Probabilistic Operator Algebra Seminar Fall 2004
Schedule for Winter 2005 Monday, November 29 4:00 - 5:30, Jeffery 225 -- NOTE CHANGE Antonia Tulino Princeton and Universita degli Studi di Napoli THE ETA AND SHANNON TRANSFORMS: A BRIDGE BETWEEN RANDOM MATRICES & WIRELESS COMMUNICATIONS Of late, random matrices have attracted great interest in the engineering community because of their applications to the communications and information theory on the fundamental limits of wireless communication noisy vector channels. These channels are characterized by random matrices that admit various statistical descriptions depending on the actual application. For each of these channels, we analyze two performance measures of engineering interest: the average mutual information (highest data rate that can be conveyed reliably per unit bandwidth) and minimum mean-square error (smallest mean-square error that can be incurred estimating the channel input based on its noisy received observations), which are determined by the distribution of the singular values of the channel matrix. In this talk we introduce the eta and the Shannon transforms, which have been motivated by the intuition drawn from various applications in communications, and we give a brief summary of their main properties and applications. These transform are very related to a more classical transform in random matrix theory, the Stieltjes transform, and turn out to be quite helpful at clarifying the exposition as well as the statement of many results. In particular, the eta-transform leads to compact definitions of other transforms used in random matrix theory such as the R and the S-transform. The eta and Shannon transforms are then used to state recent asymptotic distribution theorems. We also present an overview of the application of Voiculescu's free probability theory to wireless communication problems. In addition, recent results on the speed of convergence to the asymptotic limits are visited and used to evaluate the probability density function of the mutual information. Throughout the talk, we apply the various findings to the fundamental limits of wireless communication with focus on several classes of vector channels that arise in wireless communications: Code-division multiple-access (CDMA), with and without fading (both frequency-flat and frequency-selective) and with single and multiple receive antennas. Multi-carrier code-division multiple access (MC-CDMA), which is the time-frequency dual of CDMA Channels with multiple receive and transmit antennas, incorporating features such as antenna correlation, polarization, and line-of-sight components. Monday, November 22 4:00 - 5:30, Jeffery 319 Adam Hulcoop, Queen's University Title: Moment functionals & Orthogonal Polynomials Abstract: We will discuss the moment functional and its corresponding orthogonal polynomial system, with particular interest in the relationship between the zeros of the orthogonal polynomials and the spectrum of the moment functional.Friday, November 19 4:00 - 5:30, Jeffery 319 Ian Goulden, University of Waterloo Title: Words avoiding a reflexive acyclic relation Abstract: We consider the generating series for words in a finite alphabet, in which the pairs of consecutive elements avoid a fixed relation A. This is in general a rational function with alternating signs in the denominator, consistent with the cancellation that is a part of the classical combinatorial methods for such results. However, for certain relations A, the rational function can be transformed into one with negative signs for all terms in the denominator, except for the constant, 1. This latter form implies the existence of a combinatorial bijection with no cancellation. Such bijections are presented for various choices of A, including the case of cycle-free posets. Finally, we point to some connections between this work and recent results in commutative algebra, in which our generating series arise as Hilbert series. This is joint work with Curtis Greene. Monday, November 8 4:00 - 5:30, Jeffery 319 Ana-Maria Savu, Queen's University Title: Large deviations for random matrices. Abstract: The Wigner theorem asserts that the spectral measure of symmetric Gaussian matrices converges almost surely to the semicircular law. Moreover, it has been shown by G. Ben Arous and A. Guionnet that the spectral measure of symmetric Gaussian matrices satisfies a large deviation principle with good rate function expressed in terms of the Voiculescu's free entropy for a single noncommutative random variable. I will present G. Ben Arous and A. Guionnet's result on large deviations. Monday, November 1 4:00 - 5:30, Jeffery 319 Todd Kemp, Cornell University Title: Strong hypercontractivity in the q-Gaussian algebras Abstract: Hypercontractivity is a regularity property for semigroups of operators. It was introduced in the late 1960s, to study the spectra of energy operators for interacting quantum field theories. In the last three decades, it has become an important tool in analysis, probability theory, differential geometry, statistical mechanics, and many other areas. (For example, it plays an important role in Perelman's work on the proof of the Poincaree conjecture.) Recently, hypercontractivity has been generalized in many forms. In 1997, Biane proved a non-commutative hypercontractivity theorem for the q-Gaussian algebras Gamma_q of Bozejko, K"ummerer and Speicher, which extended earlier work of Gross, Meyer, Carlen, and Lieb. Meanwhile, in 1983, Janson introduced strong hypercontractivity, a stronger form for holomorphic function spaces. In this talk, I will introduce q-holomorphic algebras H_q which are q-deformations of the algebra of holomorphic functions in Janson's work. I will discuss their relation to the q-Gaussian algebras, a q-Segal-Bargmann transform L^2(Gamma_q)\to L^2(H_q), and a strong hypercontractivity theorem for H_q. Monday, October 25 4:00 - 5:30, Jeffery 319 Ana Maria Savu , Queen's University Title: Introduction to large deviation theory Abstract: Large deviation theory is the theory developed to calculate probabilities of events where a sum of random variables deviates from its mean by more than the normal amount, i.e. beyond what is described by the central limit theorem. The talk is meant to be at the introductory level. We will cover as time permits: Cramer's Theorem for the empirical average, Sanov's theorem for the empirical measure, large deviation principle, rate function. Monday, October 18, 4:00 - 5:30, Jeffery 319 Sarah Reznikoff, Reed College Title: Hilbert Space Representations of the Temperley-Lieb Planar Algebra Abstract: The Temperley-Lieb algebra TL_n(delta) arises naturally as the algebra of projections propagating the Jones Tower of a subfactor of index delta^2. TL_n(delta) has a pictorial representation as a quotient of the algebra of planar diagrams on n strings inside a rectangle. With planar algebras, Vaughan Jones has extended this diagrammatic approach to represent the standard invariant of II_1 subfactors. The simplest planar algebra is Temperley-Lieb itself, which is contained in any other. Exploiting this fact, it is possible to realize any planar algebra as a Hilbert module of the annular Temperley-Lieb algebra. We will give an overview of the necessary background material before outlining a theorem giving a complete characterization of the irreducible Hilbert Temperley-Lieb modules. Monday, September 27, 4:00 - 5:30, Jeffery 319 Piotr Sniady, Mathematical Institute, Wroclaw, Poland Introduction to Random Unitary Matrices Random matrices have now many applications in mathematics, mathematical physics, and even engineering. One of the most important ensembles (right after the Gaussian one) is the one of random unitary matrices. In this lecture I will explain what random unitary matrices are and show how to calculate moments. No previous knowledge of random matrices is assumed so the lecture should be accessible to all. Monday, September 20, 4:00 - 5:30, Jeffery 319 Introduction to Random Matrices J. A. Mingo, Queen's Random matrices have now many applications in mathematics, mathematical physics, and even engineering. The simplest ensemble to understand is the Gaussian. In this lecture I will explain what the Gaussian ensemble is and show how to calculate moments and eigenvalue distributions. I will not assume previous knowledge of random matrices so the lecture should be accessible to all. Monday, Sept 13, 4:30 am - 5 , Jeff 319 Michael Skeide, Universita Degli Studi del Molise, Campobasso, Italy Abstract: Let $B$ and $C$ be von Neumann algebras acting on Hilbert spaces $G$ and $L$, respectively. Then there is an isomorphism, called the commutant, between the category of (concrete) von Neumann $B$-$C$-modules and the category of (concrete) von Neumann $C'$-$B'$-modules. As a special case, we obtain a category isomorphism between the category of von Neumann $\bf C$-$B$-modules (that is, right von Neumann $B$- modules) and the category of von Neumann $B'$-$\bf C$-modules (that is, unital normal representations of $B'$ on a Hilbert space), and back. A normal $*$-functor from the von Neumann $B$-modules into the von Neuman $C$-modules is, thus, translated into a normal $*$-functor from the normal unital representations of $B'$ into those of $C'$, and back. This translation translates Blecher's Eilenberg-Watts theorem (and any of its proofs) into Rieffel's Eilenberg-Watts theorem (and any of its proofs), and back. Hereby, the von Neumann $B$-$C$-module implementing the first functor by tensoring from the right is mapped to its commutant implementing the second functor by tensoring from the left, and back. Wednesday, Sept 8, 10:30 am - 12 , Jeff 422 Michael Skeide, Campobasso, Italy Representations of $B^a$ and Blecher's Eilenberg-Watts Theorem Abstract: A strict unital representation $\theta$ of the algebra $B^a(E)$ of all adjointable operators on a Hilbert $B$-module $E$ by operators on a Hilbert $C$-module $F$ factors $F$ into the tensor product of $E$ and a Hilbert $B$-$C$-module in such a way that the action of an element of $B^a(E)$ on $F$ is given simply by amplification. We give a short proof of that fact and apply it to present a simple proof of Blecher's Eilenberg-Watts theorem. That is, every strict $*$-functor from the category of Hilbert $B$-modules to the category of Hilbert $C$-modules is naturally equivalent to tensoring with a Hilbert $B$-$C$-module from the right. Schedule for Fall 2003 and Winter 2004