Poncelet porisms and beyond: integrable billiards, hyperelliptic Jacobians and pencils of quadrics

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 ideasconnected 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 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 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 over thelast 20 years. Some of those results represent key contributions to the development of the theory, while this book provides the complete image. 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. INDICE: 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.

• ISBN: 978-3-0348-0014-3
• Editorial: Birkhaüser
• Encuadernacion: Rústica
• Páginas: 290
• Fecha Publicación: 30/11/2010
• Nº Volúmenes: 1
• Idioma: Inglés