# ESI Programme on "Bialgebras in Free Probability"

## February 1 - April 22, 2011

##### organized by

#### Marcelo Aguiar, Franz Lehner, Roland Speicher and Dan Voiculescu

The aim of this ESI-programme is to advance the cross-fertilization between the areas of free probability and algebraic combinatorics. The workshop will provide an opportunity for researchers in these areas to meet and interact, in some cases for the first time. The core groups of free probabilists and combinatorialists will be supplemented by researchers from operator algebras, random matrices, physics, and other related fields.

Free probability theory is a line of research that parallels aspects of classical probability, in a highly non-commutative context where tensor products are replaced by free products, and independent random variables are replaced by free random variables. It grew out from attempts to solve some longstanding problems about von Neumann algebras of free groups. In the twenty five years since its creation, free probability has become a subject in its own right, with connections to several other parts of mathematics: operator algebras, the theory of random matrices, classical probability and the theory of large deviations, and algebraic combinatorics.

In recent years, several bialgebra structures have occurred, in quite different ways, at a fundamental level in free probability theory and it is becoming clear that progress in the field requires a better understanding of this phenomenon. Three instances stand out at this time : the connections to

- "combinatorial Hopf algebras" ,
- bialgebras with derivation comultiplications (a.k.a. "infinitesimal bialgebras")
- the symmetries provided by certain "free" C*-algebraic quantum groups.

Furthermore we plan to accompany this exploration by an examination for possible connections of some important recent advances in the interaction of free probability with random matrices, operator algebras and some mathematical physics.

There will be introductory lectures and two workshops,

- the first workshop period February 14-25, 2011 will be about Combinatorial, Bialgebra, and Analytic Aspects
- the second workshop period April 11-21, 2011 will be about Random Matrix, Operator Algebra, and Mathematical Physics Aspects