
THE
FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

July
2013
Focus Program on Noncommutative Distributions in Free
Probability Theory
July
818, 2013
InterWorkshop
Program
Room
210, Fields Institute
Organizing Committee:
Serban Belinschi (Queen's), James Mingo (Queen's), and Alexandru
Nica (Waterloo)



Week 1 Preliminary Schedule
10:30  12:30
Location: Fields Institute, Room 230 
July 08
Monday,

Alice Guionnet, Ecole Normale Supérieure
de Lyon
Topological expansions and loop equations

July 09
Tuesday

Dima Shlyakhtenko, University of California,
Los Angeles
Free monotone transport 
July 10
Wednesday 
Natasha Blitvic, Vanderbilt University (slides)
qDeformed Probability and Beyond

July 11
Thursday 
Moritz Weber, Saarland University
Quantum groups and their relation with free probability 
Afternoon sessions
Location: Fields Institute, Room 230 
July 08, Monday

2:15  3:00
Jeffrey Kuan, Harvard University
Noncommutative random surface growth
3:30  4:15
Wiktor Ejsmont, University of Wroclaw
Noncommutative characterization of free Meixner processes 
July 10, Wednesday

2:15  3:00
Malte Gerhold, Universität Greifswald
Finite dimensional subproduct systems
3:30  4:15
Piotr Warchol, Jagiellonian University
Burgerslike equation for diffusing chiral matrices 
Week 2 Preliminary
Schedule
10:30  12:30
Location: Fields Institute, Room 230 
July 15
Monday 
Camille Male, Université ParisDiderot
Traffic and Voiculescu's asymptotic freeness theorems 
July 16
Tuesday 
Claus Koestler, University College Cork
(slides)
Distributional symmetries in free probability 
July 17
Wednesday 
Viakalathur Sunder, Institute of Mathematical
Sciences
From graphs to free probability (joint work with
Madhushree Basu and Vijay Kodiyalam) 
July 18
Thursday 
Matt Kennedy, Carleton University
An introduction to some noncommutative function
theory 
Afternoon sessions
Location: Fields Institute, Room 230 
July 15, Monday

2:15  3:00
Pierre Tarrago, Université ParisEst (Marne la Vallée)
Some stochastic computations on the free Unitary
quantum group
3:30  4:10
Jacek Grela, Krakow
Diffusion in the space of complex hermitian
matrices 
July 17, Wednesday 
2:15  3:00
Kamil Szpojankowski, Warsaw University of Technology
Dual Lukacs regressions in free probability
3:30  4:10
Robin Langer, Université MarneLaValée (slides)
Commutators in Semicircular systems 
Speaker

Title
and Abstract 
Natasha
Blitvic
Vanderbilt University 
qDeformed Probability and Beyond
A noncommutative Central Limit Theorem and a twisted Fock
space construction form the underpinnings of a rich and beautiful (noncommutative)
probability theory pioneered by Bozejko and Speicher in the early 90s,
and furthered by many thereafter. The framework at hand, which may be
viewed as an interpolation between classical, free, and fermionic probability,
is also interesting from operator algebraic and combinatorial viewpoints.
In this talk, I will introduce the nuts and bolts of the theory and
survey some of the exciting work in this multifaceted area.

Gerhold,
Malte
Universität Greifswald 
Finite dimensional subproduct systems (joint work with Michael Skeide)
A (discrete) subproduct system is a family of Hilbert spaces,
indexed by the nonnegative integers s.t. the Hilbert spaces to parameter
m+n are embedded into the tensor products of those for m and n in an
associative way. The prefix "sub" refers to the fact that
the embeddings are only isometries, not necessarily unitaries. It is
very well possible that all these Hilbert spaces are finite dimensional
and we will adress the question, which sequences of finite dimensions
are possible. Our main result is the reduction to the combinatorial
question of all possible cardinality sequences of what we call word
systems and is known in the literature on the combinatorics of words
under the name factorial languages.

Grela,
Jacek
Krakow 
Diffusion in the space of complex hermitian matrices
We study microscopic properties of both averaged characteristic
polynomials and averaged inverse characteristic polynomials. For that
we derive general diffusion equations for these objects and find their
asymptotic behavior for different scalings (Airy, Pearcey). This analysis
turn out to give complete set of solutions to Airy and Pearcey equations.

Guionnet,
Alice
Ecole Normale Supérieure de Lyon 
Topological expansions and loop equations
We will discuss loop equations in Random matrix theory, with
their applications to topological expansions and free probability.

Kennedy,
Matt
Carleton University 
An introduction to some noncommutative function theory
In this talk, I will give a brief introduction to the noncommutative
function theory developed by Arveson, Davidson, Popescu, and many others.
A central idea here is the notion of a dilation, which has roots in
the theory of completely bounded maps and abstract operator algebras.
I will outline some of the important developments in this area, and
present some motivating examples.

Koestler,
Claus
University College Cork 
Distributional symmetries in free probability
De Finetti type results in classical probability infer conditional
independence from certain distributional symmetries of random variables.
I will introduce into the recent progress on transferring such de Finetti
type results to free probability.

Kuan, Jeffrey
Harvard University 
Noncommutative random surface growth
We introduce a Markov chain on a noncommutative probability
space which arises naturally from the representation theory of the unitary
group. When restricted to the center, the Markov chain describes a random
surface growth model. This surface growth lies in the Anisotropic KardarParisiZhang
universality class from mathematical physics and can also be viewed
as a discretization of Dyson Brownian Motion.

Camille Male
Université ParisDiderot 
Traffic and Voiculescu's asymptotic freeness theorems
In this talk, I review how to use the tools of 'traffics'
to handle large random matrices. I present a proof of the asymptotic
freeness theorems for classical large matrices and a description of
limiting distributions.

Szpojankowski, Kamil
Warsaw University of Technology 
Dual Lukacs regressions in free probability
Characterizations of probability measures in free and classical
probability are closely related. Many characterizations known from classical
case have a free counterpart. In this talk I will discuss free analogues
of the Lukacs theorem, and so called dual Lukacs regressions.

Shlyakhtenko, Dima
University of California, Los Angeles 
Free monotone transport
In a joint work with A. Guionnet, we show that it is possible to
find a noncommutative analog of Brenier's monotone tranport theorem:
we are able to find noncommutative analytic functions that pushforward
the semicircle law into a arbitrary free Gibbs law which is sufficiently
close to the semicircle law. As consequence, we deduce that for small
q, qdeformed free group factors are isomorphic to free group factors.
We discuss some further developments in this area.

Sunder, V.S.
Institute of Mathematical Sciences, Chennai, India

From graphs to free probability (joint work with Madhushree Basu
and Vijay Kodiyalam)
This may be regarded as an expository one on some aspects
of the work of GuionnetJonesShlyakhtenko, where we investigate a construction
which associates a finite von Neumann algebra $M(\Gamma,\mu)$ to a finite
weighted (not necessarily bipartite) graph $(\Gamma,\mu)$. This construction
also yields a `natural' example of a Focktype model of an operator
with a free Poisson distribution.
Pleasantly, but not surprisingly, the von Neumann algebra associated
to a `flower with $n$ petals' is the von Neumann algebra of the free
group on $n$ generators.
In general, the algebra $M(\Gamma,\mu)$ is a free product, with amalgamation
over a finitedimensional abelian subalgebra corresponding to the
vertex set, over algebras associated to subgraphs `with one edge'
(actually a pair of dual edges).

Tarrago, Pierre
Université ParisEst (Marne la Vallée) 
Some stochastic computations on the free Unitary quantum group
I will describe the joint law of a family of variables defined
on the unitary quantum group. This result is a free natural counterpart
of similar statements already proven on the classical unitary group
and on other easy quantum groups. Some preliminaries are given in the
first part of the talk, and the second part is devoted to the combinatorial
proof of the statement. This is part of an ongoing work with Moritz
Weber.

Warchol, Piotr
Jagiellonian University 
Burgerslike equation for diffusing chiral matrices
We show that a ColeHopf transform of the averaged characteristic
polynomial, associated with a chiral matrix performing a random walk,
satisfies a Burgerslike partial differential equation. The inverse
size of the matrix plays the role of the viscosity. We recover the asymptotic
form of the polynomial at the critical point of the evolution. The study
is motivated by the spontaneous chiral symmetry breaking in Quantum
Chromodynamics.

Weber, Moritz
Saarland University 
Quantum groups and their relation with free probability
I will introduce the notion of a compact (matrix) quantum
group and then focus on easy quantum groups, a class with a very rich
combinatorial data which also appears in free probability.

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