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Misha Bialy (Tel Aviv University, Israel)
February 11 @ 4:00 pm - 5:00 pm CET
Speaker: Misha Bialy (Tel Aviv University, Israel)
Title: Birkhoff-Poritsky conjecture for centrally-symmetric billiards
Abstract: In this talk I shall discuss Birkhoff-Poritsky conjecture for centrally-symmetric C^2-smooth convex planar billiards. We assume that the domain A between the invariant curve of 4-periodic orbits and the boundary of the phase cylinder is foliated by C^0-invariant curves. Under this assumption we prove that the billiard curve is an ellipse. Other versions of Birkhoff-Poritsky conjecture follow from this result. For the original Birkhoff-Poritsky formulation we show that if a neighborhood of the boundary of billiard domain has a C^1-smooth foliation by convex caustics of rotation numbers in the interval (0; 1/4] then the boundary curve is an ellipse. The main ingredients of the proof are:
(1) the non-standard generating function for convex billiards;
(2) the remarkable structure of the invariant curve consisting of 4-periodic orbits; and
(3) the integral-geometry approach initiated by the author for rigidity results of circular billiards.
Surprisingly, we establish a Hopf-type rigidity for billiards in the ellipse. Based on joint work with Andrey E. Mironov (Novosibirsk).
NOTE: The seminar will be streamed live on our YouTube channel then saved there. If you ask questions, with your video feed on or off, you agree to the use of your image/spoken words for said purpose.