and Random Matrices

Winter 2014

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Schedule for Current Term
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Tuesday, **April 1**, 4:30 - 6:00, Jeff 319

Josué Vázquez-Becerra (Queen's)

Asymptotically liberating sequences of random unitary
matrices, Part II

In these lecture series, we will discuss a recent paper by
Greg W. Anderson and Brendan Farrell which exhibits some
systems of random unitary matrices that, when used for
conjugation, lead to freeness.

Thursday, **March 20**, 4:30 - 6:00, Jeff 319

Ruiming Zhang (Northwest A & F
University, Yangling)

On Complete Asymptotic Expansions of Certain $q$-Special Functions

According to Paul Turán special functions are defined as
useful mathematical functions. Clearly, the mathematical
functions $e^{az}$ and $\Gamma(z)$ are fundamental special
functions. In mathematics a function $f(q,z)$ is an
$q$-analogue of $g(z)$ means that $g(z)$ could be obtained
as $q\to1$ limit of $f(q,z)$ under suitable scalings. A
special function may have many interesting $q$-analogues,
for example, $q$-special functions $e_{q}(z),$ $E_{q}(z)$
$A_{q}(z)$ and $\mathcal{E}_{q}(\cos\alpha,\cos\beta;t)$ are
all $q$-analogues of $e^{az}$. $q$-analogues of classical
special functions have many important applications in
mathematics and physics. The Euler's $q$-exponential
function
\[
E_{q}(-z) = (z;q)_{\infty} = \prod_{k=0}^{\infty}
\left(1-zq^{k} \right), \quad q \in (0,1), z \in \mathbb{C}
\]
is an $q$-analogue of $e^{-z}$ and Jackson's $q$-gamma
function
\[
\Gamma_{q}(z) = \frac{(q;q)_{\infty}} {(q^{z};q)_{\infty}}
(1-q)^{1-z} \quad q \in (0,1), z \in \mathbb{C}
\]
is an $q$-analogue of $\Gamma(z)$. These two $q$-functions
are the cornerstones of the theory of basic hypergeometric
series (a.k.a $q$-series).
In this talk we present a derivation for the complete
asymptotic expansions of $(z;q)_{\infty}$ and
$\Gamma_{q}(z)$ via an Mellin transform. These asymptotic
formulas are valid throughout the entire complex plane,
uniformly on compact subsets. The formula for
$(z;q)_{\infty}$ suggests that the modular properties of
Dedekind eta function is a consequence of vanishing of odd
Bernoulli numbers.

Tuesday, **March 18**, 4:30 - 6:00, Jeff 319

Josue Vázquez Becerra (Queen's)

Asymptotically liberating sequences of random unitary
matrices Part I: Functions of the χχ-class.

In these lecture series, we will discuss a recent paper by
Greg W. Anderson and Brendan Farrell which exhibits some
systems of random unitary matrices that, when used for
conjugation, lead to freeness. Specifically, in this lecture
we will introduce the functions of the χχ-class and present
some of their properties related to sequences of complex
matrices.

Tuesday, **March 11**, 4:30 - 6:00, Jeff 319

Hao-Wei Huang (Queen's)

Regularization properties of free convolutions with
$\boxplus$-infinitely divisible measures II

We will continue discussing the regularity properties of the
free additive convolution. Specifically, a probability
measure $\mu$ on $\mathbb{R}$ is said to have the property
(H) if the density of $\mu\boxplus\nu$ is positive and
analytic everywhere on $\mathbb{R}$ for any probability
measure $\nu$ on $\mathbb{R}$. We will provide necessary and
sufficient conditions on $\mu$ so that $\mu$ has the
property (H) when it is $\boxplus$-infinitely divisible. We
will also give some examples which have the property (H).

Thursday, **March 6**, 4:30 - 6:00, Jeff 319

Benoît Collins (Ottawa)

Numerical range for random matrices

We review the notion of numerical range, and show that it
behaves in an almost deterministic way for very general
examples of random matrix models. By passing, we obtain norm
estimates for DT-random matrix models introduced by Dykema
and Haagerup. Joint work with Sasha Litvak, Karol Zyckowski,
Piotr Gawron.

Tuesday, **February 25**, 4:30 - 6:00, Jeff 319

Hao-Wei Huang (Queen's)

Regularization properties of free convolutions with
$\boxplus$-infinitely divisible measures

Let $\mu$ and $\nu$ be probability measures on
$\mathbb{R}$. We will discuss some regularization properties
of free additive convolution $\mu\boxplus\nu$, where $\mu$
is $\boxplus$-infinitely divisible. More precisely, we will
provide necessary and sufficient conditions on $\mu$ so that
for any $\nu$ the density of $\mu\boxplus\nu$ is positive
and analytic everywhere on $\mathbb{R}$. We will also give
necessary and sufficient conditions so that the density is
analytic at points at which the density vanishes. This is a
joint work with Jiun-Chau Wang.

Tuesday, **February 11**, 4:30 - 6:00, Jeff 319

Jerry Gu (Queen's)

Bi-freeness and left-right cumulant functionals in free
probability, III

We will first finish the proof from last week, then we will briefly
discuss a new concept in free probability, called bi-freeness, which was
introduced by D. Voiculescu last year. On a free product of Hilbert
spaces with specified unit vector, there are two actions of the
operators of the initial spaces, corresponding to a left and to a right
tensorial factorization, respectively. The notion of bi-free
independence (or bi-freeness) arises when algebras of left-operators and
algebras of right-operators on the free product space are considered at
the same time.

Tuesday, **February 4**, 4:30 - 6:00, Jeff 319

Jerry Gu (Queen's)

Joint moments of left-right canonical operators on full Fock
space, II

We will continue from last week and give an alternative
description for the ``special'' set of partitions that was
introduced last time. The definition arises from the concept
of a double-ended queue used in theoretical computer science
which, in a certain way, describes the joint action of the
left-right canonical operators on full Fock space. If time
permits, we will prove the main theorem stated last week.

Tuesday, **January 28**, 4:30 - 6:00, Jeff 319

Jerry Gu (Queen's)

Joint moments of left-right canonical operators on full Fock
space

Let $\mathcal{T}$ be the full Fock space over $\mathbb{C}^d$ and consider a $(2d)$-tuple $A_1, \dots, A_d, B_1, \dots, B_d$ of canonical operators on $\mathcal{T}$, where $A_1, \dots, A_d$ act on the left and $B_1, \dots, B_d$ act on the right. The joint moments of the $(2d)$-tuple can be computed using the family of $(l, r)$-cumulant functionals, which enlarges the family of free cumulant functionals. Moreover, let $f$ and $g$ be the joint $R$-transforms of $(A_1, \dots, A_d)$ and $(B_1, \dots, B_d)$ with respect to the vacuum-state defined on $\mathcal{B}(\mathcal{T})$, then it turns out that every joint moment of the combined $(2d)$-tuple can be written in a canonical way as a sum of products of coefficients of $f$ and $g$ combined. This talk is based on a recent paper by M. Mastnak and A. Nica.

Tuesday, **January 21**, 4:30 - 6:00, Jeff 319

Mario Diaz (Queen's)

Noncommutative functions, the Taylor-Taylor formula and
applications, II Some Analytical Aspects

This series of lectures, which are divided in three parts,
are based on the paper of Verbovetskyi and Vinnikov
"Foundations of Noncommutative Function Theory".

Tuesday, **January 14**, 4:30 - 6:00, Jeff 319

Mario Diaz (Queen's)

Noncommutative functions, the Taylor-Taylor formula and
applications

This series of lectures, which are divided in three parts,
are based on the paper of Verbovetskyi and Vinnikov
"Foundations of Noncommutative Function Theory".
Part I: Taylor-Taylor (TT) Formula
Part II: Some Analytical Aspects
Part III: Application
Part I. In this part we will introduce the noncommutative
functions and their difference-differential
operator. Motivated by the latter, we will define the so
called higher order noncommutative functions. Finally, we
will derive the TT formula, the analogue of the classical
Taylor formula in the case of noncommutative functions.

Previous Schedules