## EVENT DETAILS AND ABSTRACT

**Colloquium**
**Title:** On integrable billiards, Birkhoff conjecture, and deformational spectral rigidity.

**Speaker:** Vadim Kaloshin

**Speaker Info:**

**Brief Description:**

**Special Note**:

**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. It turns out that the method of proof
gives an insight into deformational spectral rigidity of
planar axis symmetric domains and gives a partial
answer to a question of P. Sarnak. This is based on several papers with Avila, De Simoi, G.Huang, Sorrentino, Q. Wei.

**Date:** Wednesday, November 14, 2018

**Time:** 4:10pm

**Where:** Lunt 105

**Contact Person:** Dmitry Tamarkin

**Contact email:** zelditch@math.northwestern.edu

**Contact Phone:**

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