## Probabilistic Operator Algebra Seminar Winter 2005

Schedule for Fall 2005 Monday,July 25, 10am, Jeffery 319 Xu Li, Queen's Addition and multiplication formulas for boolean independent variables in Banach Algebra Situations Abstract: Boolean independence was introduced by Speicher and Woroudi. Recently Franz and Bercovici proved these two formulas in the scalar version by different methods. In this talk, the operator-valued Banach Algebra version of these formulas will be presented. We will also shortly address the situation for free independence, where the multiplicative formula in the operator valued case was first considered by Aagard in the case of a commutative Banach algebra, and then in full generality by Dykema. Monday,July 25, 1pm Jeffery 319 Jonathan Novak, Queen's Introduction to the Longest Increasing Subsequence Problem Abstract: A classical problem in probability theory is to find the distribution of the longest increasing subsequence of a uniformly random partition from the symmetric group S_n. Numerical computations of Odlyzko and work by Baik-Deift-Johansson showed that the distribution of the longest increasing subsequence of a random permutation is asymptotically the Tracy-Widom distribution, which corresponds to the distribution of the largest eigenvalue of a random GUE matrix. Further work by Baik-Deift-Johansson and Okounkov shows that in fact the joint distribution of the first k rows of a Plancherel-random Young diagram agrees asymptotically with the joint distribution of the k largest eigenvalues of a GUE matrix. We will give an introduction to this large topic, beginning with the early work of Erdos-Szekeres in Ramsey theory and the RSK correspondence between permutations and pairs of standard Young Tableaux on the same shape, which will allow us to define the Plancherel measure on partitions as the push forward of the uniform measure on permutations. A combinatorial highlight will be the Duality Theorem for finite posets as it applies to the RSK correspondence. Tuesday,June 14, 4:00 - 5:30, Jeffery 319 Hanne Schultz UCLA and University of Odense Title: A random matrix approach to the lack of projections in C_red*(F_2) Abstract: In 1982 it was shown by Pimsner and Voiculescu that C_red*(F_k) has no projections but the trivial ones. Their proof was based on K-theory. We give a new proof using random matrices. More precisely, refining some of the techniques used by Uffe Haagerup and Steen Thorbjoernsen to prove that Ext(C_red*(F_k)) is not a group for k>1, we show that for a projection p in M_m(A) where A = C*(1,x_1, ..., x_k) is the unital C*-algebra generated by a semicircular system x_1, ..., x_k, the unnormalized trace of p belongs to {0,1,2, ..., m} (this was also shown by Pimsner and Voiculescu). As an elementary consequence of this, for a non-commutative polynomial q in k variables with coefficients in M_m(C), the spectrum of q(x_1, ..., x_k) splits into at most m connected components. We use independent random matrices X_1^(n), ...y, X_k^(n) from the Gaussian unitary ensemble to predict the behaviour of x_1, ..., x_k. With q as before, for the sequence of random matrices (q(X_1^(n), ..., X_k^(n))), we find that with probability 1 these have no eigenvalues outside (small neighbourhoods) of the above mentioned connected components of sp(q(x_1, ..., x_k)) as n tends to infinity. Monday,May 2, 4:30 - 6 pm, Jeffery 319 Cristian Ivanescu, University of Ottawa Title: On the classification of simple C*-algebras not necessarily of real rank zero. Abstract. An introduction to the Elliott classification program for C*-algebras will be given. Also I will review basic notions like C*-algebras, real rank for C*-algebras and some applications. Along these lines I will describe a classification result for simple C*-algebras not necessarily of real rank zero, namely inductive limits of continuous trace C*-algebras with spectrum [0,1]. Monday,April 4, 4:30 - 6 pm, Jeffery 319 Ana Maria Savu, Queen's Computation of the generating functions for moments of a process of Wishart matrices. Monday,March 21, 4:30 - 6 pm, Jeffery 319 Jonathan Novak, Queen's Title: Enumeration of Planar Constellations, part II Abstract: In a recent paper, Bousquet-Melou and Schaeffer discuss the problem of counting the number of transitive ordered factorizations of specified length of an arbitrary element of the symmetric group of order n. The authors determine this number explicitly via a bijection with a class of rooted planar maps. Their results generalize a well known theorem of Hurwitz which gives the number of factorizations into transpositions. In this talk we will present the result and outline the combinatorial techniques used by the authors, in particular, the bijection between planar constellations and trees. Monday,March 14, 4:30 - 6 pm, Jeffery 319 Roland Speicher, Queen's Title: Combinatorial description of the S-transform Abstract: I will give a survey talk about my old results with A. Nica on the combinatorial understanding of the S-transform in the scalar-valued case. If time permits I might also address the problem of the operator-valued situation. Monday,March 7, 4:30 - 6 pm, Jeffery 319 Xu Li, Queen's Title: On the S-transform over a Banach algebra (after K. Dykema), part II Abstract: The talk will be on a recent paper of Ken Dykema. The S-transform is shown to satisfy a specific twisted multiplicativity property for free random variables in a B-valued noncommutative probability space. Tuesday,March 1, 3:30 - 5 pm, Jeffery 115 Benoit Collins, Kyoto Title: New scaling for IZ integrals Abstract: This talk is based on joint work with Piotr Sniady. It has been proved recently by Guionnet and Zeitouni under fairly weak hypotheses that the real Itzykson-Zuber integral converges in the large limit dimension, and that the limit only depends on the asymptotic distribution of the matrices. We prove that this limit, correctly rescaled, tends towards Voiculescu's R-transform. Thus we generalize recent results of Guionnet-Maida (JFA 2004), as well as prior works of the authors. Monday,February 28, 4:30 - 6 pm, Jeffery 319 Xu Li, Queen's Title: On the S-transform over a Banach algebra (after K. Dykema) Abstract: The talk will be on a recent paper of Ken Dykema. The S-transform is shown to satisfy a specific twisted multiplicativity property for free random variables in a B-valued noncommutative probability space. Monday,February 144:30 - 6:00 pm, Jeffery 319 Jonathan Novak, Queen's Title: Enumeration of Planar Constellations Abstract: In a recent paper, Bousquet-Melou and Schaeffer discuss the problem of counting the number of transitive ordered factorizations of specified length of an arbitrary element of the symmetric group of order n. The authors determine this number explicitly via a bijection with a class of rooted planar maps. Their results generalize a well known theorem of Hurwitz which gives the number of factorizations into transpositions. In this talk we will present the result and outline the combinatorial techniques used by the authors. Monday,January 314:30 - 6:00 pm, Jeffery 319 Roland Speicher, Queen's Second-order freeness and fluctuations of random matrices Abstract: In ongoing work, Jamie Mingo and I are trying to understand the structure behind fluctuations of random matrices from a free probability perspective. We have isolated a new concept of "second order freeness" lying behind these fluctuations and are developping a beautiful combinatorial theory around this notion. This talk is intended as an appetizer for the theory (and for maybe more technical talks to come); I hope to explain what the question is, where we see the answer and (maybe) why this is so beautiful. Monday,January 244:30 - 6:00 pm, Jeffery 319 Ana Maria Savu, Queen's Title: The rate function for large deviations for Hermitian Brownian motion Abstract: I will continue with the presentation of the result of Biane, Capitaine and Guionnet about the large deviations of Hermitian Brownian motion. In particular I will discuss the derivation of the variational formula for the rate function. Monday,January 10, 4:30 - 6:00 pm, Jeffery 319 Ana Maria Savu, Queen's Title: Large deviation principle for the convergence of Hermitian Brownian motion. Abstract: Hermitian Brownian motion is a matrix-valued stochastic process. The entries of this random matrix are independent (modulo the symmetry constraint) classical Brownian motions. Under a certain scaling, the Hermitian Brownian motion converges in the appropriate sense towards a one-parameter family of non-commutative random variables (operators) known as the free Brownian motion. I will go over a result of Biane, Capitaine and Guionnet that states the large deviation principle for the convergence mentioned above. Schedule for Fall 2004