Quentin de Mourgues

The (extended) Rauzy classes are the connected components of the strata of the moduli space of abelian differentials. The classification of the extended case was proved by Kontsevich and Zorich while the non-extended case (in which one singularity of the flat surface is marked) was later demonstrated by Boissy.

The Rauzy classes also appear in the study of intervals exchange transformation on flat surfaces and can be described as equivalence classes on permutations given by a dynamic (often called the Rauzy induction) with either 2 or 4 (for the extended case) operators.

Using this description, I will give a fairly detailed combinatorial proof of the classification : first by finding a set of invariants for the dynamic and then proving that those invariants do characterize the classes. If time allows, I'll show how to extend the proof methods to similar dynamics such has the generalization to matchings (which has ties to the classification of the connected components of the strata of the moduli space of quadratic differentials done by Lanneau).

This work was done in collaboration with Andrea Sportiello.