# ERC Advanced Grant

## Non-Commutative Distributions in Free Probability

## Description of the ERC Advanced Grant NCDFP

Type of Grant: | ERC Advanced Grant |

**Aim**

The aim of the grant is to study new directions in free probability theory with high potential to lead to breakthroughs in our understanding of random matrix models and operator algebras. We will drive forward the study of *free analysis* which is intended to provide a whole new mathematical theory for variables with the highest degree of non-commutativity and which lies at the crossroad of many exciting mathematical subjects.

More specifically, the objective of the research founded by this grant is to extend our armory for dealing with non-commutative distributions and to attack some of the fundamental problems which are related to such distributions, like: the existence and properties of the limit of multi-matrix models; the isomorphism problem for free group factors, and more generally, properties of free entropy and free entropy dimension as invariants for von Neumann algebras.

**Quantum Symmetries:**

The first project deals with *quantum symmetries of non-commutative distributions*. We try to classify non-commutative symmetries and describe the effect of invariance under such quantum symmetries for non-commutative distributions. This is based on the theory of easy quantum groups.

Details

**Free Malliavin Calculus:**

In the second project we will develop the theory of *free Malliavin calculus*. This will then be used to investigate regularity properties of non-commutative distributions.

Details

**Analytic Aspects of Operator-Valued Free Probability:**

In the third project, we will study the *analytic theory of operator-valued free convolutions*. One specific goal in this context is to find and implement algorithms for calculating non-commutative distributions and asymptotic eigenvalue distributions for general random matrix problems.

Details

## Activities funded by the ERC Advanced Grant NCDFP

*C**-Algebras, Random Matrices, Free Probability, and Beyond

Conference in honour of James A. Mingo on the occasion of his 60th birthday

partially supported by the ERC Advanced Grant NCDFP

Kingston, Canada

October 16 - 17, 2015

organized by Serban Belinschi (Toulose, France), M. Ram Murty (Kingston, Canada), Roland Speicher (Saarbrücken)

George Boole Mathematical Sciences Conference

Theme 8: Quantum Probabilistic Symmetries & Quantized Boolean Algebras

partially supported by the ERC Advanced Grant NCDFP

Cork, Ireland

August 17 - 28, 2015

Quantum Groups and Quantum Information Theory

partially supported by the ERC Advanced Grant NCDFP

Herstmonceux, UK

July 13 - 17, 2015

organized by Benoit Collins (Kyoto, Japan), James Mingo (Kingston, Canada), Ashley

Montanaro (Bristol, UK), Maria Grazia Viola (Lakehead, Canada), Moritz Weber (Saarbrücken)

Free (Stochastic) Analysis

Mini-workshop on Operator Valued Free Probability, Non-Commutative Functions

and the Functions in Several Non-Commuting Random Variables

fully funded by the ERC Advanced Grant NCDFP

Saarbrücken, Germany

23 March - 27 March, 2015

organized by Tobias Mai, Roland Speicher and John Williams (all Saarbrücken)

Special session on free probability and operator theory,

part of IWOTA 2014

partially supported by the ERC Advanced Grant NCDFP

International Workshop on Operator Theory and its Applications

yearly conference, this time held in Amsterdam, Netherlands

July 14 - 18, 2014

special session organized by Serban Belinschi (Queens, Canada), Roland Speicher

and Moritz Weber (both Saarbrücken)

Quantum Symmetries in Free Probability

Mini-workshop on easy quantum groups and other symmetries in free probability

fully funded by the ERC Advanced Grant NCDFP

Saarbrücken, Germany

31 March - 4 April, 2014

organized by Roland Speicher and Moritz Weber (both Saarbrücken)

### Postal address

Saarland University

Department of Mathematics

Postfach 15 11 50

66041 Saarbrücken

Germany

### Visitors

Saarland University

Campus building E 2 4

66123 Saarbrücken

Germany