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: Products of trees, lattices and simple groups March 28

Presenter: Shahar Mozes, the Hebrew University at Jerusalem

Abstract: We study groups of automorphisms of trees and in particular lattices in the product of groups of tree automorphisms. A certain class of such lattices exhibits phenomena analogous to ones encountered in lattices in semisimple Lie groups such as superrigidity and arithmeticity. Combining these with phenomena which do not exist in the setting of Lie groups we construct the first examples of finitely presented torsion free simple groups. We obtain simple groups which are amalgamated products $A *_C B$ of two free groups thus answering a question of P.M. Neumann.

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: Random regular graphs of high degree March 29

Presenter: Benny Sudakov, Princeton University and the Institute for Advanced Study

Topic: Some problems in compensated compactness and H-measures March 29

Presenter: Luc Tartar, Carnegie-Mellon

Topic: Exceptional Dehn surgeries March 29

Presenter: Cameron Gordon, University of Texas at Austin

Abstract: If K is a hyperbolic knot in the 3-sphere, then most Dehn surgeries on K yield hyperbolic 3-manifolds. We will describe progress towards the goal of completely describing the exceptions.

Topic: Operator Valued Holomorphic Functions with Positive Real Part March 30

Presenter: Jeff Schenker, Princeton University

Abstract: The classical theory of holomorphic maps of the disk with positive real part is quite elegant. I will show how this theory can be translated to describe operator valued holomorphic functions. The arguments require only basic Hilbert space and measure theory.

Topic: Einstein spaces as attractors for the Einstein flow 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: Collective Dynamics for Brain Decisions April 2

Presenter: John Hopfield, Molecular Biology, Princeton University

Abstract: The network of nerve cells in a brain carries out computations through exploiting the dynamics of its electrical activity. A transient dynamical 'phase transition' coordinating the activity of many neurons can represent and signal a 'decision' in the brain. The phase transition can implement powerful algorithms that are part of the brain's computational resources. The use of such an algorithm is illustrated in a designed 'neural' system that recognizes a spoken word independent of the speed at which the word is spoken. Such integration over a brief 'moment' is common to many perceptual systems. No previous biology will be assumed of the audience.

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: TBA April 3

Presenter: N. Tziolas, University of Warwick

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

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: Free boundary regularity for the Poisson kernel April 5

Presenter: Tatiana Toro, University of Washington

Topic: Dirac operators and vanishing theorems on foliations 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: Stability and Dynamics of Self-similarity in Evolution Equations April 9

Presenter: Andrew Bernoff, Harvey Mudd College

Topic: TBA April 9

Presenter: A. Bertram, University of Utah

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: TBA April 11

Presenter: Igor Rodnianski, 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: Twisted Orbifold K-theory 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: Change of Variable Formula for Complex Genera and Its Applications to April 17

Higher Dimensional Flops

Presenter: Chin-Lung Wang, Harvard University

Topic: Syzygies over the exterior algebra and Chow forms April 17

Presenter: F. Schreyer, University of Bayreuth

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 24

Presenter: János Pach, New York 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 26

Presenter: B. Doug Park, McMaster University

Topic: TBA April 27

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

Topic: TBA April 30

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

Topic: TBA May 1

Presenter: Jeff Kahn, Rutgers University

Topic: TBA May 3

Presenter: Lenny Ng, MIT

Topic: TBA May 4

Presenter: Guan Bo, University of Tennessee

Topic: Time-dependent Taylor Vortices in Wide-Gap Spherical Couette Flow May 7

Presenter: Rainer Hollerbach, Geosciences, Princeton University