**Fast mixing of Sinai billiard flows
**

Sinai billiards are natural chaotic dynamical systems which are notoriously difficult to study, due to their singularities (arising from "grazing orbits"). Their first ergodic properties were obtained by Sinai over forty years ago. More recently, LSYoung obtained in 1998 exponential mixing for the Sinai billiard map (=the discrete-time dynamical system corresponding to the collision map), and Chernov and Melbourne proved independently in 2007 that the Sinai billiard flow mixes faster than any polynomial. We recently showed that two-dimensional finite-horizon Sinai billiard flow mix in fact exponentially fast. Along the way, our proof gives a description of the spectrum of the billiard flow (the Ruelle resonances). We will try to give the flavour of the method of proof, which uses transfer operators on anisotropic Banach spaces, as discussed in the first talk. (joint with Demers and Liverani)