Billiards and quadrics, Poncelet porisms and Kowalevski top

We address a cycle of questions and results connecting algebraic and differential geometry with integrable dynamics. A progress in a thirty years old programme of Griffiths and Harris of understanding of higher-dimensional analogues of Poncelet porisms and synthetic approach to higher genera addition theorems is presented. We study discrete differential geometric structures arising in billiard systems within pencils of quadrics. We give their natural higher-dimensional and higher-genera generalizations. Among several applications, a new view on the Kowalevski top and Kowalevski integration procedure is presented. It is based on our recent notion of discriminant separability.
References:
[1] V. Dragovic, Geometrization and generalization of the Kowalevski top, Comm. Math Phys, 298 (2010) no. 1, 37-64
[2] V. Dragovic, M. Radnovic, Poncelet porisms and beyond, Springer-Birkhauser,
2011, 294 p. ISBN 978-3-0348-0014-3
[3] V. Dragovic, Poncelet-Darboux curves, their complete decomposition and Marden theorem, International Math. Res. Notes, (2011), 3502–3523
[4] V. Dragovic, M. Radnovic, Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: beyond Poncelet porisms, Adv. Math. 219 (2008), 1577-1607