Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics

ISBN-13: 9783034800143 / Angielski / Miękka / 2011 / 294 str.

Vladimir Dragovi?; Milena Radnovi?

Opis

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.

Recenzje

From the reviews:

"The book provides a self-guided introduction to a set of classical gems that have to do with elliptic or hyperelliptic addition theorems; several connections were worked out originally by the authors in previous research papers. In addition, solutions to some of the integrable systems that came of age in the 1970s ... are produced in this context. The style is clear and the calculations are complete. ... it is certainly a focused, enjoyable and valuable one." (Emma Previato, Mathematical Reviews, Issue 2012 j)

"Poncelet Porisms and Beyond: Integrable Billiards, Hyperelliptic Jacobians, and Pencils of Quadrics ... gives a nice picture of both Poncelet's Porism and a number of the areas of mathematics that are close by. ... Dragovic and Radnovic do a good job of covering a wide range of material, including both citations and descriptions of classical material as well as extremely cutting-edge results. ... The exposition in the book is generally good, and ... any mathematician who picks up this book will learn something from it." (Darren Glass, The Mathematical Association of America, September, 2011)

Spis treści

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.

Blurb

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.