Quelques contributions \`a la th\'eorie de l'action de SL(2,R) sur les espaces de modules de surfaces plates

TL;DR

This paper reviews the author's research on the action of SL(2,R) on moduli spaces of flat surfaces, contributing to the understanding of their geometric and dynamical properties.

Contribution

It presents new insights into the dynamics of SL(2,R) actions on moduli spaces, expanding theoretical understanding in this area.

Findings

Analysis of orbit structures under SL(2,R)

Characterization of invariant measures

Applications to flat surface geometry

Abstract

This memoir is part of the author's `Habilitation \`a Diriger des Recherches' (HDR) dossier: it summarizes some of his researches after his PhD thesis. (The HDR diploma is general requirement for many purposes [e.g., supervising PhD students] within the French academic system.)