This collection of articles from the Independent University of Moscow is derived from the Globus seminars held there. They are given by world authorities, from Russia and elsewhere, in various areas of mathematics and are designed to introduce graduate students to some of the most dynamic areas of mathematical research. The seminars aim to be informal, wide-ranging and forward-looking, getting across the ideas and concepts rather than formal proofs, and this carries over to the articles here. Topics covered range from computational complexity, algebraic geometry, dynamics, through to number theory and quantum groups. The volume as a whole is a fascinating and exciting overview of contemporary mathematics.

1. The Independent University of Moscow and student sessions at the IUM 2. Mysterious mathematical trinities V. I. Arnold 3. The principle of topological economy in algebraic geometry V. I. Arnold 4. Rational curves, elliptic curves, and the Painlve equation Yu. I. Manin 5. The orbit method and finite groups A. A. Kirillov 6. On the development of the theory of dynamical systems during the past quarter century D. V. Anosov; New or 'renewed' directions; 'Named' problems; Some other achievements 7. Foundations of computational complexity theory A. A Razborov 8. The Schrodinger equation and symplectic geometry S. P. Novikov 9. Rings and algebraic varieties Miles Reid 10. Billiard table as a playground for a mathematician A. B. Katok 11. The Fibonacci numbers and simplicity of 2127 minus 1 A. N. Rudakov 12. On problems of computational complexity Stephen Smale 13. Values of the -function Pierre Cartier 14. Combinatorics of trees Pierre Cartier 15. What is an operad Pierre Cartier? 16. The orbit method beyond Lie groups A. A. Kirillov; Infinite-dimensional groups 17. The orbit method beyond Lie groups A. A. Kirillov; Quantum groups 18. Conformal mappings and the Ehitham equations I. M. Krichever 19. Projective differential geometry: old and new V. Yu. Ovsienko 20. 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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