Non-commutative geometry is an arsenal of tools to study non-commutative operator algebras from a topological or geometric viewpoint. Universal methods permit to treat ordinary Riemannian manifolds, spaces of non-integral dimensions (e.g. fractals and boundaries of trees) as well as spaces of leaves of a foliation and quantum groups on the same footing, using spectral triples. A finite set of points can be equipped into an interesting differential geometry where the spectral triple is represented by matrices, and studying random geometries then relates to random matrix models. Free probability and more generally non-commutative probability provide tools to study coupled systems of random matrices in the large size limit.

Topological recursion is a universal structure invented by Chekhov, Eynard and Orantin, providing a recursive procedure to compute all-order asymptotic expansions (for large size) in certain matrix models. Once formulated abstractly in terms of the geometry of spectral curves, it has found an increasing number of applications beyond matrix models: in enumerative geometry, mirror symmetry, low-dimensional quantum field theories, and more recently deformation quantization and hyperbolic geometry. Tropical geometry is an array of techniques to reduce problems of enumerative geometry to combinatorics.

Some promising bridges between these topics have been discovered in the last 5 years. This workshop therefore aims at developing further these interactions and encourage the transfer of knowledge to address problems in all four areas thanks to this broader perspective.