Wednesday, **May 23**, 3:30 - 5:00, Jeff 319

Benson Au (Berkeley)

Rigid structures in the universal enveloping traffic space

For a tracial $*$-probability space $(\mathcal{A},
\varphi)$, Cébron, Dahlqvist, and Male constructed
an enveloping traffic space $(\mathcal{G}(\mathcal{A}),
\tau_\varphi)$ that extends the trace $\varphi$. The CDM
construction provides a universal object that allows one
to appeal to the traffic probability framework in
generic situations, prioritizing an understanding of its
structure.

We show that $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$ comes equipped with a canonical free product structure, regardless of the choice of $*$-probability space $(\mathcal{A}, \varphi)$. If $(\mathcal{A}, \varphi)$ is itself a free product, then we show how this additional structure lifts into $(\mathcal{G}(\mathcal{A}), \tau_\varphi)$. Here, we find a duality between classical independence and free independence.

We apply our results to study the asymptotics of large (possibly dependent) random matrices, generalizing and providing a unifying framework for results of Bryc, Dembo, and Jiang and of Mingo and Popa. This is joint work with Camille Male.

Tuesday, **April 10**, 3:30 - 5:00, Jeff 319

Camille Male (Bordeaux)

An introduction to traffic independence

The properties of the limiting non-commutative
distribution of random matrices can be usually
understood thanks to the symmetry of the model,
e.g. Voiculescu's asymptotic free independence
occurs for random matrices invariant in law by
conjugation by unitary matrices. The study of random
matrices invariant in law by conjugation by
permutation matrices requires an extension of free
probability, which motivated the speaker to
introduce in 2011 the theory of traffics. A traffic
is a non-commutative random variable in a space with
more structure than a general non-commutative
probability space, so that the notion of traffic
distribution is richer than the notion of
non-commutative distribution. It comes with a notion
of independence which is able to encode the
different notions of non-commutative independence.

The purpose of this task is to present the motivation of this theory and to play with the notion of traffic independence.

Tuesday, **April 3**, 3:30 - 5:00, Jeff 319

Pei-Lun Tseng (Queen's)

Linearization trick of infinitesimal freeness (II)

Last week, we introduced how to find a linearization for
a given selfadjoint polynomial and showed some
properties of this linearization. We will continue our
discussion this week and introduce the operator-valued
Cauchy transform. Then, we will show the algorithm for
finding the distribution of $P$ where $P$ is a selfadjoint
polynomial with selfadjoint variables $X$ and $Y$. Based on
this method, we will discuss how to extend this
algorithm for finding infinitesimal distribution for $P$.

Tuesday, **March 27**, 3:30 - 5:00, Jeff 319

Pei-Lun Tseng (Queen's)

Linearization trick of infinitesimal freeness

For given infinitesimal distribution of selfadjoint
elements $X$, $Y$, and given a selfadjoint polynomial $P$ with
variable $X$ and $Y$. The natural question is whether we can
write down the precise formula for the infinitesimal
distribution of $P$? In 2009 Belinschi and Shlyakhtenko
gave a precise formula to solve for the infinitesimal
distribution of $P$ for $P(X,Y)=X+Y$. In the talk, we will
discuss how to find the formula for an arbitrary
polynomial by using the linearization trick.

Tuesday, **March 20**, 3:30 - 5:00, Jeff 319

Mihai Popa (University of Texas (San Antonio))

Permutations of Entries and Asymptotic Free Independence
for Gaussian Random Matrices

Since the 1980's, various classes of random matrices
with independent entries were used to approximate free
independent random variables. But asymptotic freeness of
random matrices can occur without independence of
entries: in 2012, in a joint work with James Mingo, we
showed the (then) surprising result that unitarily
invariant random matrices are asymptotically (second
order) free from their transpose. And, in a more recent
work, we showed that Wishart random matrices are
asymptotically free from some of their partial
transposes. The lecture will present a development
concerning Gaussian random matrices. More precisely, it
will describe a rather large class of permutations of
entries that induces asymptotic freeness, suggesting
that the results mentioned above are particular cases of
a more general theory.

Tuesday, **March 13**, 3:30 - 5:00, Jeff 319

Jamie Mingo (Queen's)

The Infinitesimal Law of a real Wishart Matrix, II

I will continue from last week and compute the infinitesimal
cumulants.

Tuesday, **March 6**, 3:30 - 5:00, Jeff 319

Jamie Mingo (Queen's)

The Infinitesimal Law of a real Wishart Matrix

The Wishart ensemble is the random matrix ensemble used
to estimate the covariance matrix of a random vector.
Infinitesimal freeness is a generalized independence
stronger than freeness but weaker than second order
freeness.
I will give the infinitesimal distribution of a real
Wishart matrix. It is given in terms of planar diagrams
which are ‘half’ of a non-crossing annular partition.

Tuesday, **February 13**, 3:30 - 5:00, Jeff 319

Jamie Mingo (Queen's)

The Infinitesimal Law of the GOE, Part II

If $X_N$ is the $N \times N$ Gaussian Orthogonal
Ensemble (GOE) of random matrices, we can expand
$\mathrm{E}(\mathrm{tr}(X_N^n))$ as a polynomial in
$1/N$, often called a genus expansion. Following the
celebrated formula of Harer and Zagier for the GUE,
Ledoux (2009) found a five term recurrence for the
coefficients of $\mathrm{E}(\mathrm{tr}(X_N^n))$. We
show that the coefficient of $1/N$ counts the number of
non-crossing annular pairings of a certain type.

Our method is quite elementary. A similar formula holds for the Wishart ensemble. This identification is related to the theory of infinitesimal freeness of Belinschi and Shlyakhtenko.

Tuesday, **February 6**, 3:30 - 5:00, Jeff 319

Jamie Mingo (Queen's)

The Infinitesimal Law of the GOE

If $X_N$ is the $N \times N$ Gaussian Orthogonal Ensemble
(GOE) of random matrices, we can expand
$\mathrm{E}(\mathrm{tr}(X_N^n))$ as a polynomial in
$1/N$, often called a genus expansion. Following the
celebrated formula of Harer and Zagier for the GUE,
Ledoux (2009) found a five term recurrence for the
coefficients of $\mathrm{E}(\mathrm{tr}(X_N^n))$. We show
that the coefficient of $1/N$ counts the number of
non-crossing annular pairings of a certain type.

Our method is quite elementary. A similar formula holds for the Wishart ensemble. This identification is related to the theory of infinitesimal freeness of Belinschi and Shlyakhtenko.

Tuesday, **January 30**, 3:30 - 5:00, Jeff 319

Neha Prabhu (Queen's)

Semicircle distribution in number theory, Part II

In free probability theory, the role of the semicircle
distribution is analogous to that of the normal
distribution in classical probability theory. However,
the semicircle distribution also shows up in number
theory: it governs the distribution of eigenvalues of
Hecke operators acting on spaces of modular cusp
forms. In this talk, I will give a brief introduction to
this theory of Hecke operators and sketch the proof of a
result which is a central limit type theorem from
classical probability theory, that involves the
semicircle measure.

Tuesday, **January 23**, 3:30 - 5:00, Jeff 319

Rob Martin (University of Cape Town)

A multi-variable de Branges-Rovnyak model for row contractions

In the operator-model theory of de Branges and
Rovnyak, any completely non-coisometric (CNC)
contraction on Hilbert space is represented as the
adjoint of the restriction of the backward shift to
a de Branges-Rovnyak subspace of the classical
(vector-valued) Hardy space of analytic functions in
the open unit disk.

We provide a natural extension of this model to the setting of CNC (row) contractions from several copies of a Hilbert space into itself. A canonical extension of Hardy space to several complex dimensions is the Drury-Arveson space, and the appropriate analogue of the adjoint of the restriction of the backward shift to a de Branges-Rovnyak space is a Gleason solution, a row contraction whose adjoint acts as a several-variable difference quotient. Our several-variable model completely characterizes the class of all CNC row contractions which can be represented as (extremal contractive) Gleason solutions for a multi-variable de Branges-Rovnyak subspace of (vector-valued) Drury-Arveson space.

Tuesday, **January 16**, 4:00 - 5:30, Jeff 319

Neha Prabhu (Queen's University)

Semicircle distribution in number theory

In free probability theory, the role of the semicircle
distribution is analogous to that of the normal
distribution in classical probability theory. However,
the semicircle distribution also shows up in number
theory: it governs the distribution of eigenvalues of
Hecke operators acting on spaces of modular cusp
forms. In this talk, I will give a brief introduction to
this theory of Hecke operators and sketch the proof of a
result which is a central limit type theorem from
classical probability theory, that involves the
semicircle measure.

Previous Schedules