SCIENTIFIC PROGRAMS AND ACTIVITIES

October 31, 2014
THE FIELDS INSTITUTE FOR RESEARCH IN MATHEMATICAL SCIENCES

July 2013
Focus Program on Noncommutative Distributions in Free Probability Theory

July 8-18, 2013
Inter-Workshop 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)
q-Deformed 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,
Universitšt Greifswald
Finite dimensional subproduct systems

3:30 - 4:15
Piotr Warchol, Jagiellonian University
Burgers-like equation for diffusing chiral matrices
Week 2 Preliminary Schedule
10:30 - 12:30
Location: Fields Institute, Room 230
July 15
Monday
Camille Male, Universitť Paris-Diderot
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 Paris-Est (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 Marne-La-Valée (slides)
Commutators in Semicircular systems
Speaker
Title and Abstract
Natasha Blitvic
Vanderbilt University

q-Deformed Probability and Beyond

A non-commutative Central Limit Theorem and a twisted Fock space construction form the underpinnings of a rich and beautiful (non-commutative) 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
Universitšt 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 Kardar-Parisi-Zhang universality class from mathematical physics and can also be viewed as a discretization of Dyson Brownian Motion.
Camille Male
Universitť Paris-Diderot

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 non-commutative analog of Brenier's monotone tranport theorem: we are able to find non-commutative analytic functions that push-forward 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, q-deformed 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 Guionnet-Jones-Shlyakhtenko, 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 Fock-type 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 finite-dimensional abelian subalgebra corresponding to the vertex set, over algebras associated to subgraphs `with one edge' (actually a pair of dual edges).

Tarrago, Pierre
Université Paris-Est (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

Burgers-like equation for diffusing chiral matrices

We show that a Cole-Hopf transform of the averaged characteristic polynomial, associated with a chiral matrix performing a random walk, satisfies a Burgers-like 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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