Operator Algebra Seminar Fall 2003

Schedule for Winter 2004 Monday, December 2, 2003, 4:30 - 5:30, Jeffery 319 Speaker: Claus Koestler, Queen's University

Monday, November 24, 2003, 4:30 - 5:30, Jeffery 319 Jamie Mingo, Queen's University Orthogonal Polynomials and non-crossing annular diagrams If X_n is a n W n self-adjoint Gaussian random matrix then a theorem of Johansson (1998) shows that the random variables {Tr(T_k(X_n))}_k converge, as n tends to infinity, to independent Gaussian random variables, where {T_k} are the Chebyshev polynomials of the second kind (suitably centred). Cabanal-Duvillard (2001) extended this result to the case of a pair of independent self-adjoint Gaussian random matrices X_n and Y_n, in that he showed that the random variables {Tr(T_j(X_n)), Tr(T_k(Y_n)), Tr(S_{l,m}(X_n, Y_n))}_{j,k,l,m} converge to independent Gaussian random variables, where S_{m,n} is a family of Chebyshev polynomials of the first kind in two non-commuting variables. I shall give a diagrammatic interpretation of the result of Cabanal-Duvillard motivated by a theorem of Andu Nica and me which shows that the correlation of certain ensembles of random matrices is given by non-crossing annular diagrams. Some extensions to the case of Wishart ensembles will be given. This is joint work with Tim Kusalik and Roland Speicher (Queen's).

Monday, November 17, 4:00 - 5:30, Jeffery 319 Speaker: Roland Speicher, Queen's University TITLE: q-Levy processes (after M. Anshelevich) Abstract: I will present some of the main results of a recent paper of Anshelevich on q-Levy processes. Monday, November 10, 4:00 - 5:30, Jeffery 319 Speaker: Matthias Neufang, Carleton University TITLE: Representations of multiplier algebras ABSTRACT: I shall present a common representation theoretical framework for various multiplier algebras arising in Abstract Harmonic Analysis, such as the measure algebra and the completely bounded multipliers of the Fourier algebra on a locally compact group G. The image algebras are completely characterized as certain normal completely bounded bimodule maps on B(L_2(G)). The study of our representations reveals various intriguing properties of the algebras. Furthermore, it provides a simple description of their Kac algebraic duality. I shall finally discuss several extensions of our program, especially to non-normal maps. Part of this work is joint with Zhong-Jin Ruan and Nico Spronk. Monday, November 3, 4:30 - 6:00, Jeffery 319 Speaker: Roland Speicher , Queen's University Title: Some new results about the q-deformed von Neumann algebras Abstract: I will present a recent work of Shlyakhtenko on estimates for non-microstates free entropy, with applications to q-semicircular families. Monday, October 20, 4:30- 6:00, Jeffery 319 Speaker: Michael Roth, Queen's University Title: Random Matrices and Moduli Spaces II Abstract: This is the continuation of the talk from Oct. 6th. We will review the setup from last time, and outline Kontsevich's argument for his basic identity. We will then look at Kontsevich's matrix model and see how essentially the same formula arises, allowing a proof of Witten's conjecture. Finally we will conclude with some other relations between tautological numbers on moduli spaces and numbers arising in the "large N limit" for the standard matrix model. Tuesday, October 14, 4:30 - 6:00 pm, Jeffery 422 Speaker: Benoit Collins, Kyoto University Title: Jacobi matrix ensembles, free probability and universality results Abstract: The Jacobi unitary matrix ensembles have already been widely studied in the physical and mathematical literature. Most "universality" properties that have been obtained for the local scaling of eigenvalues of this ensemble arise from the case of the dimension tending to infinity and the parameters $(a,b)$ being fixed. We present a result stating that for $\pi,\pi'$ two independent random selfadjoint projectors of fixed rank, $\pi\pi'\pi$ is a Jacobi ensemble. Therefore the Jacobi unitary ensemble with suitably varying parameters is a matrix model for the semigroup of free additive convolution for Bernoulli measures. We establish universality results for these models.

Monday, October 6, 4:30 - 6:00, Jeffery 319 Speaker: Michael Roth, Queen's University There are (according to physicists) two models for 2d quantum gravity. One coming from the moduli space of curves, and one coming from random matrix theory. There are certain numbers coming from the moduli space of curves (tautological intersection numbers) which are of interest to geometers. Based on an the generating function of these numbers as the partition function of the theory, and by analogy with the Matrix model, Witten proposed in 1991 that this generating function should satisfy the KdV hierarchy of differential equations. This conjecture was proved by Kontsevich in 1992, partly by comparison with a particular matrix model. The talk is intended as a brief survey of developments linking tautological intersection numbers and numbers coming in the "large N limit" from matrix models. I will (briefly) define the relevant moduli spaces and intersection numbers, explain Witten's conjecture, and sketch Kontsevich's proof. I also hope to have time to explain some more recent developments, including a second proof of Kontsevich's theorem, and further links between the tautological intersection numbers, Hurwitz numbers, and large N limits. These last results are largely due to Okounkov and Pandharipande.

Monday, September 29, 4:30 - 6:00, Jeffery 319 SPEAKER: Andrew Toms, University of Copenhagen Title: Simple nuclear C*-algebras: a growing menagerie Abstract: The recent advent of 'fast dimension growth' AH (approximately homogeneous) C*-algebras has led to ever more unusual examples of simple nuclear C*-algebras. We present several such examples, and discuss their implications for the classification and structure theory of C*-algebras.

Monday, September 22, Jeffery 319, 4:30-6 pm Speaker: Roland Speicher, Queen's Title: Random matrices with classical and non-commutative entries I will review some of the questions and answers about random matrices, in particular, in connection with fluctuations around the semi-circular limit. Most of the talk is on classical random matrices, but I will also contrast some points with the corresponding situation for matrices with non-commutative entries.

Thursday, September 11, 2:30 - 4 pm, Jeffery 115 SPEAKER: Professor Wlodek Bryc, University of Cincinnati TITLE: Spectral measure of large random Hankel, Markov and Toeplitz matrices. Abstract: For Hankel and Toeplitz matrices generated by i.i.d. random variables $\{X_k\}$ of zero mean and unit variance, and for symmetric Markov matrices generated by i.i.d. random variables $\{X_{i,j}\}_{j>i}$ of zero mean and unit variance, scaling the eigenvalues by $\sqrt{n}$ we prove the almost sure, weak convergence of the spectral measures to universal, non-random, symmetric distributions $\gamma_T$ and $\gamma_H$ , $\gamma_M$ of unbounded support. We show that $\gamma_H$ is not unimodal and that $\gamma_T$ is not normal. The moments of $\gamma_T$ and $\gamma_H$ are the sum of volumes of solids related to Eulerian numbers. The distribution $\gamma_M$ has a bounded smooth density given by the free convolution of the semi-circle and normal densities.

Thursday, September 11, 4:30 - 6 pm, Jeffery 115 SPEAKER: Professor Marek Bozejko, University of Wroclaw TITLE: Radial functions on the free groups and von Neumann algebras generated by q-gaussian random variables and the Schauder basis problem Abstract: We consider radial functions on the free group and on the von Neumann algebras generated by free and by q-Gaussian random variables. We show that these algebras are maximal Abelian subalgebras. Next we deal with the Schauder basis problem in L(p) spaces for the von Neumann algebras generated by the free generators and by free and q-Gaussian random variables. We show that the natural basis corresponding to trigonometric system is not a Schauder basis for p > 3 and p< 3/2. Relations with integral singular operators similar to the Hilbert transform will also be presented.

Monday, September 8, 3:00 - 4:30, Jeffery 422 SPEAKER: Claus Koestler, Queen's University TITLE: The type of a white noise ABSTRACT: In non-commutative probability theory emerged an operator algebraic notion of white noises. It captures classical examples of white noises as well as many non-commutative examples. Among the classical examples are Gaussian and Poisson white noise, the more familiar among the non-commutative examples are fermionic, free and bosonic white noises. In my talk I will prove the general result that white noises can't be of infinite type, referring to the classification scheme of von Neumann algebras.