Topic: Segregation problems in binary fluids March 7

Presenter: Rafelle Esposito, University of Rome

Abstract: We consider a system of two species of particles (colored particles), interacting via color blind collisions and long range repulsion between different colors. Under suitable assumptions the system is described by kinetic equation of Vlasov-Boltzmann type. We discuss the hydrodynamical limit in the Euler and Navier-Stokes regimes. Moreover we study the equilibria and their stability as minimizers of a free energy functional.

Topic: Free probability, free entropy and applications to von Neumann algebras March 7

Presenter: Liming Ge, University of New Hampshire

Abstract: Basic results in the theory of free probability and free entropy will be explained (e.g., free central limit theorem, limit laws of random matrices, etc.). Applications of these results to the solutions of some longstanding open questions in operator algebras will be discussed.

Topic: Cohomological equations and asymptotics of ergodic averages for March 8

horocycle flows

Presenter: Livio Flaminio, University of Lille, France

Abstract: There are infinitely many obstructions to existence of smooth solutions of the cohomological equation for horocycle flows on Riemann surfaces. We study the Sobolev regularity of these obstructions and use them to derive precise asymptotics for the ergodic averages of the horocycle flows.

Topic: Complications of spaces and polyGEMs March 8

Presenter: Wojchiech Chacholski, Yale University

Topic: TBA March 9

Presenter: Andrea Bertozzi, Duke University

Topic: Lefschetz fibrations and symplectic four-manifolds March 9

Presenter: Tian-Jun Li, Princeton University

Topic: Rank-two vector bundles over a general curve: A conjecture of March 12

Bertram-Feinberg-Mukai

Presenter: H. Clemens, University of Utah

Topic: TBA March 12

Presenter: Mei-Chi Shaw, University of Notre Dame

Topic: Passive Scalar Mixing: Averaging in Time Varying Flow March 12

Presenter: Rich Mclaughlin, University of North Carolina

Abstract: We discuss enhanced mixing induced by complex fluid motion, first overviewing the importance of these problems from a general perspective in modeling turbulence with closure coefficients, and then focusing upon rigorous averaging theories in idealized contexts to try to explicitly quantify effective mixing coefficients. We overview the idealized problem of calculating enhanced diffusivities for passive transport in steady periodic geometries, reviewing the poor dependence of these coefficients upon large scale flow parameters. Through the introduction of temporal variation into these models through rapid wind fluctuations, we present a theory which identifies regions in the Peclet-Strouhal plane for which fluctuation massages the poor coefficient dependence existing in the steady geometry, and regions with the mixing coefficients plagued by non-mononotic Peclet dependence. Two-parameter asymptotics in time varying shear geometries show that the limiting behavior of vanishing Strouhal number and infinite Peclet number varies with path, indicating different scaling regimes. Numerical simulations of the time varying cell problem for more general non-sheared geometries will be discussed.

Topic: Critical percolation in the plane is conformally invariant March 13

Presenter: Stanislav Smirnov, KTH, Stockholm and Caltech, Pasadena

Abstract: We will discuss critical site percolation on triangular lattice in the plane, derive the conformally invariant Cardy formula, and construct the (conformally invariant) continuum scaling limit.

Topic: Ramsey games and the second moment method March 13

Presenter: Jozsef Beck, Rutgers University

Topic: Families of singular rational curves March 13

Presenter: S. Kebekus, Bayreuth

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: Essential Laminations and Kneser Normal Form March 15

Presenter: David Gabai, Caltech

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: 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: Some mathematical challenges from materials science March 28

Presenter: Jean Taylor, Rutgers University

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

differential manifolds April 5

Presenter: Daniel Biss, MIT

Topic: TBA April 6

Presenter: Daniel Pollack, University of Washington

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

Presenter: Rainer Hollerbach, Geosciences, Princeton University

Topic: TBA April 10

Presenter: Michael Krivelevich, Tel Aviv University

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

Presenter: Salvatore Torquato, Princeton University

Topic: TBA April 12

Presenter: Alejandro Adem, Madison

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 18

Presenter: Benjamin Weiss, Stanford University

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