Four equivalent properties of integrable billiards
The billiard inside an ellipse is completely integrable: its interior is foliated by caustics which are confocal ellipses. The string construction reconstructs a billiard table from its caustic, in particular, the string construction on an ellipse yields a confocal ellipse (a theorem of Graves). The string construction on a convex curve defines its diffeomorphism. A curve has the Poritsky property if it can be parametrized in such a way that these string diffeomorphisms (for different lengths of the string) are shifts with respect to this parameter. In 1950, Poritsky proved that only conics have this property. The Ivory lemma asserts that the diagonals of a curvilinear quadrilateral made by arcs of confocal ellipses and hyperbolas are equal. I shall deduce Ivory lemma from the integrability of billiards in ellipses.
These constructions and properties can be extended from Euclidean plane to Riemannian surfaces, in particular, the Liouville surfaces. By a classical result of Darboux, a foliation of a Riemannian surface has the Graves property if and only if the foliation consists of the coordinate lines of a Liouville metric (a Liouville net). A similar result of Blaschke says that a pair of orthogonal foliations has the Ivory property if and only if they form a Liouville net. To this we add that a curve on a Riemannian surface has the Poritsky property if and only if it is a coordinate curve of a Liouville net.
I shall mention another classical result, Weihnacht's theorem: the only Liouville nets in the Euclidean plane are nets of confocal conics and their degenerations. This suggests the following generalization of Birkhoff's conjecture: If an interior neighborhood of a closed geodesically convex curve on a Riemannian surface is foliated by billiard caustics, then the metric in the neighborhood is Liouville, and the curve is one of the coordinate lines.
The talk will be via Zoom at: https://utoronto.zoom.us/j/99576627828