Integrable billiards, Poncelet porisms and Kowalevski top

4:00 pm Thursday, April 8, 2010
Vladimir Dragovic (Mathematical Institute SANU and University of Lisbon)


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. A set T of lines tangent to d-1 quadrics from a given confocal family in a d- dimensional space is equipped with an algebraic operation. Using it, well-known results of Donagi, Reid and Knorrer are developed further. We derive a fundamental property of T: any two lines from T can be obtained from each other by at most d-1 billiard reflections at some quadrics of the confocal family. The interrelations among billiard dynamics, linear subspaces of intersections of quadrics and hyperelliptic Jacobians enabled us to obtain higher-dimensional and higher-genera generalizations of several classical genus 1 results. Among several applications, a new view on the Kowalevski top and Kowalevski integration procedure is presented. It is based on a classical notion of Darboux coordinates, a modern concept of n-valued Buchstaber-Novikov groups and a new notion of discriminant separability. Unexpected relationship with the Great Poncelet Theorem for a triangle is established.

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