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Surveys in Modern Mathematics: The Independent University of Moscow Seminars

Victor Prasolov and Yulij Ilyashenko, editors
Cambridge University Press
Publication Date: 
Number of Pages: 
London Mathematical Society Lecture Notes Series 321
[Reviewed by
Fernando Q. Gouvêa
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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

The Fibonacci numbers and simplicity of 2127–1
A. N. Rudakov

On problems of computational complexity
S. Smale

Values of the ζ-function
P. Cartier

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