Events for the week of February 13 2012

Abstract. I will discuss Corks and Plugs (and possibly anti-corks) which are useful tools for understanding exotic smooth manifolds. A natural puzzle is to find the corks and plugs of a given small exotic manifold, such as the Dolgachev surface and the Akhmedov-Park's exotic CP^2 # 2(-CP^2), whose handle-body pictures I will describe.

Abstract. I will discuss my work with J. Kahn about the Ehrenpreis conjecture and the surface subgroup theorem for hyperbolic 3-manifolds.

Abstract. Orbital characters are symmetric functions on group algebras. They can be seen as a generalization of characters. In this talk, elementary properties of orbital characters will be presented and then some applications to modular representation theory of finite groups are given.

Abstract. In critical planar percolation, there are almost surely no infinite clusters. However, if the configuration evolves according to a continuous time Markov chain, there might be exceptional times when the origin is connected to infinity. A theorem of Christophe Garban, Oded Schramm and myself from 2008 is that such exceptional times do exist, and (for site percolation on the triangular lattice) their Hausdorff dimension is 31/36. I will talk about two results:

(1) The probability that we have to wait for the first exceptional time for more than time t is exponentially small in t.

(2) The cluster of the origin at this first time is thinner than the cluster at a typical exceptional time (which is Kesten's Incipient Infinite Cluster).

The proof of (1) uses an extension of a classical lemma of Ajtai-Komlos-Szemeredi about hitting times of simple random walk on expander graphs. The proof of (2) uses (1). I will also discuss what happens if we relax the expander condition: if we know fast decorrelation of a single event in a reversible Markov chain, can we deduce that the event is unlikely to occur continually for a long time?

These are joint works with Alan Hammond, Elchanan Mossel, and Oded Schramm.

Abstract. High performance numerical equilibrium solvers are crucial to the computational effort in plasma physics for magnetic fusion applications. The computed plasma equilibria are indeed needed as inputs to numerical analyses of the global stability and transport properties of the plasma configurations under consideration. Equilibrium solvers need to be fast, so the computational time can be primarily spent on the stability and transport calculations. Equilibrium solvers need to be accurate, to minimize uncertainties associated with error propagation.

In this talk, I will point out the fundamental difference in the mathematical and physical natures of equilibria in axisymmetric configurations and equilibria in inherently three-dimensional configurations. I will present two new equilibrium solvers we developed for the two different cases, and highlight the unique challenges faced by three-dimensional solvers.

Abstract. There has recently been a coordinated attempt to place the classification problem for various classes of separable C*-algebras within the general context of descriptive set theory of equivalence relations, in order to try to study the complexity of classifying various classes of C*-algebras. In order to successfully carry out this program, one needs to establish that many standard C*-algebra constructions, as well as central theorems, admit "Borel versions". I will give an overview of these developments, and discuss in particular a Borel version of Kirchberg's O_2 embedding theorem.

This is joint work with Ilijas Farah and Andrew Toms.

Abstract. Since the early days of Euler, at least three books and countless articles were published, solely devoted to various optimization questions, related to coverings of graphs by their circuits.

After a brief introduction, I will focus on the "Shortest Cycle Cover Problem": Find a set of circuits of a graph whose edge union is the entire graph and the sum of their lengths is minimum. I will elaborate on some techniques, results and conjectures, in particular on a recent joint paper with E. Macajova, A. Raspaud and X. Zhu. In that article we show how short cycle covers can be derived from Fano-Flows, which are certain embeddings of cubic graphs on the Fano projective plane.

Abstract. The abstract is at

http://www.math.ucla.edu/~rouquier/AbstColl2012.pdf

Abstract. Let M be some separable object of functional analysis, e.g. a von Neumann algebra, unitary group, etc... Assuming the continuum hypothesis, any two ultrapowers of M will be isomorphic. What about if we assume the negation of the continuum hypothesis? Then the answer turns out to be dependent on the model-theoretic notion of stability. In Part I, we will show that if M is unstable, then M has two nonisomorphic ultrapowers. In Part II, we will show that if M is stable, then all ultrapowers of M are isomorphic. All relevant notions will be defined.

Abstract. A cornerstone of statistical physics is the determination of the critical temperature and some critical exponents of the planar Ising spin model by Onsager (1944). In particular, these give the correlation length exponent of the FK-Ising model (a dependent bond percolation model describing the correlations between the Ising spins), which determines how fast a large finite system becomes supercritical from subcritical as the temperature is lowered.

However, Onsager's approach is very non-geometric and somewhat mysterious for mathematicians. In this talk, I will discuss a definition of the correlation length via crossing probabilities, and its computation using Smirnov's fermionic observable. Secondly, I will highlight a striking phenomenon about the near-critical behavior of FK-Ising, which is completely missing from the case of standard percolation: in any monotone coupling of FK configurations \omega_p (e.g., in the one introduced by Grimmett 1995), as one raises p near p_c, the new edges arrive in a fascinating self-organized way, so that the correlation length is not governed anymore by the amount of pivotal edges at criticality.

This is joint work with Hugo Duminil-Copin and Christophe Garban.

Abstract. Roughly speaking, Herbrand's theorem states that in a universal theory with quantifier elimination, given any formula f(x,y), there are finitely many terms t_1(x),...,t_n(x) such that whenever f(a,y) is true for some y, then f(a,t_i(a)) is true for one of the terms t_i. We will present an "approximate version" of this theorem for continuous logic and show how it implies a theorem describing definable functions in Sigma_2 theories. We will present several applications of this result, both in classical logic and continuous logic.

Abstract. We will study some modifications to the notion of an exact C*-algebra by replacing the minimal tensor product with the reduced free product. First we will demonstrate how the reduced free product of a short exact sequence of C*-algebras with another C*-algebra may be taken. It will then be demonstrated that this operation preserves exact sequences. We will also establish that adjoining arbitrary k-tuples of operators in a free way behaves well with respect to taking ultrapowers.

Abstract. We obtain the asymptotic radial distribution of zeros of orthogonal polynomials on the unit circle when the random Verblunsky coefficients decay as $O( 1/\sqrt{n} )$