Seminar on Free Probability and Random Matrices
Winter 2010

Organizers: J. Mingo and R. Speicher

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

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

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 Serban Belinschi, (Saskatoon) 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 under quantum spreading 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.

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