We plan to run the workshop in hybrid form with a considerable number of participants present in Münster and video transmission to the outside world.

This workshop intends to be a first meeting point for specialists and young researchers active in noncommutative geometry, free probability, and topological recursion. In the two first areas, one often wants to compute expectation values of a large class of noncommutative observables in random ensembles of (several) matrices of size N, in the large N limit. The motivations come from the study of various models of 2d quantum gravity via random spectral triples, or from the problem of identifying interesting factors via approximations by matrix models. Topological recursion and its generalisations provide a priori universal recipes to make and to compactly organise such computations, not only for the leading order in N, but also to all orders of expansion in 1/N. In this way connections are established to domains like enumerative geometry, tropical geometry, mirror symmetry, topological and more generally low-dimensional quantum field theories where topological recursion has also been applied.

Concretely, the last 10 years have witnessed the developement

- of analytic techniques to establish the existence of large-N asymptotic expansions,
- of applications of the topological recursion to a growing class of matrix models which now include some of direct interest in the study of random spectral triples and in noncommutative probability,
- and of connections between the combinatorics of free probability (i.e. higher order free cumulants) and the topological recursion together with symplectic transformations acting on it.

The workshop will explore the consequences of these remarkable algebraic structures axiomatised in topological recursion for problems in noncommutative geometry and free probability. Knowledge will also flow in the other direction, as the very nature of topological recursion hints at connections to (noncommutative) algebraic geometry and to Hopf algebraic structures and Connes-Kreimer renormalisation.

Mathematical models and phenomena under consideration are common to all these fields, and we wish to unite the strength of probabilistic/asymptotic, algebraic/geometric and combinatorial approaches for the benefit of all the communities involved. This interaction should in fine lead to a better geometric understanding, more powerful computational tools, and new results.

## Minicourses

*I**ntroduction to topological recursion*

*Elba Garcia-Failde (I**RIF, Université Paris Diderot)*

*Noncommutative geometry and spectral triples*

*Walter van Suijlekom (IMAPP, Radboud University Nijmegen)*

*Free probability and higher order freeness*

*Roland Speicher (Fachrichtung Mathematik, Universität des Saarlandes)*

*(Mixed) topological recursion and the two-matrix model*

*Bertrand Eynard (Institut de Physique Théorique, Paris-Saclay / IHÉS)*

## Organisers