Birkhoff Conjecture for convex planar billiards

Date
2019/07/26 Fri 14:00 - 17:00
Room
6号館609号室
Speaker
Vadim Kaloshin
Affiliation
University of Maryland
Abstract

G.D.Birkhoff introduced a mathematical billiard inside of a convex domain as the motion of a massless particle with elastic reflection at the boundary. A theorem of Poncelet says that the billiard inside an ellipse is integrable, in the sense that the neighborhood of the boundary is foliated by smooth closed curves and each billiard orbit near the boundary is tangent to one and only one such curve (in this particular case, a confocal ellipse).
A famous conjecture by Birkhoff claims that ellipses are the only domains with this property. We show a local version of this conjecture - namely, that a small perturbation of an ellipse has this property only if it is itself an ellipse.