Institut für Mathematik


Konferenz: Symposium about Pure Mathematics

On convex planar billiards, Birkhoff Conjecture and whispering galleries

Vortrag von Prof. Dr. Vadim Kaloshin
Datum: 09.02.17   Zeit: 10.10 - 10.50   Raum: Y27H28

Abstract: 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 $\Omega$ with the property that a trajectory, once tangent to it, stays tangent after every reflection (as on the left 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 true for perturbations of ellipses preserving many caustics. This consists of two main steps: perturbations of a circle (joint with A. Avila—J. De Simoi) and perturbations of an ellipse (joint with A. Sorrentino). In a somewhat different direction we prove this conjecture for perturbations of the circle preserving not so many caustics (joint w G. Huang—A. Sorrentino).