Vadim Kaloshin, Univ. of Maryland and ETH Zürich

BIRKHOFF CONJECTURE FOR CONVEX PLANAR BILLIARDS AND DEFORMATIONAL SPECTRAL RIGIDITY OF PLANAR DOMAINS

Abstract: The classical Birkhoff conjecture states that the only integrable convex planar domains are ellipses. We show that this conjecture is true for perturbations of ellipses. It consists of two steps: in a joint work with A. Avila and J. De Simoi and we study perturbations of the circle and in a joint work with A. Sorrentino we extend our analysis to perturbations of an ellipse. It turns out that the method of proof gives an insight into deformational spectral rigidity and the famous M. Kac's question can you hear the shape of a drum. The latter is a joint work with J. De Simoi and Q. Wei.