Discrete Integrable Systems: Qrt Maps and Elliptic Surfaces - J. J. Duistermaat

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Résumé :
This book is devoted to Quisped, Roberts, and Thompson (QRT) maps, considered as automorphisms of rational elliptic surfaces. The theory of QRT maps arose from problems in mathematical physics, involving difference equations. The application of QRT maps to these and other problems in the literature, including Poncelet mapping and the elliptic billiard, is examined in detail. The link between elliptic fibrations and completely integrable Hamiltonian systems is also discussed. The book begins with a comprehensive overview of the subject, including QRT maps, singularity confinement, automorphisms of rational elliptic surfaces, action on homology classes, and periodic QRT maps. Later chapters cover these topics and more in detail. While QRT maps will be familiar to specialists in algebraic geometry, the present volume makes the subject accessible to mathematicians and graduate students in a classroom setting or for self-study.

Biographie:
Hans Duistermaat was a geometric analyst, who unexpectedly passed away in March 2010. His research encompassed many different areas in mathematics: ordinary differential equations, classical mechanics, discrete integrable systems, Fourier integral operators and their application to partial differential equations and spectral problems, singularities of mappings, harmonic analysis on semisimple Lie groups, symplectic differential geometry, and algebraic geometry. He was (co-)author of eleven books. Duistermaat was affiliated to the Mathematical Institute of Utrecht University since 1974 as a Professor of Pure and Applied Mathematics. During the last five years he was honored with a special professorship at Utrecht University endowed by the Royal Netherlands Academy of Arts and Sciences. He was also a member of the Academy since 1982. He had 23 PhD students.

Sommaire:
The QRT Map.- The Pencil of Biquadratic Curves in .- The QRT surface.- Cubic Curves in the Projective Plane.- The Action of the QRT Map on Homology.- Elliptic Surfaces.- Automorphisms of Elliptic Surfaces.- Elliptic Fibrations with a Real Structure.- Rational elliptic surfaces.- Symmetric QRT Maps.- Examples from the Literature.- Appendices.

From the reviews:

The book ... takes us on a tour of the field and provides us with a wealth of applications of various concepts coming from algebraic and analytic geometry to dynamical systems. ... the book gives an exhaustive analysis of discrete algebraically integrable maps in two dimensions. The bibliography contains over 200 references, and covers recent works as well as fundamental titles, making it an inescapable encyclopaedic reference on the subject. (Claude M. Viallet, Bulletin of the London Mathematical Society, Vol. 45 (4), August, 2013)

This is an excellent book for graduates and researchers who have some knowledge of integrable systems and a rudimentary understanding of algebraic geometry. ... Overall, this book delivers an excellent overview of a geometric interpretation of this class of discrete integrable systems, and is written in a manner that is fairly accessible to a postgraduate student or keen researcher in integrable systems. (C. M. Ormerod, G. R. W. Quispel and J. A. G. Roberts, SIAM Review, Vol. 54 (1), 2012)