(September 17) Steve Zelditch: Nodal sets and ergodicity for eigenfunctions of the Laplacian — We discuss some new results on the ergodic properties of eigenfunction on Riemannian manifolds with ergodic geodesic flow. We emphasize results on nodal (zero) sets, both in the ergodic case and in general.

(October 15) Gerard Letac: The Thomae formula and the hypergeometric densities of random continued fractions — The abstract is available here.

(October 22) Pierre Colmez: The p-adic local Langlands correspondence — Wiles's proof of Fermat's last theorem has shown the power of p-adic methods to tackle questions that were considered to be part of harmonic analysis. A p-adic counterpart to the Langlands program is slowly coming to existence, and I will try to explain where its local part is coming from, what we know about it (not much, which is part of its charm), and what it is good for.

(November 19) David Harbater: Quadratic forms, patching, and local-global principles — A classical question is whether a given quadratic form has a non-trivial solution over a given field, such as the rational numbers. This can be approached using local-global principles; the terminology arises from an analogous geometric context, in which local information is used to deduce global information. After reviewing classical theorems, this talk will discuss recent results about quadratic forms and local-global principles that have been obtained using patching, an algebraic technique that is motivated by geometry and analysis. (This is joint work with Julia Hartmann and Daniel Krashen.)

(February 11) Peter Constantin: Complex Fluids — The talk will be about some of the models used to describe fluids with particulate matter suspended in them. Some of these models are very complicated. After a bit of history and a review of known results, I will try to point out some open problems, isolate some of the mathematical difficulties, and illustrate some of the phenomena on simpler didactic models.

(February 25) Andrew Majda: Mathematical Strategies for Real Time Filtering of Turbulent Dynamical Systems — An important emerging scientific issue in many practical problems ranging from climate and weather prediction to biological science involves the real time filtering and prediction through partial observations of noisy turbulent signals for complex dynamical systems with many degrees of freedom as well as the statistical accuracy of various strategies to cope with the "curse of dimensions". The speaker and his collaborators, Harlim (North Carolina State University), Gershgorin (CIMS Post doc), and Grote (University of Basel) have developed a systematic applied mathematics perspective on all of these issues. One part of these ideas blends classical stability analysis for PDE's and their finite difference approximations, suitable versions of Kalman filtering, and stochastic models from turbulence theory to deal with the large model errors in realistic systems. Many new mathematical phenomena occur. Another aspect involves the development of test suites of statistically exactly solvable models and new NEKF algorithms for filtering and prediction for slow-fast system, moist convection, and turbulent tracers. Here a stringent suite of test models for filtering and stochastic parameter estimation is developed based on NEKF algorithms in order to systematically correct both multiplicative and additive bias in an imperfect model. As briefly described in the talk, there are both significantly increased filtering and predictive skill through the NEKF stochastic parameter estimation algorithms provided that these are guided by mathematical theory. The recent paper by Majda et al (Discrete and Cont. Dyn. Systems, 2010, Vol. 2, 441-486) as well as a forthcoming introductory graduate text by Majda and Harlim (Cambridge U. Press) provide an overview of this research.

(March 11) Jose Carrillo: Keller-Segel, Fast-Diffusion and Functional Inequalities — We will show how the critical mass classical Keller-Segel system and the critical displacement convex fast-diffusion equation in two dimensions are related. On one hand, the critical fast diffusion entropy functional helps to show global existence around equilibrium states of the critical mass Keller-Segel system. On the other hand, the critical fast diffusion flow allows to show functional inequalities such as the Logarithmic HLS inequality in simple terms who is essential in the behavior of the subcritical mass Keller-Segel system. HLS inequalities can also be recovered in several dimensions using this procedure. It is crucial the relation to the GNS inequalities obtained by DelPino and Dolbeault. This talk corresponds to two works in collaboration with E. Carlen and A. Blanchet, and with E. Carlen and M. Loss.

(April 1) Federico Rodriguez-Hertz: Global Rigidity for certain actions of higher rank lattices on the torus — In this talk we will give an approach to the following theorem:

Let Γ be an irreducible lattice in a connected semi-simple Lie group with finite center, no non-trivial compact factor and of rank bigger than one. Let α: Γ → Diff(T^N) be a real analytic action on the torus preserving an ergodic large measure (large means essentially that its support is non-trivial in homotopy). α induces a representation a₀: Γ → SL(N, Z) and thus a linear action α₀. Assume further that Γ has no zero weight and no rank one factor. Then α and α₀ are conjugate by a real analytic map outside a finite α₀-invariant set.

The theorem essentially says that α is built from α₀ by blowing up finitely many points.

(April 8) Maria Schonbek: Questions on regularity of liquid crystals — The flows of nematic liquid crystals can be treated as slow moving particles where the fluid velocity and the alignment of the particles influence each other. The hydrodynamic theory of liquid crystals was established by Ericksen and Leslie in the 1960's. As Leslie points out in his 1968 paper : ``liquid crystals are states of matter which are capable of flow, and in which the molecular arrangements give rise to a preferred direction". In my talk I will consider one of the simplified models for the flow of nematic liquid crystals, and discuss the existence of regular solutions.

(April 15) Alexander Stanoyevitch: Randomized search and optimization — In this talk we will introduce the concepts of setting up and running some search and optimization algorithms that can be applied to difficult problems from mathematics and other fields. Such methods can lead to many insights about a difficult problem, and have been used to obtain state-of-the-art best solutions for some famously unsolved problems. We will begin by explaining the simple concept of simulations, and then move on to describe some examples of more sophisticated randomized algorithms. Some details will be given to describe setting up simulations for specific problems and the implementation of a randomized algorithm used to obtain a nontrivial bound on a certain Ramsey number.

(May 6) Richard Canary: Dynamics on character varieties — If S is a closed orientable surface of genus at least 2, it is natural to consider the Teichmueller space of hyperbolic metrics on S (up to isotopy). This Teichmuller space can be viewed as a component of the character variety of (conjugacy classes of) representations of the fundamental group of S into PSL(2,R). The mapping class group of S, i.e the group of isotopy classes of orientation-preserving self-homeomorphism of S, acts properly discontinuously on Teichmueller space.

In this talk, we discuss recent work on mapping class groups of 3-manifolds on PSL(2,C)-character varieties. Let M be a fixed compact 3-manifold and let X(M) be the character variety of (conjugacy classes of) representations of the fundamental group of M into PSL(2,C). The outer automorphism group of the fundamental group of M acts naturally on X(M). We will survey recent work of Canary, Lee, Minsky and Storm on the dynamics of this action.

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