Понедельник, 17 апреля 2017, 12:00 в конференц зале НМУ (ауд. 401)
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 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.
