Probabilistic Operator Algebra Seminar
Winter 2008

Organizers: J. Mingo and R. Speicher

Schedule for Fall 2008

Tuesday, May 27, 3:00 - 4:30, Jeff 422

Wend Werner (Munster)

Symmetric Spaces of Infinite Dimension

Abstract: Symmetric spaces are manifolds of the form G/H, G
a Lie group, and H (essentially) the fixed point group of an
involutory automorphism of G. In this representation, G can
usually be chosen as the automorphism group of some
geometrical object such as the isometries of a Riemannian
manifold or the automorphisms of an affine manifold.

The first encounter with a symmetric spaces usually is
provided by the open unit disk of the complex plane, D.
Within the context of a complex variable course, G would be
taken as the group of holomorphic automorphisms of D. It has
long been known that, with respect to these mappings, the
open unit ball of a C*-algebra behaves quite like the one
dimensional example.

In this talk we will discuss to what extend other
automorphism groups can be used in the infinite dimensional
situation. The possibly most non-standard approach seems to
be one in which Hilbert spaces of a Riemannian manifold are
replaced by Hilbert-C* modules. In this approach it is
important to realize how these latter objects fit into the
category of JB*-triples and to furthermore exploit their
connection with the theory of operator spaces.

Wednesday, April 30, 1:30 - 3:00, Jeff 422 Roland Speicher, Queen's University Introduction to stochastic analysis with respect to free Brownian motion Abstract: In this talk, I want to give an introduction to the beginnings of a stochastic integration theory with respect to free Brownian motion. I will explain what a free Brownian motion is, how to define integrals with respect to it and give an idea of two of the most important properties of such integrals: the free Ito formula and the free Burkholder-Gundy estimate. If time permits, I will also address free stochastic differential equations and show what the above allows to conclude on the existence and properties of solutions to such equations. I will try to make my talk independent (and free) of any prior knowledge on stochastic analysis or free probability.

Wednesday, April 23, 1:30 - 3:00, Jeff 422 Eugene Kritchevski, McGill University Poisson statistics of eigenvalues for random discrete Schrodinger operators. ABSTRACT: In this talk, I will consider random discrete Schrodinger operators of the form H = L + c V acting on the Hilbert space l^2(X), where X is a given countable set endowed with a nice homogeneous structure: e.g. the lattice Z^d, a regular tree, or a hierarchical lattice. Here L is a given self-adjoint operator on l^2(X) e.g. the discrete or the hierarchical Laplacian, V is a random potential (Vf)(x) = v(x) f(x) with v(x) i.i.d. random variables, and c>0 is a coupling constant measuring the strength of the disorder. One is interested to understand the statistical behavior of the random eigenvalues of large finite volume approximations to H in the thermodynamic limit. In some cases it is possible to prove that, after a natural rescaling, the random eigenvalues behave as a Poisson point process. After reviewing the known results for the the Anderson model on Z^d and for the regular tree, I will discuss my work on the hierarchical model. I will explain the mechanism responsible for Poisson statistics of eigenvalues and, if time permits, results on generic spectral localization.

Wednesday, April 16, 1:30 - 3:00, Jeff 422 Amna Grgar (Queen's) The Order Structure for Non-crossing Annular Permutations ABSTRACT: Non-crossing annular permutations are needed to analyze the fluctuations of random matrices, in the same way as non-crossing partitions are used to analyze the limiting eigenvalue distribution of random matrices. A very important part of the theory of non-crossing partitions is the order structure. I will present an order structure on the non-crossing annular permutations and more generally on partitioned permutations that extends the order on non-crossing partitions.

Tuesday, April 8, 4:15 - 5:45, Jeff 422 (NOTE TIME CHANGE) Manjunath Krishnapur (Toronto) Title: From Random Matrices to Random Analytic Functions Abstract: Peres and Virag proved that the zeros of the power series a_0 + z a_1 + z^2 a_2 + ... with i.i.d. standard complex Gaussian coefficients, is a determinantal point process on the unit disk, invariant in distribution under isometries of the hyperbolic plane. Extending this result, I show that the singular points of the matrix-valued power series A_0 + z A_1 + z^2 A_2 + ... where A_i are k x k matrices with i.i.d. standard complex Gaussian entries, is also a determinantal process in the disk. This gives a unified framework in which to view the result of Peres and Virag and a well-known theorem of Ginibre on Gaussian random matrices.

Wednesday, April 2, 1:30 - 3:00, Jeff 422 Nikolay Ivanov (Queen's) Free entropy dimension estimates ABSTRACT: I will present the estimates for free entropy dimension from the paper "Lower estimates on microstates free entropy dimension" of D. Shlyakhtenko.

Wednesday, March 12, 1:30 - 3:00, Jeff 422 Matthew Chivers (Queen's) Second Order Free Cumulants of Haar Unitaries ABSTRACT: We begin by summarizing a result obtained by Nica and Speicher regarding the behaviour of the free *-cumulants of Haar unitary elements in a *-probability space. Moving then into the framework of a second order *-probability space, necessary to capture the notion of fluctuations, we present an analogous result involving the second order free *-cumulants of Haar unitaries. Along the way we shall also introduce a formula for second order cumulants of products given by Mingo, Speicher and Tan, with which we will make judicious use of in proving our main result.

Wednesday, February 27, 1:30 - 3:00, Jeff 422 Nikolay Ivanov (Queen's) An Introduction to Free Entropy ABSTRACT: I will give the definition and basic properties of both the microstate and non-microstate version of Voiculescu's free entropy. I will then present the basic examples.

Wednesday, February 13, 1:30 - 3:00, Jeff 422 Roland Speicher, (Queen's) Introduction to the q-deformed commutation relations ABSTRACT: The q-deformed commutation relations provide an interpolation between the canonical anti-commutation relations (CAR) and the canonical commutation relations (CCR). They generate von Neumann algebras (which interpolate between classical and free Brownian motion), which have been in the center of interest during the last 10 years. However, we still do not know whether these vN-algebras are the same for all q or not. Recently, there has been some progress on the structure of these algebras and we plan to have a series of a few talks on this work. In this first talk on the q-relations I will give an introduction on some basic facts about them. Mainly, I will show how to realize these operators on Hilbert spaces and how to prove that the commutant of the vN-algebra is trivial, in the case of infinitely many generators. No knowledge about vN-algebras is required; my talk will, hopefully, also give an introduction to vN-algebras by doing concrete calculations (combinatorial and analytical) for an explicit example.

Wednesday, January 30, 1:30 - 3:00, Jeff 422 Emily Redelmeier (Queen's) A Genus Expansion for the Real Wishart Matrix Ensemble We examine the real Wishart matrix ensemble using methods applied to the complex case. We write a genus expansion for the expectation of products of traces which, unlike the complex case, includes nonorientable surfaces. We present a version of the Kreweras complement which can be applied to a nonorientable surface.

Wednesday, January 23, 4:00 - 5:30, Jeff 422 Jonathan Novak (Queen's) Jack symmetric functions of random matrices ABSTRACT: Let Z be a Ginibre matrix, i.e. a matrix whose entries are independent real or complex standard Gaussian random variables. In this talk we examine a paper of Hanlon, Stembridge, Stanley regarding the expected value of p_k(AZBZ'), where A,B are non-random Hermitian matrices and p_k is a Newton power sum symmetric polynomial. The average value of this expression is a polynomial with terms of the form p_i(A)p_j(B), and for certain choices of A and B has a nice combinatorial interpretation in terms of perfect matchings on graphs. We will begin with discussing the basic aspects of Gelfand pairs and zonal spherical functions, which make this analysis possible.

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