Seminar "Dynamical systems" (guided by Prof.Yulij Ilyashenko)

Vadim Kaloshin (University of Maryland, USA and ETH, Switzerland)
Monday, 17 April 2017
Conference hall, 401 Moscow center for continuous mathematical education

On convex planar billiards, Birkhoff Conjecture and whispering galleries

A mathematical billiard is a system describing the inertial motion of a point mass inside a domain with elastic reflections at the boundary. In the case of convex planar domains, this model was first introduced and studied by G.D. Birkhoff, as a paradigmatic example of a low dimensional conservative dynamical system. A very interesting aspect is represented by the presence of 'caustics', namely curves inside the domain with the property that a trajectory, once tangent to it, stays tangent after every reflection (as on the right Figure). Besides their mathematical interest, these objects can explain a fascinating acoustic phenomenon, known as "whispering galleries", which can be sometimes noticed beneath a dome or a vault. The classical Birkhoff conjecture states that the only integrable billiard, i.e., the one having a region filled with caustics, is the billiard inside an ellipse. We show that this conjecture holds near ellipses.