Program

• Au, Benson, University of California:

Rigid structures in the universal enveloping traffic space

For a tracial *-probability space (A, phi), Cébron, Dahlqvist, and Male constructed a universal enveloping traffic space (G(A), tau) that extends the trace phi. This construction provides an important foundational tool that allows us to appeal to the traffic probability framework in generic situations, prioritizing an understanding of its structure. We show that (G(A), tau) comes equipped with a canonical free product structure, regardless of the choice of *-probability space (A, phi). If (A, phi) is itself a free product, then we show how this structure lifts into (G(A), tau). Here, we find a surprising connection to classical independence. Finally, we discuss applications to random matrices.

Joint work with Camille Male.

• Banna, Marwa, Saarland University:

Operator-valued matrices with exchangeable entries

We extend the Lindeberg method to differentiable functions in non-commuting variables. As a consequence, we approximate an nxn operator-valued Wigner-type matrix with exchangeable entries by a semi-circular element with respect to the smallest W*-algebra closed under the variance mapping. We prove that the difference between the associated Cauchy transforms is of the order of 1/sqrt(n).

Joint work with G. Cébron and T. Mai.

• Capitaine, Mireille, Université Paul Sabatier (Toulouse):

Spectral properties of polynomials in independent Wigner matrices and deterministic matrices

We investigate the spectrum of Hermitian polynomials in large independent Wigner matrices and deterministic matrices. We focus on the existence of ``a deterministic equivalent" and the problem of outliers in the spectrum.

• Collins, Benoît, Kyoto university:

The strong asymptotic freeness for random permutations

Abstract: n by n permutation matrices act naturally on the (n-1)-dimensional vector subspace of C^n of vectors whose components add up to zero. We prove that random independent permutations, viewed as operators on this vector subspace, are asymptotically strongly free with high probability. While this is a counterpart of a previous result by the presenter and Male in the case of a uniform distribution on unitary matrices, the techniques required for random permutations are very different, and rely on the development of a matrix version of the theory of non-backtracking operators.

This is joint work with Charles Bordenave.

• Friedrich, Roland, Saarland university:

Moments and Cumulants revisited

In this talk I will consider the notion of moments and cumulants in non-commutative probability theories from two perspectives. The first one is algebra and the second one is geometry. These two viewpoints can be nicely combined into single entities of algebraic-geometric nature, such as, e.g. Lie groups. Further, I shall illustrate the general framework with several examples.

• Koestler, Claus, University College Cork:

Spreadability and semi-cosimplicial objects

Distributional symmetries and invariance principles are of growing interest in noncommutative probability. Among them, spreadability denotes the invariance of a distribution when passing from a sequence of random variables to a subsequence. Recently we have identified the importance of spreadability from the viewpoint of algebraic topology and homological algebra. More precisely, spreadability is equivalent to the existence of a covariant functor from the semi-simplicial category into the category of noncommutative probability spaces.

My talk will introduce to this quite surprising connection and is based on joint work with Gwion Evans and Rolf Gohm.

• Lévy, Thierry, Université Pierre et Marie Curie (Paris 6):

Two-dimensional Yang-Mills theory and the Makeenko-Migdal equations

Given a surface and a compact Lie group, the Yang-Mills measure is the distribution of a family of random variables with values in the group, indexed by the set of loops traced on the surface. The master field is, in this context, a deterministic real-valued function on the set of loops, which satisfies a remarkable set of equations called the Makeenko-Migdal equations.

I will present these objects, explain how the Yang-Mills measure can be understood as the distribution of a random homomorphism from the group of loops on the surface to the chosen compact Lie group, and how, as the rank of this Lie group tends to infinity, this random morphism converges to a deterministic representation of which the master field is the character.

• Speicher, Roland, Saarland University:

Mixtures of classical and free independence, and their cumulants and symmetries

The concept of Lambda-independence of Mlotkowski was recently reconsidered in joint work with Wysoczanski; also leading to the introduction of new quantum groups and corresponding cumulants in joint work with Weber and with Ebrahimi-Fard and Patras, respectively. I will present some of the results of those investigations.

• Yin, Sheng, Saarland University:

Non-commutative rational functions in random matrices and operators

In this talk, we will show that it is natural to go from non-commutative polynomials to rational functions when we have strongly convergent random matrices. This answers part of convergence problems arised in a recent work of Helton, Mai and Speicher, in which they developed the theory for calculating the distribution of any rational functions in free random variables. Beside the convergence problem, we will also talk about the zero divisor problem when we evaluate the rational functions at some tuple of operators.

It is a joint-work in process with Tobias Mai.