Seminar on Free Probability and Random Matrices Winter 2010

Organizers: J. Mingo and R. Speicher

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Schedule for Fall 2010

Thursday July 15, 2:00 - 4:00 Jeff 110

Rob Wang (Queen's)

The Weingarten function and the number of non-crossing
permutations on a multi-annulus

Calculating integrals over the group of unitary n x n
matrices is a recurrent problem in both pure and applied
mathematics.  A method using the symmetric group was found
over thirty years ago and involves a function now called the
Weingarten function.

I will show how the Weingarten function is related to planar
diagrams and time permitting I will present some results
based on a 1996 paper of P. W. Brouwer and C. W. J
Beenakker.

Tuesday June 29, 2:00 - 4:00, Jeff 319

Carlos Vargas (Queen's)

Describing the Cauchy transform of a sum of rectangular
random matrices using Operator Valued Free Probability
Theory

We want to describe the asymptotic behavior of the Cauchy
Transform of the matrix ensemble consisting of sums of
matrices of the form RXTX*R, where the X 's are rectangular
random matrices of different sizes with i.i.d.
entries. This problem has already been solved using standard
analytical techniques. The objective of the talk is to show
how this result follows naturally using Operator Valued Free
Probability, specifically, we employ a theorem by Nica,
Shlyakhtenko and Speicher, relating operator-valued and
scalar-valued free cumulants.

Monday June 28, 2:00 - 4:00, Jeff 319

Octavio Arizmendi Echegaray (Queen's)

k-Divisible Elements

In his combinatorial approach to Free Probability, Speicher,
introduced the notion of free cumulants. It turns out that
the combinatorics of the free cumulants involve studying the
lattice of non-crossing partitions. This work was motivated
by some results in the combinatorics of free cumulants by
Speicher and Nica. In particular by a formula for the free
cumulants of the square x^2 of an even element x in terms of
the free cumulants of x. The main object of this project is
to generalize this formula in the sense if defining
naturally the notion of a k-divisible element x and derive a
formula the free cumulants the x^{s} in terms of the free
cumulants of x.

In this talk we will give some results involving the
combinatorics of k-divisible non-crossing partitions and
explain the consequences on Free Probability. In particular
on Free Infinite Divisibility.

Thursday June 24, 1:30 - 3:00,  Jeff 422

Michael Brannan, (Queen's)

The property of rapid decay (RD) for discrete quantum
groups, Part 2

We will continue our discussion of the paper of Vergnioux on
the property of rapid decay (RD) for discrete quantum
groups.  Our main goal will be to introduce this notion, and
understand his proof of RD for the class of free orthogonal
quantum groups.

Tuesday June 22, 1:00 - 2:30,  Jeff 225

Michael Brannan, (Queen's)

The property of rapid decay (RD) for discrete quantum groups

We will look at a paper of Vergnioux on the property of
rapid decay (RD) for discrete quantum groups.  Our main goal
will be to introduce this notion, and understand his proof
of RD for the class of free orthogonal quantum groups.

Friday, June 18, 2:00 - 4:00, Jeff 422

Jonathan Novak, Waterloo

Primitive factorizations and matrix integrals.

A classical result of Hurwitz states that a cyclic
permutation of rank r can be factored into r transpositions
in exactly (r+1)^{r-1} ways.  Many different proofs of this
result are known.

A very interesting bijective proof, due to Biane, sets up a
bijection between the set of factorizations and the set of
parking functions.  This bijection is "functorial," in the
sense that it intertwines two natural actions of the
symmetric group S(r), one on factorizations and one on
parking functions.  Canonical representatives of orbits of
these actions are called "primitive" factorizations/parking
functions.

The problem of counting factorizations of a permutation into
more than the minimal number of factorizations is an old
problem related to enumerative geometry; roughly speaking,
counting minimal factorizations of a permutation corresponds
to counting almost simple ramified covers of the sphere by
itself, whereas allowing an excess of transposition factors
corresponds to counting covers of the sphere by compact
surfaces of higher genus.

We will consider the problem of counting long primitive
factorizations of permutations, i.e. we consider the whole
problem modulo Biane's action.  Using character theory for
permutation groups, we give a complete solution for cyclic
permutations; surprisingly, the "central factorial numbers"
of Riordan and Carlitz appear.  In the general case, it
turns out that this problem is equivalent to the computation
of polynomial integrals on the unitary group U(N), for N
sufficiently large.

Wednesday, May 19, 4:30 - 6:00, Jeff 422

Free infinite divisibility for q-deformations
of the classical Gaussian

Free infinite divisibility for q-deformations of the
classical Gaussian Deformations of “classical” analytic
objects have a very long history, and numerous
applications/occurrences in various branches of mathematics.
One such example is the classical normal distribution:
considering its characterization in terms of its orthogonal
polynomials (the well-known Hermite polynomials), certain
“q-deformations” of these objects (obeying a few rules) can
be shown to be orthogonal polynomials for other
distributions of interest in analysis and probability.

This presentation will discuss the free infinite
divisibility of some q-deformations of the classical
Gaussian. The results presented are part of work with
M. Anshelevich, M. Bożejko, F. Lehner and R. Speicher.

Wednesday, March 31, 4:30 - 6:00, Jeff 422

Octavio Arizmendi Echegaray (Queen's)

The free divisibility indicator

In a paper by Serban T. Belinschi and Alexandru Nica, they
consider a family of homomorphisms B_t between probability
measures relating free and boolean convolution. The free
divisibility indicator of a probability measure is defined
as the supremum of t's such that mu is the image of B_t of
some probability measure nu. In the present talk we will
explain how this family of homomorphism is related to free
divisibility, the so called boolean Bercovici-Pata
bijections and free Brownian motion.

In particular, we will focus in the role of the free
divisibility indicator (phi).  We give a bound for phi in
terms of kurtosis, find the fixed points of this family of
homomorphisms and see relations between the case
phi = infinity, the strict stable laws and domains of
attraction. We prove some continuity properties of phi.

Wednesday, March 24, 4:30 - 6:00, Jeff 422

Mike Brannan (Queen's)

Symmetrization of partitions and strong Haagerup
inequalities with operator coefficients.

In this talk we will outline the proof (due to Mikael de la
Salle) of a strong Haagerup inequality with operator
coefficients for the non self-adjoint operator algebra
generated by a *-free family of standard circular variables.
The proof is combinatorial in nature and relies on a process
of symmetrization of certain non-crossing partitions.

Wednesday, March 17, 4:30 - 6:00, Jeff 422

Jamie Mingo (Queen's)

Asymptotic Freeness of Wigner Matrices and Constant Matrices

A Wigner matrix is a self-adjoint random matrix with real
i.i.d. entries. Asymptotic freeness means that as the size
of the matrices tends to infinity they behave more and more
like free random variables.

I will show that the demonstration of asymptotic freeness
can first be reduced to the analysis of certain graphs
arising from set partitions and complete the proof by
analyzing the relevant graphs.

This is joint work with Roland Speicher.

Friday March 12, 4:00 - 5:00, Jeff 319

Claus Köstler (Aberystwyth)

Representations of the infinite symmetric group in
noncommutative probability spaces

The infinite symmetric group is the paradigm for a 'wild'
group and remarkable progress has been made in the past
decades in the study of its representation theory, in
particular by the work of Kerov, Okounkov, Olshanski and
Vershik, among many others.  The first major result on this
subject can be attributed to Thoma (1964) who characterized
the extremal characters of the infinite symmetric group.

In my talk I will introduce a new approach to the
representation theory of the infinite symmetric group which
is based on de Finetti type results in noncommutative
probability.  Quite surprisingly, our operator algebraic
approach reveals that Thoma's theorem can be thought of as a
noncommutative de Finetti theorem. This is joint work with
Rolf Gohm.

Wednesday March 3, 4:30 - 6:00, Jeff 422

Stephen Curran (Berkeley)

A characterization of freeness by invariance

De Finetti's famous theorem characterizes sequences of
random variables whose joint distribution is invariant under
permutations as conditionally i.i.d.  It was later shown by
Ryll-Nardzewski that this in fact holds under the seemingly
weaker assumption that the distribution be invariant under
"spreading", i.e. taking subsequences.  Beginning with the
breakthrough work of Claus Koestler and Roland Speicher,
there have been a number of recent results in free
probability around de Finetti type theorems where the class
of symmetries comes from a quantum group.  In this talk we
will introduce quantum objects which play the role of spaces
of increasing sequences taking values in the set {1,...,n}.
Using these objects, we introduce a notion of "quantum
spreadability" for a sequence of noncommutative random
variables, and establish a free analogue of the
Ryll-Nardzewski theorem.

Wednesday February 17, 4:30 - 6:00, Jeff 422

Octavio Arizmendi Echegaray (Queen's)

On the Non-Classical Infinite Divisibility of Power
Semicircle Distributions

Power semicircle laws appear as the marginals of uniform
distributions on spheres in high-dimensional Euclidean
spaces. A review of some results is presented including a
genesis and the so-called Poincaré's theorem.  The moments
of these distributions are related to the super Catalan
numbers and their Cauchy transforms are derived in terms of
hypergeometric functions.  Some members of this class of
distributions play the role of the Gaussian distribution
with respect to additive convolutions in non-commutative
probability, such as the free, the monotone, the
anti-monotone and the Boolean convolutions. The infinite
divisibility of other members of the class of power
semicircle distributions with respect to these convolutions
is studied.

This is joint work with Victor Perez-Abreu.

Wednesday February 10, 4:30 - 6:00, Jeff 422

Mike Brannan (Queen's)

Title: Representations of free orthogonal quantum groups.

The free orthogonal quantum group O(n)_+ can be described as
the universal noncommutative analogue of the orthogonal Lie
group O(n).  In this talk, we will construct all of the
irreducible representations of O(n)_+ and determine their
fusion rules.  Interestingly, it turns out that for any n>1,
the irreducible representations of O(n)_+ are indexed by the
non-negative integers and satisfy the same fusion rules as
the irreducible representations of the special unitary group
SU(2).

Monday, January 18, 5:30 - 6:30, Jeff 319

Orr Shalit (Waterloo)

Classifying universal operator algebras by subproduct
systems

Given a set of (noncommutative) polynomial relations, one
can consider the universal operator algebra generated by
operators subject to these relations. The existence of such
a universal algebra has had important consequences in
multivariable operator theory.  Natural questions arise
regarding these universal algebras: What sort of algebras
are obtainable this way? What are the symmetries of these
algebras? How are the polynomial relations reflected in the
metric and algebraic structures of these algebras?  In this
talk I will describe how subproduct systems - certain
objects that arose in the study of semigroups of completely
positive maps - give a handle on these universal
algebras. The ideas will be illustrated by solving the
classification problem of the universal algebra generated by
a q-commuting tuple.

Thursday, January 14, 2:30 - 3:30, Jeff 222

Ken Dykema (Texas A & M)

Matrices of unitary moments

Motivated by Kirchberg's simplification of Connes' embedding
problem, we consider matrices of second--order unitary
moments, and discover a few facts about them.  (Joint work
with Kate Juschenko.)

Monday, January 11, 4:30 - 6:00, Jeff 319

Roland Speicher (Queen's)

Quantum Symmetries in Free Probability

In recent years it has become increasingly apparent that
quantum symmetries (i.e., the invariance under the action of
some quantum group) play an important role in free
probability. The starting point of this was my joint work
with Claus Koestler on a free de Finetti Theorem, where we
showed that invariance under the action of quantum
permutations characterizes freeness with amalgamation, for
an infinite sequence of random variables. More general
results in this direction for a recently introduced class of
quantum groups (called "easy") were obtained in joint work
with Teo Banica and Stephen Curran. I will survey some of
these developments.

Previous Schedules

Schedule for Fall 2009

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
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