and Random Matrices

Fall 2016

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Schedule for Current Term
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Wednesday, **December 21**, 2:30 - 4:00, Jeff 222

Claus Koestler (University College Cork)

Representations of the Thompson group F in free probability

After giving an elementary approach to the
construction of unitary representations of the
Thompson group F, I show how the underlying ideas
transfer to the setting of free probability.

Roughly speaking, this transfer yields a Markov type perturbation of shifts on infinite free products as they arise from quantum spreadability. My talk is in parts based on joint work with Gwion Evans, Rajarama Bhat, Rolf Gohm, and Stephen Wills.

Thursday, **December 15**, 2:30 - 4:00, Jeff 222

Ian Charlesworth (UCLA)

An alternating moment condition and liberation for bi-freeness

Bi-free probability is a generalization of free
probability to study pairs of left and right faces in
a non-commutative probability space. In this talk, I
will demonstrate a characterization of bi-free
independence inspired by the “vanishing of
alternating centred moments” condition from free
probability. I will also show how these ideas can be
used to introduce a bi-free unitary Brownian motion
and a liberation process which asymptotically creates
bi-free independence.

Thursday, **December 1**, 2:30 - 3:30, Jeff 222

Mario Diaz (Queen's)

On the Fluctuations of Polynomials in Gaussian Matrices

About a decade ago, Mingo, Nica, and Speicher studied
the fluctuations of the moments of Gaussian matrices
from a combinatorial perspective. Based on their work,
in this talk we will study the fluctuations of the
moments of block Gaussian matrices. In particular, we
will find a semi-explicit formula for a matricial
version of the second-order Cauchy transform. Using
the linearization technique, this formula will provide
the second-order Cauchy transform of polynomials in
Gaussian matrices. This is joint work with Serban
Belinschi and James Mingo.

Monday, **November 28**, 10:30 - 11:30, Jeff 222

Pei-Lun Tseng (Queen's)

The Combinatorics of Subordination, III

Thursday, **November 23**, 2:30 - 3:30, Jeff 222

Pei-Lun Tseng (Queen's)

The Combinatorics of Subordination, III

Monday, **November 21**, 10:30 - 11:30, Jeff 222

Pei-Lun Tseng (Queen's)

The Combinatorics of Subordination, II

Thursday, **November 17**, 2:30 - 3:30, Jeff 222

Pei-Lun Tseng (Queen's)

The Combinatorics of Subordination

Given two analytic functions $f$ and $g$, we say that
$f$ is subordinate to $g$, if we can find $h$ such
that $f = g \circ h$ for some analytic function $h$
(defined on a suitable domain).
We investigate this when $f$ and $g$ are the Cauchy
transforms of probability measures on $\mathbb{R}$;
and give a combinatorial interpretation of this using
the non-commutative derivative.

Thursday, **November 10**, 2:30 - 3:30, Jeff 222

Jamie Mingo (Queen's)

Freeness and the vanishing of mixed cumulants, III

Monday, **November 7**, 10:30 - 11:30, Jeff 222

Jamie Mingo (Queen's)

Freeness and the vanishing of mixed cumulants, II

Thursday, **November 3**, 2:30 - 3:30, Jeff 222

Jamie Mingo (Queen's)

Freeness and the vanishing of mixed cumulants

Freeness was defined in terms the centredness of
certain moments. There is also a formulation in terms
the vanishing of mixed cumulants which is frequently
much easier to work with. In many extensions of
freeness this is the only formulation. I will show the
equivalence of freeness and the vanishing of mixed
free cumulants.

Monday, **October 31**, 10:30 - 11:30, Jeff 222

Josué Daniel Vázquez Becerra (Queen's)

Free Cumulants and their generating functions

We show that in the free case the relation between the
ordinary generating functions of moments and free
cumulants is given by the Cauchy transform.

Thursday, **October 27**, 2:30 - 3:30, Jeff 222

Josué Daniel Vázquez Becerra (Queen's)

Classical Cumulants and their generating functions

We show that the exponential generating functions of
the classical cumulants and moments are related by the
logarithm.

Monday, **October 24**, 10:30 - 11:30, Jeff 222

Josué Daniel Vázquez Becerra (Queen's)

The combinatorics of Several Gaussian random matrices, II

We use our understanding of the combinatorics of a
single Gaussian random matrix to study the interaction
of several Gaussian random matrices.

Thursday, **October 20**, 2:30 - 3:30, Jeff 222

Josué Daniel Vázquez Becerra (Queen's)

The combinatorics of Several Gaussian random matrices

We use our understanding of the combinatorics of a
single Gaussian random matrix to study the interaction
of several Gaussian random matrices.

Monday, **October 17**, 10:30 - 11:30, Jeff 222

Pei-Lun Tseng (Queen's)

The combinatorics of Gaussian random matrices, III

Our understanding of the combinatorics of Gaussian
random variables will be applied to Gaussian random
matrices.

Thursday, **October 13**, 2:30 - 3:30, Jeff 222

Pei-Lun Tseng (Queen's)

The combinatorics of Gaussian random matrices, II

Our understanding of the combinatorics of Gaussian
random variables will be applied to Gaussian random
matrices.

Thursday, **October 6**, 2:30 - 3:30, Jeff 222

Pei-Lun Tseng (Queen's)

The combinatorics of Gaussian random matrices

Our understanding of the combinatorics of Gaussian
random variables will be applied to Gaussian random
matrices.

Monday, **October 3**, 2:30 - 3:30, Jeff 202

Jamie Mingo (Queen's)

The combinatorics of Gaussian random variables II

Thursday, **September 29**, 2:30 - 3:30, Jeff 222

Jamie Mingo (Queen's)

The combinatorics of Gaussian random variables

A new phenomenon in random matrix theory was found by
Voiculescu twenty five years ago, now called the
theory of free probability. I will give the first few
lectures of a learning seminar starting with the
combinatorics of Gaussian random variables. The
seminar will follow the first chapter of a new book by
Roland Speicher and me on random matrix theory and
free probability.

Thursday, **September 22**, 2:30 - 3:30, Jeff 222

Yinzheng Gu (Queen's)

Operator-valued bi-free probability

Free probability is a non-commutative probability
theory where the classical notion of independence is
replaced by free independence and operator-valued free
probability is a generalization of free probability
where the field of complex numbers is replaced by a
unital algebra. On the other hand, c-free probability
is an extension of free probability where the notion
of c-free independence is given with respect to two
states instead of one, and bi-free probability is an
extension of free probability where systems of left
and right random variables are considered
simultaneously. In this talk, we will review the
operator-valued generalization of bi-free probability
and discuss a possible extension where the
corresponding notion of independence is given with
respect to two conditional expectations, hence
generalizing everything mentioned above. This is joint
work with P. Skoufranis.

Thursday, **September 15**, 1:00 - 2:30, Jeff 222

Jamie Mingo (Queen's)

Asymptotic Fluctuations of Wigner and Constant Matrices

A Wigner random matrix is a self-adjoint random matrix
with i.i.d. entries. It is known that independent
Wigner matrices are asymptotically free and that
Wigner matrices and constant matrices are
asymptotically free.

In this talk I shall show that this is only partially true at the level of fluctuations. Thus an extension to second order freeness is required to capture this phenomenon. While the precise statement remains open it is clear the definition should involve the action of a graphical operad on an algebra.

I will also demonstrate the second order relation between Wigner and constant matrices. This is joint work with Roland Speicher.

Previous Schedules