Seminar on Free Probability and Random Matrices
Fall 2008

Organizers: J. Mingo and R. Speicher

Schedule for Winter 2009

Tuesday, December 9, 4:30 - 6:00, Jeff 422 (Note room change)

Emily Peters, University of California at Berkeley

Planar algebras and the Haagerup subfactor

The strongest invariant of a subfactor is its tower of
relative commutants, which has the structure of a 'planar
algebra' -- a tower of algebras which are acted on by planar
diagrams.  The point of planar algebras is that they let one
think about subfactors using combinatorics and easy
topology.  As an example of the usefulness of this point of
view, we will discuss a new construction of the Haagerup
subfactor (this is the `smallest exotic' subfactor).  No
prior knowledge of planar algebras will be assumed, and the
Temperley-Lieb algebra will make a guest appearance.

Tuesday, November 25, 4:30 - 6:00, Jeff 319 Michael Brannan, Queen's University Operator Spaces and Ideals in Fourier Algebras Over the past decade or so, the theory of operator spaces has played a significant role in furthering our understanding of various function and operator algebras arising in abstract harmonic analysis. In this talk I will discuss how techniques from operator spaces can be used to study complemented ideals in the Fourier algebra, A(G), of a locally compact group G. Our focus in this talk will be on answering the following question: Given a locally compact group G and a closed ideal J in A(G), under what conditions on G and J does there exist another closed ideal K in A(G) such that A(G) decomposes into the Banach space (or operator space) direct sum A(G) = J + K?

Tuesday, November 18, 4:30 - 6:00, Jeff 319 Jonathan Novak, Queen's University Complete Symmetric Polynomials in Jucys-Murphy Elements ABSTRACT: The JM elements J_1,...,J_n in the symmetric group algebra C[S(n)] have many remarkable properties. Okounkov-Vershik proved that they generate the Gelfand-Zetlin algebra of the symmetric group algebra. Earlier, Jucys showed that the center of C[S(n)] is generated by symmetric polynomials in the JM elements. He proved this by showing that e_r(J_1,...,J_n) is the indicator function of permutations with exactly n-r cycles. Here we will consider the "dual" problem of computing the resolution of h_r(J_1,...,J_n) with respect to the conjugacy class basis of the center of C[S(n)]. It turns out that, via Schur-Weyl duality, this is equivalent to a problem in random matrix theory, namely the computation of the full asymptotic expansion of the Weingarten function.

Tuesday, November 11, 4:30 - 6:00, Jeff 319 Claus Koestler (St. Lawrence University) De Finetti theorems in noncommutative probability Distributional symmetries and invariance principles of sequences of random variables lead to deep structural results in probability theory. For example, exchangeability means that the joint distributions of a sequence is invariant under finite permutations of the random variables. Now the classical de Finetti theorem characterizes exchangeable infinite sequences to be conditionally i.i.d. A natural question is to ask for noncommutative versions of this fundamental result. I will report on recent progress in this matter. A very recent result is the free version of de Finetti's theorem, obtained in joint work with Roland Speicher. Replacing the role of permutations by quantum permutations we obtain a new characterization of Voiculescu's freeness with amalgamation. If time permits, I will also discuss how this free version fits into the scheme of a general noncommutative de Finetti's theorem, recently proven by the speaker.

Tuesday, November 4, 4:30 - 6:00, Jeff 319 Nikolay Ivanov (Queen's) We find a necessary and sufficient condition for simplicity and uniqueness of trace for reduced free products of finite families of finite dimensional C*-Algebras with specified traces on them.

Tuesday, October 28, 4:30 - 5:20, Jeff 319 Andrew Toms (York) Dynamics, C*-algebras, and K-theoretic rigidity Abstract: Dynamics provides some of the most interesting and ubiquitous examples in the theory of operator algebras. It is typically difficult, however, to understand their fine structure. A conjecture of Elliott (c. 1990) predicts that the C*-algebras associated to minimal dynamical systems on compact metric spaces (among others) will be classified up to isomorphism by their K-theory and tracial state spaces. In the case of uniquely ergodic systems, K-theory alone should suffice. In this talk I will explain how characteristic class obstructions can be used to prove that the conjecture can only hold for metric spaces of finite covering dimension. Modulo this necessary condition, I will present the solution of Elliott's conjecture in the uniquely ergodic case. (Joint work with Wilhelm Winter.)

Tuesday, October 14, 4:30 - 6:00, Jeff 319 Jamie Mingo (Queen's) Title: The Free Cycle Lemma Abstract: Suppose k, m and n are positive integers and e_1 , . . . , e_{m + n} is a string of '+1's and '-1's with m '+1's and n '-1's. The string is said to be k-dominating if for each 1 <= i <= m+n the number of '+1's in the substring e_1 , . . . , e_i is more than k times the number of '-1's in e_1 , . . . , e_i. The Cycle Lemma of Dvoretsky and Motzkin [1947] asserts that if m - kn >= 0 then there are exactly m - kn cyclic permutations of the string e_1 , . . . , e_{m+n} which are k-dominating. In this talk I shall prove a version of the cycle lemma for free groups, which when the group has only one generator reduces to the result of Dvoretsky and Motzkin. This is joint work with Craig Armstrong, Roland Speicher, and Jenny Wilson.

Tuesday, October 7, 4:30 - 6:00, Jeff 319 Jiun-Chau Wang (Queen's) Title: Analytic subordination and freely indecomposable measures, Part II

Tuesday, September 23, 4:30 - 6:00, Jeff 319 Jiun-Chau Wang (Queen's) Analytic subordination and freely indecomposable measures In this talk we will first discuss a new approach, based on the theory of Denjoy-Wolff fixed points, to subordination results in free probability. This is due to Belinschi and Bercovici. Then we will review several regularity results that were proved using analytic subordination. Finally, I will present a result on decomposability of measures relative to the free convolution. This is joint work with Hari Bercovici.

Tuesday, September 16, 4:30 - 6:00, Jeff 319 Craig Armstrong and Jenny Wilson (Queen's) The Fluctuations of the Sum of Freely Independent Arcsine Operators Let U be a Haar-distributed n x n random unitary matrix. As n tends to infinity, U + U* converges to an arcsine operator. We will discuss techniques for computing the moments and fluctuations of the sum of two or more freely-independent arcsine operators.

Tuesday, September 9, 4:30 - 6:00, Jeff 319 Ricky Tang (Queen's) An example of an unbounded operator on q-Gaussian von Neumann algebras Abstract: We introduced the derivation operator delta_j on a twisted Fock space of L^2(R) which maps from H_1 = C [X_1 , ... , X_N ] to H_2 = C [X_1 , ... , X_N ] x C [X_1 , ... , X_N ]. Where the operators l and l^* on H_1 satisfy the relations (for a fixed 1 <= q <= 1) l(e) l^*(f ) - q l^*(f ) l(e) = < e, f > I . e, f in L^2(R) We show that this unbounded operator is actually closable by showing that its adjoint delta^*_j is densely defned. We then make some estimates on certain operators on H_1 in the one variable case and indicate a range of q for which the power series Xi(X) given by delta^* converges.

Tuesday, September 2, 4:00 - 5:30, Jeff 422 Roland Speicher (Queen's) Quantum groups and liberation of orthogonal matrix groups ABSTRACT: I will recall the notion and basic results about quantum groups, with particular emphasis on Woronowicz's approach to compact quantum groups. The main goal is to liberate (i.e., to find the right kind of quantum version of) classical orthogonal matrix groups. Tanaka-Krein duality says that such (quantum) groups can be identified by its representations, in the form of the tensor category of its intertwiners. In the case of subgroups of orthogonal matrix groups these intertwiners can be described in terms of partitions. Liberation corresponds then to going over to non-crossing partitions. Naturally, there is some relation between such free quantum groups and free probability theory. I will try to explain all this and, in particular, I will present the results of a joint work with Teodor Banica where we classify some special classes of classical and corresponding free quantum groups. 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