Seminar on Free Probability and Random Matrices
Fall 2009

Organizers: J. Mingo and R. Speicher

Schedule for Winter 2010

Monday, November 30, 4:30 - 6:00, Jeff 319

Matthew Lewis (Queen's)

Distributional Derivatives of log|x - y|

I will speak on an analytic proof for a formula which
appears in a couple of papers on the fluctuations of large
matrices.  This leads into some interesting results
involving the weak limit of derivatives and generalized

Monday, November 23, 4:30 - 6:00, Jeff 319 Emily Redelmeier (Queen's) Real Second-Order Freeness In this talk I will suggest a definition of real second-order freeness and discuss the second-order freeness of real Wishart matrices.

Monday, November 16, 4:30 - 6:00, Jeff 319 Natasa Blitvic (MIT) The combinatorics of the q-circular operator In the q-commutative world, the q-circular operator is the natural deformation of the complex Gaussian and of the latter's free-probabilistic counterpart, given by Voiculescu's circular operator. Though it is well-known that the moments of the q-circular operator are given by the generating function for parity-reversing pairings, the challenge of the underlying combinatorics leaves open a number of questions relating to the properties of the q-circular operator. In this talk, I will discuss my recent work pertaining to the combinatorics of the parity-reversing pairings, their ties to other combinatorial constructs, and their implications on the properties of q-circular operators. In particular, I will present a surprising result that casts doubt on the analyticity of the norm of the q-circular operator as a function of complex q.

Monday, November 9; 4:30 - 6:00, Jeff 319 Michael Skeide (Campobasso) E_0--Semigroups and other things that can be classified by product systems E_0--Semigroups (semigroups of unital endomorphisms) or completely positive semigroups on a quantum system can be classified by product systems of Hilbert modules. We intend to give a partial overview of what is known and what is not known.

Monday, November 2, 4:30 - 6:00, Jeff 319 Jonathan Novak (Queen's) Jucys-Murphy elements and unitary matrix integrals The JM-elements are a family of commuting elements in the group algebra of the symmetric group. Although they are non-central, symmetric functions of JM-elements are central. It turns out that certain unitary matrix integrals, namely the "correlation functions'' of matrix elements, can be expanded perturbatively in the small parameter 1/N and that the coefficients in this series encode conjugacy class multiplicities in complete symmetric functions of JM-elements. A wealth of information follows from this fact, including previously inaccessible sub-leading asymptotics of correlation functions, and there is surprising contact with Vershik-Kerov's "polynomial functions on Young diagrams." This is joint work with Sho Matsumoto from Nagoya University. Monday October 26, 4:30 - 6:00, Jef 319 Michael Brannan (Queen's) The Similarity Problem for Representations of Fourier Algebras. Let G be a locally compact group, let VN(G) denote the von Neumann algebra generated by the left regular representation of G, and let A(G) = VN(G)_* denote its (unique) predual. A(G) is a Banach algebra of continuous functions on G, called the Fourier algebra of G. In this talk we consider the following question: Is very bounded representation of A(G) on a Hilbert space similar to a *-representation? We will show (among other things) that for a large class of groups including SIN groups, maximally almost periodic groups and totally disconnected groups, a representation U of A(G) is similar to a *-representation if and only if U is completely bounded (i.e. U defines a co-representation of the Hopf-von Neumann algebra VN(G).) This is part of a joint work with Ebrahim Samei (University of Saskatchewan).

October 19, 4:30 - 6:00, Jef 319 Octavio Arizmendi Echegaray (Queen's) The S-transform of symmetric probability measures with unbounded supports The Voiculescu S-transform is an analytic tool for computing free multiplicative convolutions of probability measures. It has been studied for probability measures with non-negative support and for probability measures having all moments and zero mean. In this talk I will explain how to extend the S-transform to symmetric probability measures with unbounded support and without moments. As an application, a representation of symmetric free stable measures is derived as a multiplicative convolution of the semicircle measure with a positive free stable measure.

Monday October 5, 4:30 - 6:00, Jef 319 Roland Speicher (Queen's) Easy Quantum Groups I will give a survey on my recent three preprints with Teo Banica and Steve Curran about various aspects of easy quantum groups. I will recall what easy quantum groups are, say where we stand in our program to classify them, and then concentrate on the stochastic aspects of these quantum groups; in particular, the corresponding de Finetti Theorems and the analogues of results of Diaconis and Shahshahani.

Monday, September 28, 4:30 - 6:00, Jeff 319 Jamie Mingo, (Queen's) Free Additive Convolution and the Subordination of Analytic Functions One the central constructions of free probability is the free additive convolution of two probability measures on the real line, in which one obtains a third --- regarded as a 'convolution' of the first two. I will present a method for doing this found by Hans Maassen using the subordination (or factorization) of the corresponding Cauchy transforms of the given probability measures.

Monday, September 21, 4:30 - 6:00, Jeff 319 Ion Nechita, (Ottawa) Random matrix models in quantum information theory We study random matrix models inspired by quantum information theory. Our main tool is a graphical calculus based on a diagrammatic notation for tensors, inspired by ideas of Penrose and Coecke. We introduce Wick and Weingarten calculus in our formalism and we describe a method for computing expectation values of diagrams which contain Gaussian or unitary, Haar-distributed random matrices. This is done by the means of a graph-expansion of diagrams. The graphical computations are intuitive and give insight on the dominating terms via combinatorics on permutations and non-crossing partitions. Finally, applications of these results to additivity conjectures are discussed. This is joint work with Benoît Collins (University of Ottawa).

Previous Schedules Schedule for Winter 2009 Schedule for Fall 2008 Schedule for Winter 2008

Schedule for Winter 2007

Schedule for Fall 2006

Schedule for Winter 2006

Schedule for Fall 2005

Schedule for Winter 2005

Schedule for Fall 2004

Schedule for Winter 2004

Schedule for Fall 2003