Surveys in Modern Mathematics - The Independent University of Moscow Seminars

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. Independent University Of Moscow And Student Sessions At The Ium -- Mysterious Mathematical Trinities / V. I. Arnold -- Principle Of Topological Economy In Algebraic Geometry / V. I. Arnold -- Rational Curves, Elliptic Curves, And The Painleve Equation / Yu. I. Manin -- 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 -- Schrodinger Equation And Symplectic Geometry / S. P. Novikov -- Rings And Algebraic Varieties / M. Reid -- Billiard Table As A Playground For A Mathematician / A. B. Katok -- Fibonacci Numbers And Simplicity Of 2[superscript 127] -- 1 / A. N. Rudakov -- On Problems Of Computational Complexity / S. Smale -- Values Of The [zeta]-function / P. Cartier -- Combinatorics Of Trees / P. Cartier -- What Is An Operad? / P. Cartier -- Orbit Method Beyond Lie Groups : Infinite-dimensional Groups / A. A. Kirillov -- 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. Edited By Victor Prasolov, Yulij Ilyashenko. Includes Bibliographical References. 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 "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 an overview of contemporary mathematics."--BOOK JACKET