Topic: Integrability and Near Integrability in Infinite Dimensions February 16

Presenter: P. Deift, University of Pennsylvania

Abstract: This is joint work with Xin Zhou. We consider a model problem illustrating various novel features of near integrable systems in infinite dimensions. In particular we consider perturbations of the Nonlinear Schroedinger Equation on the line and show that solutions of the associated Cauchy problem have universal behavior as $t\goto\infty$ and are completely integrable on open, invariant subsets of phase space.

Topic: Adiabatic Pistons as a Dynamical System February 17

Presenter: Ya G. Sinai, Princeton University

Topic: Vector field methods, Strichart type inequalities and applications February 17

Presenter: Sergiu Klainerman, Princeton University

Topic: Lefschetz fibration on $S^1\times M^3$ February 17

Presenter: Weimin Chen, University of Wisconsin at Madison

Topic: On Branched Covers of the Torus by Surfaces of Genus 2 February 18

Presenter: David Goldberg, Princeton University

Abstract: There is a natural topology on the set of branched covers of the torus by surfaces of genus 2 which allows us to construct a parameter space for such covers. How many components will the parameter space have in each degree? What other invariants will distinguish the components? Is it possible to tell when two maps are in the same component? Answers shall be forthcoming.

Topic: Completely positive matrices, graphs with no long odd cycles February 18

and graphs with no short odd cycles.

Presenter: Avi Berman, Technioin, Israel

Abstract: A matrix A is completely positive if it can be decomposed as A=BB^t, where B is a (not necessarily square) elementwise nonnegative matrix. An obvious necessary condition for a symmetric nonnegative matrix to becompletely positive is that it is positive semi definite. This condition is not sufficient. A sufficient condition for a symmetric nonnegative matrix to be completely positive is that its comparison matrix is positive semi definite. This condition, due to Drew, Johnson and Loewy, is not necessary. The sufficient condition is necessary if the graph of the matrix is triangle free(contains no short odd cycles). The necessary condition is sufficient if the graph contains no odd cycle of length greater than 4 (long odd cycle). We will discuss the relationship between these two results.

The smallest number of columns of B in the decomposition A=BB^t is called the cp-rank of A. We will discuss some results and conjectures on bounds for the cp-rank.

Topic: On the parabolic Monge-Ampere equation February 18

Presenter: Cristian Gutierrez, Temple University

Topic: Absolute continuity of elliptic measure February 21

Presenter: Jill Pipher, Brown University

Topic: The ecology and evolution of communities February 21

Presenter: Simon Levin, PACM & EEB, Princeton University

Abstract: Ecological communities, just as economic markets, exhibit patterns that emerge from the collective dynamics of individual agents. Implications will be given for the theory of ecological competition, and for the self-organization of ecological systems.

Topic: Mirror Symmetry and Singularities February 21

Presenter: Richard Thomas, Harvard University

Topic: Remarks on irregular varieties February 22

Presenter: Christopher Hacon, University of Utah

Topic: The Metaplectic Group, Harmonic Oscillators, Transformation of February 23

Theta Functions, Representation Theory, Orthogonal Polynomials,

and Multivariate Statistics.

Presenter: John Stalker, Princeton University

Abstract: For some time I have been interested in the connections between the classical mechanics of harmonic scillators, the corresponding quantum systems, the Heisenberg group, the inhomogeneous metaplectic group, the Schwarz class of functions and tempered distributions, and the transformation law for theta functions. All of this material is well-known, but there doesn't seems to a be a single source that puts all the connections together. My original interest in these matters came from a problem in singular perturbation theory in quantum mechanics. Recently, through some work of Bert Kostant, I realized that there are further connections with the representation theory of the universal cover of the etaplectic group and with multivariate statistics. There should also be a connection with some known families of symmetric orthogonal polynomials. Siddharta Sahi and I are currently trying to understand that connection. That sounds like rather a lot, and I may have to skip a few of the more interesting digressions, but I hope to get through the essentials of all of it.

Topic: Newton Interpolation Polynomials and Growth of number of periodic February 24

points for prevalent diffeomorphisms (joint with B.Hunt).

Presenter: Vadim Kaloshin, Princeton University

Abstract: We shall describe a new general approach to attact a class of problems about generic properties of dynamical systems. This approach develops a new perturbative technic based on perturbation of dynamical systems by Newton Interpolation Polynomials. As the by-product this approach gives that for any $\delta>0$ the number of periodic points of a prevalent diffeomorphism $f$ of a compact manifold $M$ satisfy $$ \#\{x \in M: f^n(x)=x\}\leq \exp(C n^{1+\delta}) for some C>0. $$ This result is the opposite to the result of the author which says that on a Baire generic set of diffeomorphisms the number of periodic points can grow arbitrarily fast.

Topic: An equation of Monge-Ampere type in coformal geometry, and February 24

four-manifolds of positive Ricci curvature

Presenter: Paul Yang, Princeton University

Topic: Temperley-Lieb algebras and Four Color theorem February 25

Presenter: Robin Thomas, Georgia Institute of Technology

Abstract: The Temperley-Lieb algebra T_n with parameter 2 is the associative algebra over Q generated by 1, e_0, e_1,..., e_n, where the generators satisfy the relations e_i^2=2e_i, e_ie_je_i=e_i if |i-j|=1 and e_ie_j=e_je_i if |i-j|>1. We use the Four Color Theorem to give a necessary and sufficient condition for certain elements of T_n to be nonzero. It turns out that the characterization is, in fact, equivalent to the Four Color Theorem. This is joint work with L.H.Kauffman.

Topic: TBA February 25

Presenter: W. Mueller, Univ. of Bonn and IAS

Topic: TBA February 28

Presenter: Jared Wunsch, Columbia University

Topic: Conformal maps and the Whitham equations March 1

Presenter: I. Krichever, Columbia University

Abstract: The Whitham equations are a core stone of the perturbation theory of the soliton equations. They are deeply connected with structures of topological quantum field theories (WDVV equations), and with the Seiberg-Witten solution of N=2 supersymmetric gauge models. Recently, it was discovered that special solutions of the Whitham equations describe conformal maps.

Topic: Newton Interpolation Polynomials and Growth of number of periodic March 2

points for prevalent diffeomorphisms Modularity of elliptic curve.

(joint with B.Hunt).

Presenter: Vadim Kaloshin, Princeton University

Abstract: We shall describe a new general approach to attact a class of problems about generic properties of dynamical systems. This approach develops a new perturbative technic based on perturbation of dynamical systems by Newton Interpolation Polynomials. As the by-product this approach gives that for any $\delta>0$ the number of periodic points of a prevalent diffeomorphism $f$ of a compact manifold $M$ satisfy $$ \#\{x \in M: f^n(x)=x\}\leq \exp(C n^{1+\delta}) for some C>0. $$ This result is the opposite to the result of the author which says that on a Baire generic set of diffeomorphisms the number of periodic points can grow arbitrarily fast.

Topic: TBA March 2

Presenter: Daniel Tataru, Northwestern University

Topic: Long-time evolution in general relativity and geometrization of 3-manifolds March 3

Presenter: Michael Anderson, SUNY Stony Brook

Abstract: We will discuss some surprising relations between the geometrization of 3-manifolds (Thurston conjecture) and issues in general relativity. The relation comes from examining the long-time asymptotics for the evolution of space (i.e. space-like hypersurfaces) under the vacuum Einstein equations. The detailed relationship between these topics is completely conjectural, and involves very hard issues for the vacuum Einstein evolution. Thus, we will discuss some of these conjectures, and present a few initial results giving perhaps some credence to these relations.

Topic: Periodic complexes and group actions March 6

Presenter: Alejandro Adem, University of Wisconsin at Madison

Topic: How to put guesswork back into computing March 6

Presenter: Alexandre J. Chorin, University of California, Berkeley

Abstract: Many problems in science are described by equations whose solutions are too complicated to be solved reliably on any computer; the question is what is the best one can do in such circumstances. One often has some idea about a family of possible outcomes of a computation, and I will explain how such knowledge can be used to find a most likely solution given the limitations on computing power. It turns out that often the most mathematically likely solution looks very unlikely to the naked eye. The reason is related to uncertainty principles that are well understood in physics; I will give examples and show how the paradoxes can be resolved.

Topic: Modularity of elliptic curves March 7

Presenter: B. Conrad, Harvard University

Topic: TBA March 8

Presenter: G. Tian, M.I.T.

Topic: TBA March 9

Presenter: Frank Merle, Departement de Mathematiques, Ecole Normale Superieure, France

Topic: Random Walks and the Gittins Index March 10

Presenter: Peter Winkler, Bell Labs

Topic: Stochastic Navier Stokes Equations and Wiener Chaos March 16

Presenter: B.L. Rozovskii, University of Southern California, Los Angeles

Abstract: In this talk we are concerned with fluid dynamics described by stochastic flows of diffeomorphisms. Stochastic Euler and Navier-Stokes equations will be derived from the conservation laws of mass and momentum. Well-posedness of these equations shall be discussed. A Wiener chaos expansion of the velocity field will be presented and formulas for the statistical moments of this field will be derived.