Probabilistic Operator Algebra Seminar
Fall 2004

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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

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