## Probabilistic Operator Algebra Seminar Fall 2005

Schedule for Winter 2006 Monday,December 4, 4:30 - 6:00, Jeff 102 James Mingo, Queen's TITLE: Second Order Freeness and Compound Wishart Matrices In 1986 D. Voiculescu introduced the R-transform of a random variable. The coefficients of this power series, now called free cumulants, were shown in 1994 by R. Speicher to have a combinatorial interpretation using non-crossing partitions. In 2004 A. Nica and I showed that the covariance, when suitably magnified, of the traces of some standard families of random matrices has a limiting value that can be interpreted in terms of non-crossing annular partitions. Indeed the order of elements in a block of a partition is now significant and the relevant concept is a non-crossing permutation. These non-crossing annular permutations give rise to second order cumulants which in turn form the basis of second order freeness. We shall give a number of natural examples of families of matrices that exhibit second order freeness. This work was done in collaboration with B. Collins, A. Nica, P. Sniady, and R. Speicher.Monday,

November 21, 4:30 - 6:00, Jeff 102 Jonathan Novak, Queen's TITLE: The Lagrange Inversion Formula ABSTRACT: Many enumerative problems have a recursive structure that is naturally encoded as a functional equation in the ring of formal power series, via generating series techniques. The Lagrange inversion formula is a tool for solving certain functional equations in C[[x]]. We will give a derivation of the Lagrange formula, and use it to derive Cayley's classic formula for the number of vertex-labelled trees on n vertices, and discuss some other interesting applications.Monday,

November 7, 4:30 - 6:00, Jeff 102 Jamie Mingo, Queen's Title: Two Point Functions and Fluctuations of Random Matrices Abstract: The correlation between eigenvalues of a random matrix is given by the two point function which gives the probability of finding a pair of eigenvalues in a given region. For certain standard random matrices explicit (but complicated) formulas are known. From these we can obtain an expression for the 'global' fluctuations of the eigenvalues. However a simple approximation exists when we assume that the size of the matrix if large and becomes exact in the limit. I will review some earlier results and give a combinatorial interpretation of the limiting value in terms of planar diagrams.Monday,

October 31, 4:30 - 6:00, Jeff 102 Emily Redelmeier, Queen's Title: Stembridge's generalization of the Gessel-Viennot Theorem I will continue the discussion from last week explaining how the theorem can be generalized using Pfaffians.Monday,

October 24, 4:30 - 6:00, Jeff 102 Emily Redelmeier, Queen's Title: The Gessel-Viennot Theorem Part IMonday,

October 17, 4:30 - 6:00, Jeff 102 Maria Grazia Viola, Queen's University Title: Non-outer conjugate actions on a free product factor Abstract: In the '70s A. Connes gave a classification of automorphism on the hypefinite II_1 factor, up to outer conjugacy. Two automorphisms are said to be outer conjugate if they are conjugate up to an inner automorphism. Connes' classification was based on two invariants, the outer period and the obstruction to lifting. He showed that two automorphism are outer conjugate iff they have the same outer period and obstruction to lifting. We will show that a completely different situation appears in the case of free product factors (so, in particular for free group factors). We will prove the existence of two actions of a cyclic group on a free product factor, whose associated automorphisms have the seme outer period and obstruction to lifting, but are not outer conjugate.Monday,

October 3, 4:30 - 6:00, Jeff 102 Jonathan Novak, Queen's University Title: Longest increasing subsequences and unitary integrals. Abstract: We will prove a formula of Diaconis and Rains that enumerates permutations in S_n with longest increasing subsequence of length at most k as an integral of a polynomial of degree 2n over the unitary group U(k) (against Haar measure). This is an exact formula, valid for all n and k. We will also define a special class function on the symmetric group called the Weingarten function, and discuss how the formula of Diaconis and Rains leads to an expression of the longest increasing subsequence problem in terms of the Weingarten function.Monday,

September 26, 4:30 - 6:00, Jeff 102 Ben Turnbull, Queen's University Limiting Distributions of Wishart Random Matrices: A Combinatorial Interpretation Abstract: In this talk, I will introduce a limiting distribution associated with Wishart random matrices as well as orthogonal and companion polynomials or this distribution. I will then present the combinatorial interpretation of certain generating functions associated with one set of polynomials as found by Kusalik, Mingo, and Speicher in 2005. I will then modify and extend this interpretation to the generating functions relating to the second set of polynomials.Wednesday,

September 21, 2:00 - 3:00, Jeff 202 Maria Grazia Viola, Queen's University An Introduction to Planar Algebras The course will be an introduction to planar algebras and their relation to subfactors. Planar algebras were introduced by V. Jones in the early '90s as a new axomatization of the standard invariant which appears for subfactors. Planar algebras can be explained in terms of operads, but we will use a more pictorial approach. This is because one of the more interesting applications of planar algebras is to give pictorial proofs for theorems about subfactors; and this is the aspect on which we will focus in this course. We will also see how to describe in terms of planar algebras some of the most relavant examples of algebras, such as the Temperley-Lieb algebras, the Fuss-Catalan algebras, etc. We also plan to cover in detail the theory of annular planar algebras.Monday,

September 19, 4:30 - 6:00, Jeff 102 Piotr Sniady, University of Wroclav Representations of unitary groups and random matrices (joint work with Benoit Collins) In my talk I will concentrate on the asymptotics of the representations of the unitary groups U(d) when the representation tends to infinity. I am interested both in the limit when the group dimension d is fixed and when it tends to infinity. I will show that the asymptotic behavior of the rows of the corresponding Young diagrams can be described asymptotically by certain random matrices. Schedule for Winter 2005