Bicentennial of the Great Poncelet Theorem (1813–2013): Current advances

Abstract

We present very recent results related to the Poncelet Theorem on the occasion of its bicentennial. We are telling the story of one of the most beautiful theorems of geometry, recalling for general mathematical audiences the dramatic historic circumstances which led to its discovery, a glimpse of its intrinsic appeal, and the importance of its relationship to dynamics of billiards within confocal conics. We focus on the three main issues: A) The case of pseudo-Euclidean spaces, for which we present a recent notion of relativistic quadrics and apply it to the description of periodic trajectories of billiards within quadrics. B) The relationship between so-called billiard algebra and the foundations of modern discrete differential geometry which leads to double-reflection nets. C) We present a new class of dynamical systems—pseudo-integrable billiards generated by a boundary composed of several arcs of confocal conics having nonconvex angles. The dynamics of such billiards have several extraordinary properties, which are related to interval exchange transformations and which generate families of flows that are minimal but not uniquely ergodic. This type of dynamics provides a novel type of Poncelet porisms—the local ones.