SCIENTIFIC PROGRAMS AND ACTIVITIES

Natasha Blitvic, Vanderbilt University (slides)
q-Deformed Probability and Beyond

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.

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.

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.

Topological expansions and loop equations

We will discuss loop equations in Random matrix theory, with their applications to topological expansions and free probability.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.