Book contents
- Frontmatter
- Contents
- The Independent University of Moscow and Student Sessions at the IUM
- Mysterious mathematical trinities
- The principle of topological economy in algebraic geometry
- Rational curves, elliptic curves, and the Painlevé equation
- The orbit method and finite groups
- On the development of the theory of dynamical systems during the past quarter century
- Foundations of computational complexity theory
- The Schrödinger equation and symplectic geometry
- Rings and algebraic varieties
- Billiard table as a playground for a mathematician
- The Fibonacci numbers and simplicity of 2127 – 1
- On problems of computational complexity
- Values of the ζ-function
- Combinatorics of trees
- What is an operad?
- The orbit method beyond Lie groups. Infinite-dimensional groups
- The orbit method beyond Lie groups. Quantum groups
- Conformal mappings and the Whitham equations
- Projective differential geometry: old and new
- Haken's method of normal surfaces and its applications to classification problem for 3-dimensional manifolds – the life story of one theorem
Billiard table as a playground for a mathematician
Published online by Cambridge University Press: 18 December 2009
- Frontmatter
- Contents
- The Independent University of Moscow and Student Sessions at the IUM
- Mysterious mathematical trinities
- The principle of topological economy in algebraic geometry
- Rational curves, elliptic curves, and the Painlevé equation
- The orbit method and finite groups
- On the development of the theory of dynamical systems during the past quarter century
- Foundations of computational complexity theory
- The Schrödinger equation and symplectic geometry
- Rings and algebraic varieties
- Billiard table as a playground for a mathematician
- The Fibonacci numbers and simplicity of 2127 – 1
- On problems of computational complexity
- Values of the ζ-function
- Combinatorics of trees
- What is an operad?
- The orbit method beyond Lie groups. Infinite-dimensional groups
- The orbit method beyond Lie groups. Quantum groups
- Conformal mappings and the Whitham equations
- Projective differential geometry: old and new
- Haken's method of normal surfaces and its applications to classification problem for 3-dimensional manifolds – the life story of one theorem
Summary
The title of this lecture can be understood in two ways. Literally, in a somewhat facetious sense: mathematicians are playing by launching billiard balls on tables of various forms and observing (and also trying to predict) what happens. In a more serious sense, the expression “playground” should be understood as “testing area”: various questions, conjectures, methods of solution, etc. in the theory of dynamical systems are “tested” on various types of billiard problems. I hope to demonstrate convincingly that at least the second interpretation deserves serious attention.
The literature concerning billiards is rather large, including scientific papers as well as monographs, textbooks, and popular literature. Short brochures by G. A. Galperin and A. N. Zemlyakov and by G. A. Galperin and N. I. Chernov are written in a rather accessible manner, and touch a broad circle of questions. An introduction to problems related with billiards for a more advanced reader is contained in Chapter 6 of the book. The next level is represented by a very well written book of S. Tabachnikov, whose publication in Russian is unfortunately delayed. The book by the author and B. Hasselblatt contains a rather detailed modern exposition of the theory of convex billiards and twisting maps. A serious but rather accessible exposition of modern state of the theory of parabolic billiards is contained in a survey paper by H. Masur and S. Tabachnikov which will be published (in English) in spring 2002. The collection of papers contains rich material on hyperbolic billiards and related questions. More special references will be given below during the exposition.
- Type
- Chapter
- Information
- Surveys in Modern Mathematics , pp. 216 - 242Publisher: Cambridge University PressPrint publication year: 2005
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