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Publisher:

Cambridge University Press

Publication Date:

2005

Number of Pages:

348

Format:

Paperback

Series:

London Mathematical Society Lecture Notes Series 321

Price:

70.00

ISBN:

0-521-54793-8

Category:

Anthology

[Reviewed by , on ]

Fernando Q. Gouvêa

09/18/2005

This volume is a collection of articles based on lectures given at the Independent University of Moscow, a fairly new institution founded in 1991. The seminars, originally called the "Student Sessions", began in 1997. For the first few years (1997 to 2000), the lectures were aimed at a very broad audience, from mathematics students to professional researchers, and so had the character of surveys. These are the lectures collected in this volume.

(After 2000, the "Student Sessions" were renamed "Globus" and became, according to the introduction, "a regular mathematics research seminar". I can't help but feel that this was a loss to the mathematical world. Volumes collecting the Globus lectures are planned.)

Rather than asking the speakers to write up their talks, the lectures at the Student Sessions were videotaped, transcribed, translated into English, and provided to the authors for editing; as a result, what we have here is quite close to the spoken word. The opening lecture by V. I. Arnold, for example, is entitled "Mysterious Mathematical Trinities". It opens thus: "I shall try to tell about some phenomena in mathematics that make me surprised." That's quite a hook!

Mathematical articles of this kind are rare, and here we have a collection of such articles by some very big names. In addition to Arnold, the list of authors includes A. Kirillov, D. V. Anosov, S. P. Novikov, S. Smale, and P. Cartier. And the topics range far and wide, from the sedate "On the development of the theory of dynamical systems during the past quarter century" (Anosov) to the enticing "Billiard table as a playground for a mathematician" (A. B. Katok).

As that last title suggests, some Russianisms remain in the translation, and the level of the essays varies widely. But any problems are far outweighed by the overall interest of these surveys. The editors have done a heroic job of preserving the "feel" of the lecture hall, and the authors have useful insights to share. This one is very much worth a look.

Fernando Q. Gouvêa is Professor of Mathematics at Colby College.

*The Independent University of Moscow and Student Sessions at the IUM*

**Mysterious mathematical trinities***V. I. Arnold*

**The principle of topological economy in algebraic geometry***V. I. Arnold*

**Rational curves, elliptic curves, and the Painlevé equation***Yu. I. Manin*

**The orbit method and finite groups***A. A. Kirillov*

**On the development of the theory of dynamical systems during the past quarter century***D. V. Anosov*

**Foundations of computational complexity theory***A. A. Razborov*

**The Schrödinger equation and symplectic geometry***S. P. Novikov*

**Rings and algebraic varieties***M. Reid*

**Billiard table as a playground for a mathematician***A. B. Katok*

**On problems of computational complexity***S. Smale*

**Values of the ζ-function**

**Combinatorics of trees***P. Cartier*

**What is an operad?***P. Cartier*

**The orbit method beyond Lie groups. Infinite-dimensional groups***A. A. Kirillov*

**The orbit method beyond Lie groups. Quantum groups***A. A. Kirillov*

**Conformal mappings and the Whitham equations***I. M. Krichever*

**Projective differential geometry: old and new***V. Yu. Ovsienko*

**Haken’s method of normal surfaces and its applications to classification problem for 3-dimensional manifolds – the life story of one theorem***S. V. Matveev*

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