Geometric recursion is a procedure developed in 2017 by J.E. Andersen, G. Borot and N. Orantin, which generalizes topological recursion. For specific choices of the initial data and of the target theory on which the recursion runs, it allows to recursively construct objects that capture geometric properties of surfaces that are useful in mathematical physics. Together with J.E. Andersen, G. Borot, A. Giacchetto, D. Lewański and C. Wheeler, we have established a series of results allowing to promote the combinatorial Teichmüller space to a target theory for geometric recursion. I will first describe the combinatorial Teichmüller space and some of its properties; second I will define geometric recursion (GR) on this space. I will then give two instances of this recursion: the first one is akin to Mirzakhani–McShane identity, the second one is a recursive formula for the count of multicurves on combinatorial surfaces. Last, I will expose a set of coordinates on the combinatorial Teichmüller space that is well-suited for geometric recursion. Those coordinates allow to recover topological recursion via a procedure of integration: in particular for the 2 instances of the talk, we get another proof of Witten’s conjecture and a recursive formula for Masur–Veech volumes. [The talk will also be streamed online, please ask the organizers for the link.]

IRIF, CNRS/Université de Paris