Topic: Fluctuations for stochastic lattice gases March 14

Presenter: Rosanna Marra, University of Rome

Abstract: We discuss the hydrodynamic incompressible limit in d> 3 for a thermal lattice gas, and prove a law of large numbers for the density, velocity field and energy. We study the equilibrium fluctuations for this model and prove a central limit theorem for a suitable modification of the vector fluctuation field whose components are the density, velocity and energy fluctuations fields.

Topic: Rational and integral points on algebraic varieties March 14

Presenter: Yuri Tschinkel, Princeton University

Abstract: One of the central problems in modern number theory is to explore the relationship between the global geometry and arithmetic properties of algebraic varieties. In particular, one is interested in distribution properties of rational and integral points in Zariski topology and with respect to heights. I will explain some ideas and techniques from algebraic geometry and harmonic analysis on adelic groups used in the study of these distributions.

Topic: Dynamincal bounds for the Fibonacci Hamiltonian March 15

Presenter: Alexander Kiselev, University of Chicago

Abstract: We prove new lower and upper dynamical bounds for the Fibonacci operator, a popular model of one-dimensional quasicrystals. It is given by a discrete Schr\"odinger operator on $l^2(\integers),$ \[ h_v u(n) = u(n+1)+u(n-1) + \lambda ([(n+1)\omega] - [n\omega])u(n), \] where $\omega = (\sqrt{5}-1)/2$ is the golden mean. The spectrum of this operator is known to be purely singular continuous. The bounds show that dynamics is intermediate between localization and ballistic transport. Roughly speaking, the upper bound shows that a fixed part of the wavepacket at time $T$ remains inside the ball of radius $R_1(T) \sim T^{C_1 (\log \lambda)^{-1}},$ where $\lambda$ is the strength of coupling. At the same time the lower bound shows that most of the wavepacket leaves the ball of radius $R_2(T)\sim T^{C_2 (\log \lambda)^{-1}},$ $C_1>C_2$ are universal constants. The main new element of the proof is a general upper bound criterion which is derived using ideas from subordinacy theory.

Topic: Hydrodynamic limits of the Boltzman equations March 15

Presenter: F. Golse, ParisVI and Ens. Paris

Topic: Essential Laminations and Kneser Normal Form March 15

Presenter: David Gabai, Caltech

Topic: Uniform distribution of some sequences from cryptography March 15 IAS

Presenter: John Friedlander, University of Toronto

Topic: Gross-Zagier formula with characters March 23

Presenter: Shou-Wu Zhang, Columbia University

Topic: TBA March 26

Presenter: Linda Rothschild, University of California-San Diego

Topic: Stochastic Growth Models on Lattices and Trees March 26

Presenter: Thomas Liggett, University of California, Los Angeles

Abstract: For the past thirty years, probabilists have studied a number of stochastic growth models that were motivated by problems in physics and biology. One of the most important of these is known as the contact process -- growth occurs as the result of "contact" with existing individuals. Such models often exhibit phase transitions, and this is the feature that leads to most of our interest in them. Until a decade ago, the contact process was studied almost exclusively on Euclidean lattices, leading to a rather complete theory in that context. Since then, it has been discovered that the behavior of the process can be quite different on exponentially growing structures such as homogeneous trees. In particular, the phase structure is richer than it is in the lattice case. In this lecture, I will briefly describe the most important results about the contact process on Z^d, and then the contrasting results for the process on a tree. I will then discuss a variant of the contact process on a tree that has the appealing property that the critical value for the phase transition can be computed explicitly. One of the ingredients in the computation is a collection of combinatorial identities satisfied by the d-ary Catalan numbers.

Topic: TBA March 27

Presenter: Linda Rothschild, University of California-San Diego

Topic: Rational curves on hypersurfaces of low degree March 27

Presenter: J. Starr, MIT

Topic: Fluctuations for stochastic lattice gases March 28

Presenter: Michael Kiessling, Rutgers University

Abstract: In this talk I prove that the level sets of the semi-classical particle densities of an infinitely long, stationary beam of relativistic electrons and H+ ions with finite electrical current and unbounded cross section are concentric circular cylinders. Neither uniqueness-by-convexity arguments nor minimization-of-energy-by-radial-rearrangement arguments work in this case. Instead, I use the classical isoperimetric inequality to show that a hypothetical beam with non-radially symmetric cross section necessarily violates the virial theorem which any stationary beam has to obey.

Topic: Flowing crystals: some mathematical challenges from materials science March 28

Presenter: Jean Taylor, Rutgers University

Abstract: During the past 30 years, there has been much mathematical activity concerning minimal surfaces, soap films and soap bubble clusters. There has also been much mathematical activity concerning motion by mean curvature and other motions which reduce surface area. Simultaneously, the interaction of mathematics and materials science has grown. Most metals, semiconductors, and ceramics are made up of atoms in locally ordered arrays, called crystals, which make interfaces with other materials or with crystals having different orientations. The interfaces are often regarded as two-dimensional surfaces which are somewhat like a soap bubble cluster. Furthermore, crystals can grow or shrink, with reduction of surface energy often a major influence on that motion. The ways in which crystals are both like and unlike soap films leads to some fascinating mathematical problems. Results that have been proved, new problems that have been recently formulated, and long-standing open problems will be surveyed.

Topic: Infinite random matrices and ergodic measures March 29

Presenter: Alexei Borodin, University of Pennsylvania

Abstract: We introduce and study a 2-parameter family of unitarily invariant probability measures on the space of infinite Hermitian matrices. We show that the decomposition of a measure from this family on ergodic components is described by a determinantal point process on the real line. The correlation kernel for this process is explicitly computed. At certain values of parameters the kernel turns into the well-known sine kernel which describes the local correlations in Circular and Gaussian Unitary Ensembles. Thus, the random point configuration of the sine process is interpreted as the random set of ``eigenvalues'' of infinite Hermitian matrices distributed according to the corresponding measure. This is a joint work with Grigori Olshanski.

Topic: Einstein spaces as attractors for the Einstein flow March 29

Presenter: Vincent Moncrief, Yale University

Topic: TBA March 30

Presenter: Vincent Moncrief, Yale University

Topic: A discrete analogue of a fractional integral operator April 2

Presenter: Steve Wainger, University of Wisconsin

Topic: TBA April 2

Presenter: Eric Vanden-Eijnden, CIMS, New York University

Topic: Basic facts about wavelets that the cognoscenti are sure they know April 3

but I have doubts about this

Presenter: Guido Weiss, Washington University

Topic: Ground States in Non-relativistic Quantum mechanics April 4

Presenter: Michael Loss, Georgia Tech

Abstract: The excited states of an atom (consisting of an electron interacting with the quantized electromagnetic field and an external potential) all decay with time, but such a system should have a true *ground state* --- one that minimizes the energy and satisfies the Schr\"odinger equation. This conjecture lies at the basis of much of quantum mechanics, but had not been proved except with the assumption of small coupling to the electromagnetic field (essentially perturbation theory). The obstacle was the "infrared problem". It can now be proved quite generally that a ground state exists for *all values* of the coupling. We also show the same thing for a many electron atom under physically natural conditions.

Topic: Holomorphic disks and invariants for 3-manifolds and smooth 4-manifolds April 4

Presenter: Zoltan Szabo, Princeton University

Abstract: We will introduce and study topological invariants for closed 3-manifolds and smooth 4-manifolds. The 3-manifold construction uses Heegaard diagrams and a version of Lagrangian Floer homology. The 4-manifold invariant uses the previous construction, a pairing on Floer-homology and a handle decomposition of the 4-manifold. We will also present some applications in three and 4-manifold topology. This is a joint result with Peter Ozsvath.

Topic: Existence of a Family of Periodic Orbits in Hill's Problem Near the Zero April 5

Velocity Curves

Presenter: Ed Belbruno, Princeton University and IOD

Abstract: Hill's problem is considered. It was formulated by G. Hill back in 1865 and is sometimes called the lunar problem. It is highly nonintegrable, and models the motion of a zero mass point about the smaller of the two primaries in the restricted three-body problem. Little is proven about this problem, unless the zero mass is assumed to move very close to the primary. It is assumed here that the initial conditions of the zero mass lie far from the primary, and near the zero-velocity curves, which is very far from integrable. A method is presented which proves that a family of periodic orbits exists with these initial conditions provided the Jacobi energy is very near to the value of 3^(4/3). A small part of this proof is numerically assisted. Numerical work indicates that this family is part of a bifurcated branch of the so called classical Hill's family shown to exist numerically by Henon in 1969, he labeled g'. It is surprising to me that this proof would find this particular family.

Topic: A matroid-theoretic construction of BO(n) and the topology of combinatorial April 5

differential manifolds April 5

Presenter: Daniel Biss, MIT

Abstract: MacPherson's combinatorial differential (CD) manifolds are an attempt to bridge the gap between the category of smooth manifolds and the category of simplicial complexes. That is, they are purely combinatorial objects which, one hopes, provide a good model for smooth manifolds. We will present a result which gives evidence for the case that CD manifolds succeed in capturing much of the structure of the smooth category. More precisely, the world of CD manifolds has a natural (purely combinatorial) bundle theory, and we show that the classifying space of this bundle theory is homotopy equivalent to BO(n), that is, that these combinatorial vector bundles are precisely the same as ordinary vector bundles. This result has applications to the topology of CD manifolds and to the computation of characteristic classes.

Topic: TBA April 5

Presenter: Weiping Zhang, MIT

Topic: TBA April 6

Presenter: Daniel Pollack, University of Washington

Topic: L^p and dispersive estimates for the wave equation with the inverse-square April 9

potential

Presenter: Shadi Tahvildar-Zadeh, Rutgers University

Topic: Time-dependent Taylor Vortices in Wide-Gap Spherical Couette Flow April 9

Presenter: Rainer Hollerbach, Geosciences, Princeton University

Topic: Stories about pseudo-random graphs April 10

Presenter: Michael Krivelevich, Tel Aviv University

Topic: Relative Gromov-Witten invariants and the mirror formula April 10

Presenter: A. Gathmann, Harvard University

Topic: Revisting an Old Concept: Random Close Packing of Hard Spheres April 11

Presenter: Salvatore Torquato, Princeton University

Topic: Recent results about Orbit Equivalence for actions of non-amenable groups April 12

Presenter: Alex Furman, University of Illinois at Chicago

Abstract: Let G be a discrete group acting ergodically by m.p.t. on a probability space (X,m) and let R_G denote the equivalence relation on X defined by the G-orbits (mod 0). How much information about the group G and its action (X,m,G) is encoded in the relation R_G ? What can be said about the Out(R_G) - the group of measurable maps of X permuting the G-orbits? These purely measure-theoretical questions in Ergodic Theory turn out to be connected to Geometry and to rigidity of lattices in semisimple Lie groups. In the talk we shall survey recent developments in this theory.

Topic: TBA April 12

Presenter: Alejandro Adem, Madison

Topic: TBA April 13 IAS

Presenter: Bob Vaughan, PSU

Topic: The generalized KdV equation on the half-line April 16

Presenter: Jim Colliander, University of California - Berkeley

Topic: Creating Stability from Instability April 16

Presenter: Christopher Jones, Brown University

Abstract: The current state-of-the-art technology in optical communications is based on the use of Dispersion Managed Solitons (DMS). These propagate on fibers with dispersion compensating itself periodically. Using variational methods and averaging, a full mathematical theory for DMS will be given. Surprisingly, it is shown that the strategy can be pushed to the point where the "pulse" is oscillating between unstable states and yet remains stable itself. Another case in which two unstable objects are put together to make a stable pulse is exhibited in the FitzHugh-Nagumo system, originally derived as a model of nerve impulse propagation. While these two phenomena are unrelated, mathematically and scientifically, they both suggest that two "wrongs" can make a "right."

Topic: TBA April 17

Presenter: John Conway, Princeton University

Topic: Amenable groups and their actions April 18

Presenter: Benjamin Weiss, Hebrew University of Jerusalem

Abstract: After explaining what amenable groups are and why they are the natural setting for ergodic theory I will survey some new developments related to entropy, uniform mixing, and limit theorems.

Topic: Gibbsian Dynamics and Ergodicity for some Stochastically Forced Dissipative April 19

Equations

Presenter: Di Liu, Princeton University

Abstract: We study the stationary measures for the stochastically perturbed one dimensional Ginzburg-Landau equation, Cahn-Hilliard equation and Kuramoto-Sivashingski equation with periodic boundary conditions. We proved the uniqueness of the stationary measures of these equations under the condition that all ``determining modes'' are forced. The main idea behind the proof is to study the Gibbsian dynamics of the low modes obtained by representing the high modes as functionals of the time-history of the low modes.

Topic: On KdV and completely integrable systems April 23

Presenter: François Trèves, Rutgers University

Topic: TBA April 23

Presenter: John Hopfield, Molecular Biology, Princeton University

Topic: Hyperbolicity,diophantine approximation and complex two ball quotients April 24

Presenter: S.K.Yueng, Purdue University

Topic: TBA April 25

Presenter: Yosi Avron, Technion, Haifa

Topic: TBA April 27

Presenter: Joel Hass, Institute for Advanced Study and University of California at Davis

Topic: TBA May 1

Presenter: Jeff Kahn, Rutgers University

Topic: TBA May 4

Presenter: Guan Bo, University of Tennessee