| Michael Anshelevich: A Bochner-Pearson type class (February 5, 10:15-11:00) |
| Abstract: This talk is about the operator $L_\mu[f] = \int_{\mathbb{R}} \frac{f(x) - f(y)}{x - y} \,d\mu(y)$ (here $\mu$ is a measure), or more precisely about the related Sturm-Liouville-type operator $Q_\mu = p(x) L_\mu^2 + q(x) L_\mu$. I will describe when such an operator has polynomial eigenfunctions; for the SL operators, the corresponding class is the Bochner-Pearson class. The operator has orthogonal polynomial eigenfunctions only if $\mu$ is a semicircular distribution. More generally, the operator $p(x) L_\mu L_\nu + q(x) L_\mu$ has orthogonal polynomial eigenfunctions only if $\mu$ and $\nu$ are related by a Jacobi shift. |
| Jason Asher: Free Diffusions and von Neumann Algebras (February 4, 15:30-16:00) |
| Abstract: We establish technical properties of von Neumann algebras that are generated by the sationary laws of certain free stochastic differential equations. In particular, we consider the free diffusion equation $dX_t = dS_t - \frac{1}{2} DV(X_t) dt$ for a suitably locally convex self-adjoint multivariate polynomial $V$. We will make use of results of Guionnet and Shlyakhtenko that give existence and uniqueness of, and asymptotic norm convergence to, stationary solutions of such SDE. |
| Gerard Ben Arous: Complexity of Random Morse functions, in the large N limit (February 4, 9:30-10:15) |
| Abstract: We relate the computation of the complexity of general Gaussian functions on the Sphere in large dimensions with Random Matrix theory. Using this link we compute their complexity, i.e their number of critical points, of given index on any level sets. We relate this question to the description of the the low energy states of spherical spin glasses. THis joint work with A.Auffinger and J.Cerny |
| Philippe Biane: Kerov polynomials and large N limit (February 1, 9:30-10:15) |
| Abstract: I will review Kerov polynomials which express characters of symmetric groups in terms of free cumulants and are convenient for describing the asymptotics of symmetric group representations. |
| Charles Bordenave: Spectrum of non-hermitian heavy tailed random matrices (February 2, 11:00-11:45) |
| Abstract: Consider a $n \times n$ matrix with independent and identically distributed entries. If the variance is finite, the circular law Theorem asserts that the empirical spectral distribution converges to the uniform distribution on the unit complex disc. This theorem was recently proved by Tao and Vu after important breakthroughs of Girko, Bai and others. In this talk, we will consider the infinite variance case. We will state the convergence of the empirical spectral measure and give some properties of the limiting distribution. This is a joint work with Djalil Chafaï and Pietro Caputo. |
| Thierry Cabanal-Duvillard: A matrix model for a new continuum between classical and free infinite divisibility (February 5, 11:15-12:00) |
| Abstract: A new continuum between classical and free infinitely divisible (FID) laws with compact support has been recently established by Florent Benaych-Georges. It is obtained using conjugation of "infinitesimal" FID laws by the free unitary Brownian motion. A random matrix approximation is constructed in order to extend the continuum to non-compact FID laws. |
| Benoit Collins: Free Probability and Quantum Information Theory (February 4, 11:00-11:45) |
| Abstract: I will explain how Weingarten calculus and results about the asymptotic norm of products of random matrices can help to improve our understanding of the violation of the minimum output entropy for random quantum channels (joint work with Ion Nechita and Serban Belinschi). |
| Stephen Curran: Probabilistic aspects of easy quantum groups (February 1, 11:00-11:45) |
| Abstract: The class of "easy" quantum groups was introduced by Banica and Speicher to provide a framework for studying certain common probabilistic and representation theoretic aspects of $S_n, O_n$ and their "free versions". In this talk we will survey some recent results obtained in this framework, including de Finetti theorems and an extension of some results of Diaconis-Shahshahani. This is joint work with Teodor Banica and Roland Speicher. |
| Yoann Dabrowski: A non-commutative path space approach to stationary free SDEs (February 4, 14:30-15:00) |
| Abstract: We construct stationary solutions to certain free SDEs by interpreting such a solution as a trace-state on a non-commutative analog of a path space. We discuss the analytic problems that arise from this approach. |
| Amir Dembo: Spectral measure of heavy tailed band and covariance random matrices (February 4, 13:30-14:15) |
| Abstract: We study the asymptotic behavior of the appropriately scaled spectral measure of large random real symmetric matrices with entries based on independent random variables whose distribution is in the domain of attraction of a stable law. As a special case, we analyze the limiting spectral density for empirical covariance matrices with heavy tailed entries. The talk is based on a joint work with Serban Belinschi and Alice Guionnet. |
| Nizar Demni: Ultraspehrical type generating functions and generalized Cauchy Stieltjes transforms (February 2, 15:00-15:45) |
| Abstract: we characterize generating functions of ultraspherical type via differential equations. As by product, we relate generalized Cauchy-Stieltjes transforms of some Beta distributions and powers of ordinary Cauchy-Stieltjes transforms. We also recover another characterization of the free Meixner family. Then, we focus on a deformation of the Wigner distribution and compute its moments and free cumulants. Finally, some similarities with generalized Gamma convolutions are shown. |
| Ken Dykema: Matrices of unitary moments (February 5, 9:00-9:45) |
| Abstract: Motivated by Kirchberg's simplification of Connes' embedding problem, we consider matrices of second--order unitary moments, (inside the set of correlation matrices) and discover a few facts about them. (Joint work with Kate Juschenko.) |
| Alice Guionnet: The single ring theorem (February 2, 13:30-14:15) |
| Abstract: will talk about a recent work with O. Zeitouni and M. Krishnapur. We study the empirical measure $L_{A_n}$ of the eigenvalues of non-normal square matrices of the form $A_n=U_nD_nV_n$ with $U_n,V_n$ independent Haar distributed on the unitary group and $D_n$ real diagonal. We show that when the empirical measure of the eigenvalues of $D_n$ converges, and $D_n$ satisfies some technical conditions, $L_{A_n}$ converges towards a rotationally invariant measure on the complex plan whose support is a single ring and which was first described by Haagerup and Larsen. In particular, we provide a complete proof of Feinberg-Zee single ring theorem. |
| Mylene Maida: Central limit theorem for unitary Brownian motion (February 2, 16:00-16:45) |
| Abstract: This is joint work with Thierry L\'evy (CNRS and Universit\'e de Gen\`eve). P. Diaconis and S. Evans established a central limit theorem for traces of functions of uniformly distributed unitary matrices and the covariance involved the $H^{1/2}$-scalar product. We consider the case of unitary Brownian motion, for which P. Biane described the limiting process as the dimension goes to infinity, the so-called free multiplicative Brownian motion and we looked in this work at the fluctuations. I will discuss both combinatorial and probabilistic approaches of this problem, try to explain the form of the covariance we get (via a martingale approach) and if time permits, show that this covariance converges as time grows to the $H^{1/2}$-scalar product pointed out by Diaconis and Evans. |
| Alexandru Nica: Infinitesimal non-crossing cumulants and free probability of type B (February 3, 10:15-11:00) |
| Abstract: I will present a joint work with Maxime Fevrier on the relation between noncrossing partitions of type B and the framework of infinitesimal free probability considered by Belinschi and Shlyakhtenko in 2009. We introduce and study a concept of infinitesimal noncrossing cumulants. We prove that infinitesimal freeness is equivalent to a vanishing condition for mixed cumulants; this gives an infinitesimal version for a theorem of Speicher from usual free probability. We discuss situations when usual freeness can be naturally upgraded to infinitesimal freeness. In connection to that, we observe how one gets examples of asymptotic infinitesimal freeness for large random matrices with entries in a 2-dimensional Grassman algebra. |
| Jonathan Novak: An asymptotic version of a theorem of Knuth (February 4, 16:15-17:00) |
| Abstract: Three results on the enumeration of permutations without long decreasing subsequences serve to illustrate the connection of this topic with random matrix theory: Knuth's enumeration of permutations with no decreasing subsequence of length three (Catalan numbers), Regev's asymptotic enumeration in the single-scaling limit (Dyson-Mehta-Selberg integral), and Baik-Deift-Johansson's asymptotic enumeration in the double scaling limit (Tracy-Widom distribution). Using a simple symmetry of the Coulomb gas, we will interpret Regev's result as an asymptotic version of Knuth's theorem rather than a degeneration of the Baik-Deift-Johansson theorem. |
| Dimitri Shlyakhtenko: A II$_\infty$ factor associated to a planar algebra (February 1, 16:00-16:45) |
| Abstract: We discuss some von Neumann algebras associated to a planar algebra. |
| Terence Tao: Brown measure and the circular law (February 2, 9:30-10:15) |
| Abstract: The circular law asserts that the normalised spectral distribution of a large square random matrix with iid entries of mean zero and variance one, is asymptotically given by the uniform measure on the unit disk. A lot of effort over many decades has gone into the proof of this law. On the other hand, it is relatively easy (though still not trivial) to show that the free probability limit of such matrices (assuming all moments are finite) has the Brown measure predicted by the circular law. As is well known, Brown measure does not always correspond to spectral measure in general. Nevertheless, by using tools from the most recent work on the circular law by Van Vu and myself, one can show that Brown measure and asymptotic spectral measure in fact coincide for any random matrix ensemble with independent (but not necessarily identically distributed) entries, even when the mean and variance of the entries are allowed to vary (but obeying some uniform boundedness conditions). We discuss this result in this talk. |
| Dan Voiculescu: Free Analysis: the analogue of the Riemann sphere (February 3, 9:00-9:45) |
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| Jiun-Chau Wang: An identity relative to the free central limit theorem (February 3, 11:15-12:00) |
| Abstract: We present and identity that gives a new interpretation of the free central limit theorem and discuss further its applications on the superconvergence to the semicircle law. |
| Feng Xu: Introduction to CFT and subfactors (February 1, 14:30-15:15) |
| Abstract: I will give an introduction to operator algebraic approach to chiral two dimensional conformal field theory (CFT) motivated by subfactors, and discuss some large $N$ aspects. |
| Paul Zinn-Justin: Weingarten matrices and Jucys-Murphy elements (February 1, 13:30-14:15) |
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22 abstracts submitted.