# Seminar on Free Probability and Random Matrices Winter 2014

## Organizers: J. Mingo and S. Belinschi <!-- document.write("(Last Modified: "+document.lastModified+")") //--> Schedule for Current Term

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.

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