```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 I

Monday,  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
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

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

```