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On algebraically integrable Birkhoff and angular billiards. (arXiv:1706.04030v1 [math.DS])

We present a solution of the algebraic version of Birkhoff Conjecture on integrable billiards. Namely we show that every polynomially integrable real bounded convex planar billiard with smooth boundary is an ellipse. We also extend this result to the case of piecewise-smooth and not necessarily convex polynomially integrable billiards: we show that the boundary is a union of confocal conical arcs and straight-line segments lying in some special lines defined by the foci. The proof, which is obtained by Mikhail Bialy, Andrey Mironov and the author, is split into two parts. The first part is the paper by Bialy and Mironov, where they prove the following theorems: 1) the polar duality transforms a polynomially integrable planar billiard to a rationally integrable angular billiard; 2) the singularities and inflection points of each irreducible component of the complexified curve polar-dual to the billiard boundary lie in the two complex isotropic lines through the origin; 3) the Hessian Fo查看全文

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