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.
Inhaltsverzeichnis
Introduction to Poncelet Porisms. - Billiards First Examples. - Hyper-Elliptic Curves and Their Jacobians. - Projective geometry. - Poncelet Theorem and Cayley s Condition. - Poncelet Darboux Curves and Siebeck Marden 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.
