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.
Author(s): Victor Prasolov, Yulij Ilyashenko
Series: London Mathematical Society Lecture Note Series 321
Publisher: Cambridge University Press
Year: 2005
Language: English
Pages: 361
Cover......Page 1
LONDON MATHEMATICAL SOCIETY LECTURE NOTE SERIES......Page 2
Surveys in Modern Mathematics......Page 4
Copyright......Page 5
Contents......Page 6
The Independent University of Moscow and Student Sessions at the IUM......Page 8
Mysterious mathematical trinities......Page 14
The Ternarity Phenomenon......Page 20
The principle of topological economy in algebraic geometry......Page 26
Rational curves, elliptic curves, and the Painleve equation......Page 37
Bibliography......Page 46
1......Page 47
2......Page 67
Preface......Page 83
1.1 Symplectic geometry......Page 87
1.2 Conformal dynamics......Page 96
1.3 Nonclassical transformation groups......Page 102
1.4 Bifurcations......Page 124
2.1 Structurally stable systems......Page 139
2.2 Hilbert’s 21st problem......Page 144
2.3 The Dulac conjecture......Page 149
2.4 Homogeneous flows and the Raghunathan conjecture......Page 152
2.5 The Seifert conjecture......Page 156
3.1......Page 159
3.2 The theory of singular perturbations......Page 162
3.5 Integrable and nonintegrable systems......Page 168
3.6 The Conley theory......Page 174
3.7 Singularities in the n-body problem......Page 176
3.8 Stable ergodicity......Page 177
3.9 “Abstract” (purely metric) ergodic theory......Page 178
3.10 New periodic trajectories in the three-body problem......Page 187
References......Page 188
1 Prehistory......Page 199
2 Point of departure......Page 201
3 Foundations of the theory: basic notions......Page 202
4 The speed-up theorem......Page 204
5 Complexity classes......Page 205
6 The hierarchy theorem and complexity of elementary geometry......Page 207
7 Reducibility and completeness......Page 208
8 Are all polynomial algorithms good?......Page 210
9 Nondeterministic computations......Page 211
Acknowledgments......Page 215
1 Quantum scattering......Page 216
2 Schrodinger operators on graphs......Page 218
Rings and algebraic varieties......Page 224
Applications......Page 227
1 Elliptic, parabolic, and hyperbolic phenomena in dynamics......Page 229
2 Billiards in smooth convex domains......Page 234
3 Parabolic behavior: billiards in polygons......Page 243
4 Hyperbolic behavior: billiards of Sinai, Bunimovich, Wojtkowski and other authors......Page 250
References......Page 255
2 The Fibonacci numbers and the main theorem......Page 256
3 Complex numbers in finite arithmetics......Page 258
4 Complex conjugation for numbers modulo p......Page 260
5 The square root of 5 modulo p......Page 262
6 Proof of the main theorem......Page 264
7 Organization of computation. Examples......Page 265
References......Page 267
On problems of computational complexity......Page 268
Values of the ζ-function......Page 273
2 Polylogarithmic functions......Page 277
3 Generalizations of polylogarithmic functions......Page 280
4 Conjectures on the nature of some numbers......Page 284
Combinatorics of trees......Page 287
1 The universal algebra......Page 290
2 The Stasheff polyhedra......Page 291
3 The space of (real) configurations......Page 294
1 What is an operad?......Page 296
2 (Co)homological operations......Page 301
3 Returning to the configuration space......Page 303
4 Conclusion: physics......Page 304
The orbit method beyond Lie groups. Infinite-dimensional groups......Page 305
The orbit method beyond Lie groups. Quantum groups......Page 318
Conformal mappings and the Whitham equations......Page 329
1 Symmetry groups and differential invariants......Page 341
2 The Schwarz derivative......Page 343
3 A remarkable property of curvatures......Page 345
4 The Ghys theorem and the zeros of the Schwarz derivative......Page 346
5 The relation to the Lorentzian geometry......Page 347
6 The decisive intervention of projective geometry......Page 348
7 Discretization......Page 349
Acknowledgments......Page 350
Haken’s method of normal surfaces and its applications to classification problem for 3-dimensional manifolds – the life story of one theorem......Page 351
