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.

Self-contained, with an elementary introduction

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.

The goal of the book is to present, in a complete and comprehensive way, a few areas of mathematics interlacing around the Poncelet porism: dynamics of integrable billiards, algebraic
geometry of hyperelliptic Jacobians, and classical projective geometry of pencils of quadrics.
Most important results and ideas connected to Poncelet theorem are presented: classical as well as modern ones, together with historical overview with analysis of classical ideas, their natural generalizations, and natural generalizations.
Special attention is payed to realization of the Griffiths and Harris programme about Poncelet-type problems and addition theorems. This programme, formulated three decades ago, is aimed
to understanding of higher-dimensional analogues of Poncelet problems and realization of synthetic approach of higher genus addition theorems. This problem is realized in a sequence of papers of the authors, published in last 20 years. Some of those results represent the key contribution to the development of the theory, while this book gives the complete image.