Current info: http://www.math.princeton.edu/~web/seminar.html

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: On the models of Skyrme and Faddeev March 20

Presenter: Lev Kapitanski, Kansas State University

Abstract: The Skyrme model (1961) was one of the first (and best) attempts to describe elementary particles as localized in space solutions of nonlinear PDEs. The fields take their values in $SU(2)=S^3$ and stabilize at spatial infinity. Thus, the configuration space splits into different sectors (homotopy classes) with a constant (integer) topological charge (the degree) in each sector. Faddeev's model (1975) was designed to provide additional internal structure (knottedness) to the localized solutions. The fields take values in $S^2$ and the topological charge is the Hopf invariant. In this talk I will discuss some old and new results for these models.

Title: On Internet-related scaling phenomena, and what they tell March 20

us about the Internet

Speaker: Walter Willinger, A&T Labs-Research

Abstract: Compared to the Public Switched Telephone Network, the Internet is a prime example of a truly large-scale complex system. To illustrate how various aspects of the Internet's complexity are directly reflected in the nature of the traffic that it carries, we discuss some of the recently observed scaling phenomena in measured Internet traffic (e.g., self-similarity, multifractal scaling), and comment on the few things that they can tell us and on the many things that they may tell us (in due time) about the Internet.

Topic: The Uniqueness Problem for Neural Nets March 21 Science Bldg.

Presenter: Charles Fefferman, Princeton University

Topic: The Kirillov conjecture March 21

Presenter: M. Baruch, University of California at Santa Cruz

Topic: Fourier Coefficients and Hecke Algebras March 21

Presenter: G. Savin, University of Utah

Topic: Waves in Nonlinear Periodic Structures March 22

Presenter: Michael I. Weinstein, Bell Laboratories

Topic: New solutions of the vacuum Einstein equations March 22

Presenter: R. Schoen, Stanford University

Abstract: This talk will give a brief introduction to the Cauchy problem for the vacuum Einstein equations. We will describe new constructions of Cauchy data which produce asymptotically flat solutions with particularly simple asymptotic behavior. For example, we will explain why solutions which are identical with a Kerr solution in a neighborhood of spatial infinity are dense in a natural topology on the space of initial data.

Presenter: A. Corti, Cambridge University

Y. Kachi, The John Hopkins University

T. Kuwata, Tokyo Denki

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

Presenter: Jacques Fejoz, Northwestern University

Topic: Ample line bundles on symplectic manifolds and their March 23

almost holomorphic sections

Presenter: Steve Zelditch, John Hopkins University

Abstract: The space H^0(M, L^N) of holomorphic sections of powers of an ample line bundle over a compact Kahler manifold has a generalization to almost-complex symplectic manifolds. We study the limits as N tends to infinity of the probabilities that the sections do various things. The main theme is that in an asymptotic sense, almost holomorphic sections behave very much like holomorphic ones, hence like homogeneous polynomials of degree N.

Topic: Complex Monge-Ampere equation and Ricci curvature March 23

Presenter: Gang Tian, M.I.T. & Princeton University

Topic: Random Walks and the Gittins Index March 24

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: Embedded minimal tori and applications March 24

Presenter: Tobias Colding, New York University

Topic: TBA March 27

Presenter: Robert Glassey, Indiana University

Topic: TBA March 29

Presenter: R. Stanton, Ohio State University

Topic: TBA March 30

Presenter: Ilya Ustilovsky, New York University

Topic: TBA March 31

Presenter: Yu Yuan, University of Texas

Topic: Uniqueness of tangent connections of Yang-Mills connections March 31

of isolated singularities

Presenter: Yang Boazhong, M.I.T.

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

Topic: Some Insights of Computational Complexity Theory April 12

Presenter: Avi Wigderson, I.A.S. & Hebrew University, Jerusalem

Abstract: Computational complexity theory has been one of the most exciting fields of scientific research over the last few decades. This research studies the power of feasible computation, and is guided by a few clear and focused questions, deeply motivated on scientific, practical and philosophical grounds, like the P vs NP problem, and the questions on the power of randomized and quantum computation. While these problems are far from resolved, Complexity Theory was able to offer fresh rigorous definitions to some central notions which naturally (or less so) arise from these questions, and unveil many rich and beautiful connections between them. In this general survey, I would like to probe some of the unique features and insights of the complexity theory viewpoint. This will be done by considering how (and why) notions which intrigued people for centuries or even millenia (like Knowledge, Randomness, Cryptography, Learning, Proof, and naturally, Computation), reveal new dimensions, and are suprisingly linked together, when viewed from our special Computational Complexity glasses.

Topic: Strong form of Poincare duality April 13

Presenter: Edgar Brown, Brandeis University