Séminaire de Géométrie, Groupes et Dynamiques

Alfonso Sorrentino (Università degli Studi di Roma Tor Vergata) : On the integrability of mathematical billiards

A mathematical billiard is a system describing the inertial motion of a point mass inside a domain, with elastic reflections at the boundary. This simple model has been first proposed by G.D. Birkhoff as a mathematical playground where ``it the formal side, usually so formidable in dynamics, almost completely disappears and only the interesting qualitative questions need to be considered''. Since then billiards have captured much attention in many different contexts, becoming a very popular subject of investigation. Despite their apparently simple (local) dynamics, their qualitative dynamical properties are extremely non-local. This global influence on the dynamics translates into several intriguing rigidity phenomena, which are at the basis of several unanswered questions and conjectures. In this talk I shall focus on several of these questions. In particular, I shall describe some recent results related to the classification of integrable billiards (also known as Birkhoff conjecture). This talk is based on works in collaboration with V. Kaloshin and with G. Huang and V. Kaloshin.
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