Poincare, Relativity, Billiards and Symmetry

Abstract

This review is made of two parts which are related to Poincar\'e in different ways. The first part reviews the work of Poincar\'e on the Theory of (Special) Relativity. One emphasizes both the remarkable achievements of Poincar\'e, and the fact that he never came close to what is the essential conceptual achievement of Einstein: changing the concept of time. The second part reviews a topic which probably would have appealed to Poincar\'e because it involves several mathematical structures he worked on: chaotic dynamics, discrete reflection groups, and Lobachevskii space. This topic is the hidden role of Kac-Moody algebras in the billiard description of the asymptotic behaviour of certain Einstein-matter systems near a cosmological singularity. Of particular interest are the Einstein-matter systems arising in the low-energy limit of superstring theory. These systems seem to exhibit the highest-rank hyperbolic Kac-Moody algebras, and notably E(10), as hidden symmetries.