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

periodic 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: Local well-posedness for nonlinear hyperbolic equations March 2

Topic: Symplectic isotopy and braids March 2

Presenter: Bernard Siebert, University Bochum

Topic: A simple description of a loop group March 3

Presenter: David Nadler, Princeton University

Title: On a class of algebraically defined graphs March 3

Speaker: Felix Lazebnik, University of Delaware

Abstract: Let $F^n$ denote be the $n$-dimensional vector spaces over a field $F$. For $n\ge 2$ and each $i=1,2,\ldots, n-1$, let $f_i: F^{2i}\to F$ be a function of $2i$ variables. We consider a bipartite graph whose vertex partitions $P$ and $L$ are copies of $F^n$ with $p = (p_1,p_2,\ldots, p_n)\in P$ and $l = (l_1,l_2,\ldots, l_n)\in L$ being joined by an edge if and only if the following $n-1$ equalities are satisfied: $$\eqalign{& l_2 + p_2 = 1(p_1,l_1)\cr &l_3 + p_3 = f_2(p_1,l_1, p_2,l_2)\cr &\ldots\ldots\ldots\ldots\ldots\ldots\ldots\cr&l_n + p_n = f_{n-1}(p_1,l_1, p_2,l_2, \ldots, p_{n-1},l_{n-1})\cr}$$ For particular fields $F$ and particular functions $f_i$'s, the families of graphs defined this way (or slightly modified) posses many remarkable properties. They are concerned with forbidden cycles, girth, graph homomorphism, eigenvalues, edge-decompositions of complete graphs and complete bipartite graphs, and some Ramsey type problems. In this talk we survey some published results, and present several new ones.

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: The Challenge of Complex Systems: How will Science be Changed? March 6

Presenter: Brian D. Josephson, Cavendish Lab, Cambridge University

Abstract: It is only gradually becoming recognised that complex systems are more than complicated versions of ordinary systems or even chaotic systems. They have their own laws and their own kinds of regularities, and instead of reductionistic derivations we have to think, in this context, in terms of interrelated emergent patterns.

These ideas have been well established by workers such as Robert Rosen, but even more interesting are possibilities opened up by the fact that current science may be only an approximation that ignores and smooths out the details of a deeper underlying structure with the nature of an organised complex system. Some frequently reported anomalies may find rational explanations in such terms.

Topic: How to put guesswork back into computing March 6 Fine Hall

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: Computational Pseudo-Randomness March 7

Presenter: Avi Wigderson, IAS, Princeton & Hebrew University, Jerusalem

Abstract: A fresh view of the question of randomness was taken in the theory of computing: It has been postulated that a distribution is pseudorandom if it cannot be distinguished from the uniform distribution by any efficient procedure. This lead to a beautiful theory tightly relating probabilistic algorithms to computational intractability. In this talk I will describe the evolution of ideas and notions that lead to this understanding, and some of the side-benefits of these ideas to other central problems of computational complexity theory. The talk is completely self-contained.

Topic: Modularity of elliptic curves March 7

Presenter: B. Conrad, Harvard University

Title: Existence of solutions for the vortex sheet problem. March 8

Speaker: Zhouping Xin, Courant Institute, New York University and

Chinese University of Hong Kong.

Topic: Elliptic Yang-Mills equation March 8

Presenter: Gang Tian, M.I.T.

Abstract: The Yang-Mills equation has played a very important role in the study of geometry and topology in the last few decades.

Its regularity theory is crucial to applications. In this talk, I will give a brief tour of recent progress on regularity theory of

the Yang-Mills equation on Riemannian manifolds. Some geometric applications and open problems will be also

discussed.

Topic: Adhesion Dynamics and Random Walks March 9

Presenter: Toufic Suidan, Princeton University

Abstract: Consider N equally spaced point masses on the unit interval of the real line; each point mass is given a random initial velocity. The interparticle interaction is one dimensional gravity and binary collisions of particles are perfectly inelastic. We will be interested in the statistics of the mass aggregation process in the continuum limit. It turns out that this aggregation process is not gradual; it is abrupt and occurs at a nonrandom time. We also comment on the long time behavior of the ballistic case for which gravity is set to zero. These problems can be understood in terms of statistical properties of random walks; several of these properties will be discussed.

Topic: Soliton stability and blowup in modified KdV with critical nonlinearity March 9

Presenter: Frank Merle, Departement de Mathematiques

Ecole Normale Superieure, France

Topic: Floer homology and homology cobordism invariants March 9

Presenter: Kim Froyshev, Harvard University

Topic: Random Walks and the Gittins Index March 10

Presenter: Peter Winkler, Bell Labs

Abstract: Let $G$ be a fixed finite graph with a distinguished target node, and suppose that two tokens reside initially at nodes $x$ and $y$ of $G$. At each tick of a clock you may select either token, which then takes a uniformly random step to a neighboring node. Your object is to get one token to the target in minimum expected time. Say "$x>y$" if your correct strategy begins with selecting the token at $x$. If $x>y$ and $y>z$, is $x>z$?

Topic: Existence results for some fully non-linear elliptic equations March 10

on Riemannian manifolds

Presenter: Jeff Viaclovsky, University of Texas and M.I.T.

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.

Topic: TBA March 20

Presenter: Lev Kapitanski, Kansas State University

Topic: TBA March 22

Presenter: R. Schoen, Stanford University

Topic: Global secular dynamics in the planar three-body problem March 23

Presenter: Jacques Fejoz, Northwestern University

Topic: TBA March 23

Presenter: Steve Zelditch, John Hopkins University

Topic: TBA March 29

Presenter: R. Stanton, Ohio State University

Topic: TBA March 30

Presenter: Ilya Ustilovsky, New York University

Topic: TBA April 3

Presenter: Stephen Wainger, University of Wisconson

Topic: TBA April 3

Presenter: Mark Gross, University of Warwick

Topic: Random Colorings of a Cayley Tree April 5

Presenter: Peter Winkler, Bell Labs

Abstract: Probability measures on the space of proper colorings of a Cayley tree (that is, an infinite regular connected graph with no

cycles) are of interest not only in combinatorics but also in statistical physics, as states of the antiferromagnetic Potts model at zero temperature, on the ``Bethe lattice''. We concentrate on a particularly nice class of such measures which remain invariant under parity-preserving automorphisms of the tree. Using branching random walks, we determine when more than one such measure exists. This talk (on joint work with Graham Brightwell, of the London School of Economics) will provide, we hope, a helpful glimpse into the rapidly expanding intersection of combinatorics and statistical physics.

Topic: TBA April 10

Presenter: Hart Smith, University of Washington