Probabilistic Operator Algebra Seminar, Fall 2006

Schedule for Winter 2007 Tuesday, November 28, 3:30 - 5:00, Jeff 422 Jamie Mingo, Queen's Free Cumulants and the R-transform Abstract: Voiculescu's R-transform does for free independence what the the logarithm of the Fourier transform does for ordinary independence of random variables. The R-transform is a compositional inverse of the Cauchy or Stieltjes transform (up to a small adjustment). I will present both the analytical and combinatorial approach to the R-transform.

Tuesday, November 21, 3:30 - 5:00, Jeff 422 Kyle Lepage, (Queen's) Title: Time Series Analysis and Random Matrices ABSTRACT: In this talk I will present methods of detecting a polarized signal in vector valued time-series, and discuss potential applications of random matrix theory to spectrum estimation. In particular, singular spectrum analysis (SSA) and a modification of the multitaper method of spectrum estimation (MTM) will be discussed. An attempt will be made to cover relevant background material.

Tuesday, November 14, 3:30 - 5:00, Jeff 422 Emily Redelmeier, (Queen's) ABSTRACT: The noncrossing partitions on n elements, whose combinatorics reflect those of certain random matrix problems, naturally correspond to noncrossing pairings on 2n elements. The Temperley-Lieb algebra is a vector space over these noncrossing pairings, with some further structure, including an associative multiplication, which reflects the graph structure of these noncrossing pairings. In this talk, we consider the representations of the Temperley-Lieb algebra, and we adapt the construction of an orthogonal basis for the algebra to give us an orthogonal basis for the representations. This orthogonal basis allows us to construct a *-homomorphism between the Temperley-Lieb algebra and a direct sum of matrix algebras under certain conditions, demonstrating that the Templery-Lieb algebra is semisimple.

Tuesday, October 31, 2:30 - 3:30, Jeff 422 Note Early Start Edward Tan (Queen's) Second Order Cumulants of the Semicircle-Squared ABSTRACT: Free cumulants (of first order) can be very helpful in computing the moments of sums of free random variables. However, when dealing with fluctuations, we must bring second order cumulants into the picture. In this talk, we will briefly review free cumulants, and discuss an approach that was used to compute the second order cumulants of the semi-circular distribution. We will see that these second order cumulants correspond to certain annular noncrossing partitions.

Tuesday, October 24, 3:30 - 5:00, Jeffery 422 Jonathan Novak (Queen's) Random Unitary Matrices and Symmetric Functions II: Enumeration of Magic Squares ABSTRACT: An order k magic square to type j is a k by k matrix with nonnegative integer entries all of whose rows and columns sum to j. Very little is known about the number H_k(j) of order k magic squares of type j. However, recent work of Diaconis and collaborators shows that H_k(j) can be expressed EXACTLY as an integral over the unitary group. We will prove this theorem and give applications. As usual, the underlying principle is rooted in the fact that the irreducible characters of the unitary group are Schur functions.

Tuesday, October 17, 3:30 - 5:00, Jeffery 422 Maria Grazia Viola, (Queen's) Moments and Cumulants Abstract: One of the most important concepts in the statistical study of a random variable is that of its distribution. However, computing the moment of a random variable is not an easy task. In this talk we will introduce the notion of classical cumulants (or semi-invariants) and show how they can be used to compute the moments of a random variable. Apart from the more common formulation of classical cumulants in terms of the Fourier transform, we will also provide a combinatorial description of them.

Tuesday, October 10, 2:30 - 4:00, Jeffery 422 Note early start Ben Turnbull, (Queen's) Exact Expressions for Fluctuations of Gaussian and Wishart Random Matrices Abstract: When considering finite Gaussian and Wishart Random Matrices, we can formulate expressions involving orthogonal polynomials for the expectations. This idea can be extended to the case of the fluctuations. I will outline how these formulae are obtained and discuss the limits of the fluctuations as the size of the matrices approach infinity. If time permits, I will also briefly outline a very different approach that leads to the same result.

Tuesday, October 3, 3:30 - 5:00, Jeffery 422 Jonathan Novak, (Queen's) Title: Random Unitary Matrices and Symmetric Functions ABSTRACT: We consider unitary matrix valued random variables which are Haar distributed. Using tools from the theory of symmetric functions and representation theory of the symmetric group, we find an explicit formula for the moments of the trace of a random unitary matrix, and for the moments of the trace of a truncation of a random unitary matrix. As a simple corollary, we recover a well known theorem of Eric Rains which shows that the moments of the trace enumerate special pairs of standard Young tableaux.

Tuesday, September 26, 3:30 - 5:00, Jeffery 422 Jamie Mingo, (Queen's) Title: An introduction to Gaussian random matrices, Part 2 I will continue our study of Gaussian and Wishart random matrices and in particular the calculation of the moments of the trace of a power. This is achieved via an expression known in the quantum gravity literature as a genus expansion as it gives the link between maps on surfaces and moduli spaces (c.f. Harer and Zagier (1986) and Okounkov (random matrices and random permutations (2000)) on one hand and random matrices, free probability and their application to wireless communication, statistics, and signal processing on the other hand.

Tuesday, September 12, 4:00 - 5:30, Jeffery 422 Jamie Mingo, (Queen's) Title: An introduction to Gaussian random matrices This term the seminar will consist of a sequence of expository talks and reports on recent work. The expository talks will be under the rubric free probability and random matrices. I will open the seminar with a lecture on Gaussian random matrices. I will give the salient facts about Gaussian random variables, in particular Wick's formula, that are used in the study of Gaussian random matrices. These moments and cumulants are simple but quite important as a model for what happens in free probability.

Schedule for Winter 2006

Schedule for Fall 2005

Schedule for Winter 2005

Schedule for Fall 2004

Schedule for Winter 2004

Schedule for Fall 2003