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DTSTAMP:20221004T000642Z
UID:627b89a87d2b1596122624@ist.ac.at
DTSTART:20220523T140000
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DESCRIPTION:Speaker: Edmond Koudjinan\nhosted by Kaloshin Group\nAbstract:
A famous Birkhoff conjecture states that the only integrable convex planar
billiards are billiards in an ellipse. We examined two closely related ri
gidity questions. A rational caustic is a caustic associated to a family o
f periodic orbits of the same period and the same rotation number. For exa
mple\, a convex domain with a rational caustic of period two is a domain o
f a constant width. Elliptic billiard table admit rational caustic of any
period greater than 2. Baryshnikov and Zharnitsky proved that an ellipse c
an be deformed so as to preserve any given rational caustic. The following
question has been then proposed by Tabachnikov: are there nearly circular
domains other than discs with two rational caustics of a prime period p a
nd q? In this talk\, I will discuss the following results:(rigidity) There
are no nearly circular domains with two coexisting rational caustics of p
eriod two and three.(no super-rigidity) There may be infinitely many defor
mations of the circular domains with two coexisting rational caustics of p
eriod three and five with error given by the 3rd power of the perturbation
parameter. Baryshnikov and Zharnitsky did prove that a properly chosen pa
rametrization of the family D_n of billiard table with a rational caustic
of period $n$ give rise to a Hilbert sub-manifold of an appropriate Hilber
t manifold. One can then wonder whether this manifold is a graph. Using a
Nash-Moser-Zehnder generalized Implicit function Theorem\, We showed that
there exists an embedded continuous graph into D_m.This is based on a join
t work with Vadim Kaloshin & Ke Zhang.
LOCATION:Mondi Seminar Room 2\, Central Building\, ISTA
ORGANIZER:jdeanton@ist.ac.at
SUMMARY:Edmond Koudjinan: On non-coexistence of 2- &\; 3-rational causti
cs in nearly circular billiard tables
URL:https://talks-calendar.app.ist.ac.at/events/3784
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