Abstract: Let X_n be a GUE random matrix of size n, scaled such that the absolute varians of each entry is 1/n, and let f,g be C_infinity functions on the real line of at most polynomial groth. It is proved that the mean value and the covariance E(Tr(f(Xn)/n) and Cov(Tr(f(X_n),Tr(g(X_n)) can both be asymptoticly expanded in powers of 1/n^2. The proof relies on the Harer-Zagier recursion formula for the moments of a GUE random matrix together with an explicit formula for the covariance due to Pastur and Shcherbina. (This is joint work with Steen Thorbjornsen).

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Abstract: A survey of some applications of random matrices to physics and of some basic techniques (diagrammatic methods, topological expansion, saddle point evaluation...) used there.

Abstract: We study matrix models with 'locally convex' (but eventually complex) potentials. We show convergence towards a law which is characterized as the solution of Schwinger-Dyson's equation, or equivalently by the fact that's its conjuguate variable is the cyclic gradient of the potential. We show that the limit law depends analytically in the potential's parameters in the domain of convexity. We prove that the algebras related with this limit law are much alike the algebras generated by free semi-circular variables( in particular they are projectionless).

Abstract: I will speak about the moment method to prove some universality results for the largest eigenvalue of various ensembles of Hermitian random matrices.

Abstract: We explain how existence of certain one-parameter families of embeddings of a von Neumann algebra into its free product with a free group factor can be used to prove lower estimates for microstates free entropy dimension. Examples of such one parameter families come from exponentiation of certain derivations as well as solutions to free stochastic differential equations.

Abstract: We discuss a multi-variable counterpart for a bijection found by Bercovici and Pata in their study of the relations between infinite divisibility in free and in Boolean probability. We put into evidence a semigroup {B_t | t \geq 0} of transformations of the space of distributions, where for t=1 one has the Bercovici-Pata bijection mentioned above. The transformations B_t are defined by a simple formula involving convolution powers with respect to free additive and to Boolean convolution. They turn out to have an interesting multiplicativity property with respect to free multiplicative convolution. In the 1-variable case and for a centered probability measure \mu of variance 1, we observe a direct relation between the process {B_t (\mu) | t \geq 0} and the free Brownian motion. The talk is based on two recent papers done jointly with Serban Belinschi.

Abstract: This will be a short review of orthogonal polynomials and of the underlying classical integrable hierarchies occurring in matrix models. Applications will include the double scaling limit of the one-matrix model.

Abstract: (joint work with Nate Brown and Kenley Jung). We compute the microstates free entropy dimension of natural generating sets in amalgamated free products $M_1 *_B M_2$ of von Neumann algebras, where the algebra $B$ over which we amalgamate is assumed to be hyperfinite. This allows us to compute the free entropy dimension of certain generating sets of free group factors $L(F_s)$, which are exotic from a C*-algebra perspective. (No prizes will be given for guessing the value of this quantity.) Also, the microstates free entropy dimension of generating sets for each in a large class of groups can be computed using our results, and this is equal to the non--microstates free entropy dimension in these cases.

Abstract: We introduce a notion of rank completion for bi-modules over a finite tracial von Neumann algebra. We show that the functor of rank completion is exact and that the category of complete modules is abelian with enough projective objects. This leads to interesting computations in the L^2-homology for tracial algebras, as defined by A. Connes and D. Shlyakhtenko. In particular, we show that all L^2-Betti numbers of a von Neumann algebra coincide with those of weakly dense C*-subalgebras. As an application, we also give a new proof of a Theorem of Gaboriau on invariance of L^2-Betti numbers under orbit equivalence.

Abstract: A highly noncommutative extension of the spectral theory of resolvents is emerging from free probability theory. I will discuss the possibility of using this spectral theory in relativistic quantum physics.

Abstract: I will mainly talk about a recent work with W. Helton and R. Rashidi Far, which addresses the question whether the equation which describes the Cauchy transform of an operator-valued semicircular element has a unique solution with the 'right' positivity property. If time permits I might also make a few remarks and pose some questions around Berry-Esseen type of estimates for the multivariate free central limit theorem.

Abstract: An account of two dimensional U(N) quantum gauge theory will be presented, along with results and questions concerning the limit of this theory as N goes to infinity.

Abstract: We review the basic properties of infinitesimal bialgebras by analogy with the more familiar Lie bialgebras. We discuss the associative Yang-Baxter equation and the Drinfeld double. We discuss relations to other algebraic structures such as non-symmetric operads and pre-Lie algebras.

Abstract: It is known that the free additive convolution with a nontrivial probability measure has a strongly regularizing effect: atoms usually disappear, the density with respect to the Lebesgue measure is analytic wherever positive, and singular continuous part is nonexistent. In our talk we will discuss when the density can be guaranteed to be positive (and hence analytic) everywhere. More recently Florent Benaych-Georges defined the rectangular free convolution, a "deformation" of free additive convolution, defined for symmetric probability measures on the real line. This operation, roughly speaking, models the asymptotic distribution of sums of rectangular random matrices as their dimension tends to infinity and the ratio of their dimension tends to a fixed number between zero and one (in the same sense in which the usual free additive convolution models the addition of large square random matrices). We use the analytic tools developed by Benaych-Georges to describe some instances in which the rectangular free convolution of two probability measures is absolutely continuous with respect to the Lebesgue measure and the density is reasonably tame. We will also discuss the behaviour of such convolutions near the origin and conditions for the existence of atoms at zero. This is joint work with Florent Benaych-Georges and Alice Guionnet.

Abstract: It is well known that the free entropy of Voiculescu has a degenerate convexity property (it is infinite unless the state is a factor state), in sharp contrast with the usual concavity property of entropic quantities usually considered in statistical mechanics. In this talk I will present a slight modification of Voiculescu´s definition of free entropy which yields a concave function of the state, and coincides with Voiculescu´s quantity for factor states. I will also discuss connection with Hiai´s Legendre transform approach and with the microstate-free free entropy.

Abstract: We introduce the Schwinger-Dyson equation associated to a polynomial potential V on the set of tracial states of the free *-algebra generated by X_1=X_1^*, ...,X_n=X_n^* and unitaries U_i. We prove that, for a prescribed spectral measure of the X_i's and for V small enough, the solution of this equation exists, is unique and even analytic in V. As a byproduct, we prove that the matrix integral associated to the potential V converges in the large N limit and that its real and formal limits are the same. This is joint work with A. Guionnet and E. Maurel-Segala.

Abstract: Second order cumulants do for fluctuations what first order cumulants do for moments. An important tool in the analysis of classical cumulants is the formula of Shiryaev and Leonov (1969) which expresses the cumulants of products in terms of cumulants of the factors. In 2000 Krawczyk and Speicher gave the free version of this formula. In this talk we will show how the formula can be extended to second order cumulants. This is joint work with Roland Speicher and Edward Tan.

Abstract: In this talk we review classical information theory (a la Amari and Nagaoka) and then apply it to a random matrix model to obtain a Cramer-Rao theorem and calculate the Legendre transform of pressure. We verify that the quantities converge (to their free counterparts when available) for large N.

Abstract: We will discuss general limit theorems for additive and multiplicative free convolution, as well as (time permitting) a result on decomposability of measures. The results are joint with J. C. Wang.

Abstract: The computation of asymptotic quantities related to the heat kernel measure on the unitary group U(N) as N tends to infinity is one of the basic problems of large N Yang-Mills theory, and a natural problem in the framework of large random matrices. P. Biane (1995) and F. Xu (1997) have independently computed limiting distributions and proved asymptotic freeness results. Recently, A. Sengupta has reformulated Xu's computation of the limiting distribution in a very clear and attractive way. In this talk, I will explain how physical ideas related to "string theories" developed in the context of two-dimensional Yang-Mills theory by D. Gross and W. Taylor (1993) shed some light on the approach of Xu and Sengupta, in particular on its combinatorial aspects. More concretely, I will explain how elementary computations related to the Schur-Weyl duality allow one to relate the Brownian motion on the unitary group and the most natural random walk on the symmetric group. Then I will derive and discuss convergent series expansions for expectations of products of traces of unitary matrices under the heat kernel measure.

Abstract: A survey of some applications of random matrices to physics and of some basic techniques (diagrammatic methods, topological expansion, saddle point evaluation...) used there.

Abstract: A variant of microstate free entropy for self-adjoint random variables is introduced. It depends only on the von Neumann algebras generated by individual random variables, and thus can be regarded as a kind of free analog of mutual information. This is a recent joint work with Hiai and Miyamoto.

Abstract: We will discuss small perturbation of the GUE and how their asymptotics can be related to the enumeration of combinatorial structure embedded on surfaces. The limit to these model is related to the enumeration of planar objects while the g-th correction to this convergence counts graphs on a surface of genus g. We will try to see what can be said beyond the purely formal identification which is often used in physics. Our main tool is the use of Schynger Dyson's equation which highlights the parallel between the recursive decomposition of graphs and the induction relations between moments of the matrix model.

Abstract: I will report on recent progress on free quantum groups (with Bichon and Collins), notably on the construction of a 4-th example. The first 3 examples, due to Wang, correspond in the large N limit to the semicircular, circular and free Poisson laws. The 4-th one, that we construct, is related to a more complicated law, that I will describe in detail.

Abstract: Free Meixner distributions are a class of probability measures on the real line, which includes the semicircular, Marchenko-Pastur, double Wishart, and Bernoulli distributions. I will describe a few properties that these distributions share. Time permitting, I will also construct multi-variate versions of these distributions.

Abstract: Some random matrices have been introduced a few years ago as models for free infinitely divisible laws on the real line. I will discuss analoguous matricial constructions for free max-stable laws, for free multiplicative Lévy laws and for free Poisson point processes. This is joint work with F. Benaych-Georges.

Abstract: I will discuss closable derivations on a finite von Neumann algebra which by the work of Sauvageot give rise to semigroups of completely positive maps. I will then discuss how one can apply the deformation/rigidity techniques of Popa to these semigroups in order to obtain information about the von Neumann algebra, e.g. primeness, solidity. Of particular interest will be derivations into the Hilbert-Schmidt operators which (philosophically at least) are connected with the first L^2-Betti number.