Geometric recursion learning seminar

Given some initial data the GR is a procedure that gives mapping class group invariants. We will define the GR and give a proof that this in fact well defined. (Notes by Campbell)

We will sketch the general proof of the good definition of the GR formula. Moreover, we will check that the target theory provided by continuous functions on the Teichmüller space indeed satisfies the required axioms. (Notes by Danilo)

This talk goes through the details of the proof of Mirzakhani-McShane identity via hyperbolic trigonometry, and relates such identity to GR. Moreover, generalisations of Mirzakhani-McShane identity in the context of GR involving statistics of lenghts of simple closed curves are discussed.

In this talk, we introduce the notion of topological recursion. Moreover, we relate it to several enumerative problems, such as generalized Catalan numbers and variants of Hurwitz numbers.

In previous talks, GR was introduced, as well as the principle of TR, for which some examples were furnished. Both procedures allow to produce, from initial data, an infinite family of "invariants". The question we tackle here is the relation between those families of invariants and between the initial data. The aim of this talk is to show that, starting from TR with a certain class of spectral curves, one can reproduce the same invariants by GR.

The Masur-Veech volume b_{g,n} of M_{g,n} can be defined in two ways. Firstly, as the integral over M_{g,n} of a certain analytic function B_{g,n}. Secondly as the coefficient of the leading term of some enumerative problem (here a certain type of quadrangulations in genus g). There is a formula for the Masur-Veech volume b_{g,n} as a polynomial of integrals of psi-classes (on various M_{g',n'}). This formula can be derived from any of the two definitions (that was respectively done by Mirzakhani and Delecroix-Goujard-Zograf-Zorich). It is likely that the Masur-Veech volumes b_{g,n} satisfy TR and that the functions B_{g,n} satisfies GR. However it is not (yet) transparent from the formulas.

We show that the Masur-Veech volumes and area Siegel-Veech constants can be obtained by intersection numbers on the strata of Abelian differentials with prescribed orders of zeros. As applications, we evaluate their large genus limits and compute the saddle connection Siegel-Veech constants for all strata.

In this first talk, we discuss generalities about intersection theory on the moduli space of stable curves, with particular emphasis on computational tools like the Kontsevich model for psi intersections, Mirzakhani's recursion and Mumford formula for lambda classes. (Notes by Alessandro)

In this second talk, we investigate the connection between topological recursion (TR) and intersection theory on the moduli space of stable curves. In particular, we will show how to compute TR amplitudes associated to a spectral curve in terms of intersections of psi classes and a certain class built from the spectral curve itself. Examples will include Kontsevich and Mirzakhani's recursion, ELSV formula, and Masur-Veech volumes of the space of quadratic differentials. (Notes by Alessandro)

We give a combinatorial model for r-spin surfaces with parametrised boundary. The r-spin structure is encoded in terms of ℤᵣ-valued indices assigned to the edges of a polygonal decomposition. With the help of this model we count the number of mapping class group orbits on r-spin surfaces with parametrised boundary and fixed r-spin structure on each boundary component.