Poncelet Porisms and Beyond

Here is a comprehensive exposition of the key results and ideas connected to the Poncelet theorem, focusing on the mathematics and including the dynamics of integrable billiards and the algebraic geometry of hyperelliptic Jacobians, among other material.

The goal of the book is to present, in a complete and comprehensive way, areas of current research interlacing around the Poncelet porism: dynamics of integrable billiards, algebraic geometry of hyperelliptic Jacobians, and classical projective geometry of pencils of quadrics. The most important results and ideas, classical as well as modern, connected to the Poncelet theorem are presented, together with a historical overview analyzing the classical ideas and their natural generalizations. Special attention is paid to the realization of the Griffiths and Harris programme about Poncelet-type problems and addition theorems. This programme, formulated three decades ago, is aimed to understanding the higher-dimensional analogues of Poncelet problems and the realization of the synthetic approach of higher genus addition theorems.

Self-contained, with an elementary introduction Contains a complete account of higher-dimensional generalizations of Poncelet theorem, the corresponding Cayley-type conditions, and the Poncelet-Darboux grids. Gives introspection into exciting interlacing of different areas of mathematics around the Poncelet porism. Contains most modern results connected with the problems of Poncelet type. Includes supplementary material: sn.pub/extras

Inhalt
Introduction to Poncelet Porisms.- Billiards First Examples.- Hyper-Elliptic Curves and Their Jacobians.- Projective geometry.- Poncelet Theorem and Cayley's Condition.- PonceletDarboux Curves and SiebeckMarden Theorem.- Ellipsoidal Billiards and their Periodical Trajectories.- Billiard Law and Hyper-Elliptic Curves.- Poncelet Theorem and Continued Fractions.- Quantum Yang-Baxter equation and (2-2)-correspondences.- Bibliography.- Index.