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

Presenter: Salvatore Torquato, Princeton University

Abstract: Bernal (1965) has remarked that "heaps (random close-packed arrangements of particles) were the first things that were ever measured in the form of basketfuls of grain for the purpose of trading or of collection of taxes." Random packings of identical spheres have been studied by biologists, materials scientists, engineers, chemists and physicists to understand the structure of living cells, liquids, granular media, glasses and amorphous solids, to mention but a few examples. Despite its long history, there are many fundamental issues concerning random packings of spheres that remain elusive, including a precise definition of random close packing (RCP). If such a definition of the RCP state could be presented, then one might go about quantifying the problem with the rigor that has been used very recently to prove that the densest possible packing fraction of spheres in three-dimensional space is 0.7405..., corresponding to the close-packed face-centered cubic (FCC) lattice or its stacking variants. It is shown that the current picture of the RCP state cannot be made mathematically precise and support this conclusion via a molecular dynamics study of hard spheres. We suggest that this impasse can be broken by introducing the new concept of a "maximally random jammed" state, which can be made precise.

Topic: Regularity properties of nonlinear wave equations April 11

Presenter: Igor Rodnianski, Princeton University

Abstract: I will discuss recent interactions of Fourier and geometric analysis in the well posedness theory for quasilinear wave equations, wave maps, and the Einstein equations.

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: A fully nonlinear equation in conformal geometry and 4-manifolds of positive April 12

Ricci curvatures

Presenter: Matthew Gursky, Indiana University and Princeton University

Topic: A generalized divisor problem April 12

Presenter: Robert Vaughan, Penn State University

Topic: Concentration compactness and geometric flows April 13

Presenter: Hartmut Schwetlick, MPI

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: Segregation in the Falicov-Kimball model April 18

Presenter: Daniel Ueltschi, Princeton University

Abstract: The Falicov-Kimball model is a simple quantum lattice model that describes light and heavy electrons interacting with on site repulsion. It can be seen as a simplification of the Hubbard model. By neglecting the kinetic (hopping) energy of particles of spin up, one gets Falicov-Kimball. We show that away from half-filling, i.e. if the densities of both kinds of particles are chosen not to add to 1, the particles segregate for large enough repulsion. In the language of the Hubbard model, this means creating two regions with positive and negative magnetization. The key is a lower bound for the sum of eigenvalues of the discrete Laplace operator (hopping matrix) in an arbitrary domain, with Dirichlet boundary conditions. The lower bound consists of a bulk term, independent of the shape of the domain, and of a term proportional to the boundary. Therefore, one lowers the kinetic energy of the itinerant particles by choosing a domain with small boundary. For Falicov- Kimball, this corresponds to having a unique and ``compact'' domain that is empty of heavy particles.

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: TBA April 19

Presenter: Andras Nemethi, Ohio State University

Topic: TBA April 19 IAS

Presenter: Ben Green, Princeton University

Topic: Complex vector bundles and applications to linear partial differential equations April 20

Presenter: Howard Jacobowitz

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: Concordant sequences and integer-valued entire functions April 24

Presenter: Jonathan Pila, Melbourne, Australia

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

Presenter: S.K.Yueng, Purdue University

Topic: Quantum Pumps April 25

Presenter: Yosi Avron, Technion, Haifa

Abstract: Quantum pumps transfer charge, energy etc. beween electron baths. A useful description is in terms of adiabatic scattering theory. I shall discuss notions of optimal pumps and give their geometric characterization. This is joint work with A. Elgart, G.M. Garf and L. Sadun.

Topic: TBA April 26

Presenter: B. Doug Park, McMaster University

Topic: TBA April 26

Presenter: Cristian Popescu, Johns Hopkins

Topic: Negative sectional curvature metrics on 3-manifolds with boundary 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 3

Presenter: Johan de Jong, M.I.T.

Topic: TBA May 4

Presenter: Guan Bo, University of Tennessee

Topic: TBA May 4

Presenter: C. Margerin, Ecole Polytechnique

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

Presenter: Rainer Hollerbach, Geosciences, Princeton University