Seminar on Free Probability
and Random Matrices
Fall 2012

Organizers: J. Mingo and S. Belinschi

Schedule for Current Term

Wednesday,  November 28, 4:30 - 6:00, Jeff 422

Michael Hartglass (Berkeley)

Rigid C* tensor categories of bimodules over interpolated
free group factors

The notion of a fantastic planar algebra will be presented
and some examples will be given.  I will then show how such
an object can be used to diagrammatically describe a rigid,
countably generated C* tensor category C.  Following in the
steps of Guionnet, Jones, and Shlyakhtenko, I will present a
diagrammatic construction of a II1 factor M and a category of 
bimodules over M which is equivalent to C. Finally, I will 
show that the factor M is an interpolated free group factor 
and can always be made to be isomorphic to L(F).  Therefore 
we will deduce that every rigid, countably generated C*
tensor category is equivalent to a category of bimodules over 
L(F). This is joint work with Arnaud Brothier and David Penneys.

Wednesday, November 21, 4:30 - 6:00, Jeff 422 Jerry Gu (Queen's) Non-crossing partitions and the Kreweras complement This talk will be an introduction to the combinatorics of non-crossing partitions and the Kreweras complement which has some important applications to the theory of free probability. I will start by discussing some basic definitions and properties of the topic that lead to an important theorem by Philippe Biane which states that “the lattice of non-crossing partitions of a cycle can be embedded in the Cayley graph of the permutation group with transpositions as generators.” If time permits, I will talk about some additional theorems and/or propositions related to the topic. This talk is mainly based on Lectures 9 and 18 of the book “Lectures on the Combinatorics of Free Probability” by A. Nica and R. Speicher, and the paper “Some properties of crossings and partitions” by Philippe Biane.

Wednesday, November 14, 4:30 - 6:00, Jeff 422 Mihai Stoiciu (Williams College) Transition in the Microscopic Eigenvalue Distribution for Random CMV Matrices We consider random CMV matrices, the unitary analogues of discrete one-dimensional random Schrodinger operators, and study the microscopic statistical distribution of their eigenvalues. For CMV matrices with slowly decreasing random coefficients, we show that the microscopic eigenvalue distribution is Poisson. For rapidly decreasing coefficients, the eigenvalues have rigid spacing (clock distribution). For a certain critical rate of decay we obtain the circular beta distribution, which interpolates between the Poisson and the clock distributions.

Wednesday, November 6, 5:00 - 6:00, Jeff 422 [Note delayed start to allow for Department Meeting] Mihai Popa (Queen's) On the relation between complex and real second order free independence, Part II

Wednesday, October 31, 4:30 - 6:00, Jeff 422 Mihai Popa (Queen's) On the relation between complex and real second order free independence The notions of complex and real second order freeness were introduced as ways to describe the asymptotic behavior of fluctuations for some important classes of random matrices. More precisely, second order freeness is connected to the asymptotic behavior of Haar unitary and unitarily invariant random matrices, while a similar theory for Haar orthogonal and orthogonally invariant random matrices leads to real second order freeness. We show that real second order freeness also appears in the study of Haar unitary and unitarily invariant random matrices when transposes are also considered. Moreover, it appears that the cases of complex second order freeness from the existing literature are in fact particular cases of real second order freeness.

Wednesday, October 24, 4:30 - 6:00, Jeff 422 Serban Belinschi (Queen's) Analytic view of the semicircle law This will be an expository talk about analytic properties of the semicircle law, from the point of view of free probability. We will look at some results of Voiculescu, Biane, Speicher et. al., and Soltanifar, regarding the role of the semicircular distribution, both scalar and operator valued, as free central limit, as regularizer (through free additive convolution) and as tool for encoding continued fraction expansions.

Wednesday, October 17, 4:30 - 6:00, Jeff 422 Jamie Mingo (Queen's) Associativity of Second Order Freeness A basic property of free independence is that free subalgebras of free subalgebras are themselves free. We thus say that freeness is associative. I will show that the same conclusion holds for second order freeness. This is part of project with Mihai Popa and Emily Redelmeier on second order freeness and the orthogonal group.

Wednesday October 10, 4:30-6:00, Jeff 422 Tobias Mai (University of the Saarlands) Faber polynomials in free probability theory In 1903, G. Faber generalized the series expansion of holomorphic functions from closed discs to more general compact subsets of the complex plane. He replaced the family of shifted monomials by another family of polynomials, depending on the given compact set. Those polynomials became known as Faber polynomials and they play an important role in many areas of mathematics. In this talk, I will give a short introduction to the theory of Faber polynomials and I will explain, how they come into play in free probability theory.

Wednesday, October 3, 2012, 4:30 - 6:00, Jeff 422 Mihai Popa (Queen's) A Fock Space Model for Multiplication of C-Free Random Variables The lecture will present a Fock space model suitable for constructions of c-free algebras. An immediate application is a direct proof for the multiplicative property of the c-free S-transform. The presentation will be based on an existent preprint.

Wednesday, September 26, 2012, 4:30 - 6:00, Jeff 422 Serban Belinschi (Queen's) The subordination phenomenon for operator valued multiplicative free convolutions, part II

Wednesday September 19, 4:30 - 6:00, Jeff 422 Serban Belinschi, (Queen's) The subordination phenomenon for operator valued multiplicative free convolutions For the computation of distributions of various (mostly selfadjoint) polynomials in free variables, operator valued free convolutions are, in principle, providing a universal "practical" recipe. However, for practical computational purposes, the quotation marks around the word "practical" cannot be generally removed. Motivated by a specific problem in random matrix theory, in this joint work with Speicher, Treilhard and Vargas, we show that Voiculescu's subordination holds for very general free multiplicative convolutions of operator-valued distributions, and we find the subordination functions as limits of composition iterations of a certain initial data. These iterations turn out to be reasonably easy to perform with the help of an ordinary computer, and in some significant cases the convergence seems to be quite fast.

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