally invariant operator arising February 19

in spectral theory, and some applications

Presenter: Matt Gursky, Indiana University

Topic: Fully nonlinear curvature equation in conformal geometry February 19

Presenter: Jeff Viaclovsky, Princeton University

Topic: Elementary proofs of the existence and uniqueness theorems February 19

for Navier-Stokes systems

Presenter: Yakov Sinai, Princeton.University

Topic: Self-similar solutions and the Cauchy Problem for a non-linear February 22

Schrodinger Equation

Presenter: Fabrice Planchon, CNRS and Universiti Paris VI

Abstract: Self-similar solutions (solutions which are invariant under a proper rescaling) are of interest for many evolution equations, as examples of singular solutions or as possible candidates for describing the asymptotics (for large time or for blow-up). Recently such solutions were constructed for the Schrodinger equation $\partial_t u+\Delta u=\pm |u|^{\alpha-1}u$ by T. Cazenave and F. Weissler, using very simple techniques, but for which the set of admissible initial data is not well understood. We intend to provide a different approach by solving the usual Cauchy problem in a Besov space bigger than the Sobolev space where the problem is well-posed, which contains homogeneous data, thus allowing self-similar solutions in that Besov space. This provides a better understanding of such solutions, as well as extending the known results for Sobolev spaces. **Please Note Room Change

Topic: Oversampled A/D conversion: obtaining improved bounds February 23

through exponential sun estimates

Presenter: Sinan Gunturk, Princeton University

Topic: Effective Interactions in Lattice Systems Due to Quantum Fluctuations February 24

Presenter: Daniel Ueltschi, Rutgers 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: The Mattingly-Sinai proof for existence of smooth solutions February 25

to 2D Navier Stokes

Presenter: Vadim Kaloshin, Princeton University

Topic: Applications of Homogeneous Dynamics to Diophantine February 25

Approximation on Manifolds

Presenter: Dmitry Kleinbock, Rutgers University

Topic: Crystal bases for quantum superalgebras February 25

Presenter: Seok-Jin Kang, Seoul National University and MIT

Topic: The Nash Conjective for Threefolds February 25

Presenter: Janos Kollar, University of Utah

Topic: Stochastic Models for perception and possible implications March 3

about the way we think

Presenter: David Mumford, Brown University

Abstract: The development of computer vision in particular and AI in general has led further and further from logic-based deductions and more and more towards Bayesian statistical methods. But how can such algorithms work in the face of the exponential explosion of variables and their interactions? I want to describe some of the latest computational experiments and some of the mathematical issues they have raised. Very intriguing for us is to ask: does this suggest something about what goes on in our heads? RECEPTION TO FOLLOW COMMON ROOM, THIRD FLOOR, FINE HALL

Topic: Lanford's Program March 4

Presenter: Michael Yampolsky, Yale University

Topic: Graded algebras and a theorem of p-descent for log-schemes March 9

Presenter: Pierre Lorenzon, of Muenster University

Abstract: After discussing gradings by sheaves of degrees, we associate to any log scheme a canonical invertible sheaf endowed with a certain multiplicative structure, which we call its associated graded algebra. In the relative case we construct a canonical connection on this algebra. In the log smooth case over a base of positive characteristic p, we study integrable and p-integrable graded modules over this algebra, and establish a Cartier type p-descent theorem, generalizing previous results of Ogus. We apply it to give an alternate proof of a result of Tsuji on closed forms fixed by the Cartier operator.

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

nonhyperbolic systems

Presenter: Michiko Yuri, Sapporo University

Abstract (See Attached)