Visions in Mathematics (eBook)

Table of Contents 
6 
Foreword 
8 
Visions in Mathematics - Towards 2000 Scientific Program 
11 
GAUB-MANIN DETERMINANT CONNECTIONS AND PERIODS FOR IRREGULAR CONNECTIONS 
16 
1 Introduction 
16 
2 The Conjecture 
19 
3 The Fourier Transform 
27 
4 Kloosterman Sheaves 
32 
5 Periods 
37 
PROBLEMS IN HAMILTONIAN PDE'S 
47 
1 Introduction 
47 
2 Two Hamiltonian Models 
48 
3 The Cauchy Problem 
49 
4 Singularity Formation for NLS 
54 
5 Periodic Boundary Conditions 
57 
6 Invariant Measures 
59 
7 KAM Tori and Nekhoroshev Stability 
62 
References 
69 
ON A CLASSICAL LIMIT OF QUANTUM THEORY AND THE NON-LINEAR HARTREE EQUATION 
72 
1 General Perspective, Description of the Problem 
72 
2 Large Systems of Non-Relativistic Bosons 
77 
3 NL Hartree Equations as a Hamiltonian System 
82 
4 The Point-Particle Limit of the NL Hartree Equations with Attractive Two-Body Forces 
84 
5 More About Open Problems, Conclusions 
89 
References 
92 
ROUGH STRUCTURE AND CLASSIFICATION 
94 
1 Introduction 
94 
2 Will Mathematics Exist in 2009? 95
3 Rough Structure and Classification, and Related Problems 
105 
4 A Further Selection of Problems 
120 
References 
129 
SPACES AND QUESTIONS 
133 
1 Dawn of Space 
133 
2 Spirit of Riemann 
136 
3 K > <
145 
4 Life Without Metric 
156 
5 Symbolization and Randomization 
170 
References 
174 
CLASSIFICATION OF INFINITE-DIMENSIONAL SIMPLE GROUPS OF SUPERSYMMETRIES AND QUANTUM FIELD THEORY 
177 
Introduction 
177 
1 Lie's Problem and Cartan's Theorem 
178 
2 Statement of the Main Theorem 
180 
3 Explanation of the Proof of Theorem 1 
185 
4 Construction of Exceptional Linearly Compact Lie Superalgebras 
187 
5 Classification of Superconformal Algebras 
189 
6 Representations of Linearly Compact Lie Superalgebras 
191 
7 Fundamental Particle Multiplets 
192 
8 Speculations and Visions 
194 
References 
196 
GEOMETRIZATION IN REPRESENTATION THEORY 
199 
References 
202 
GEOMETRIC AND UNIPOTENT CRYSTALS 
203 
1 Introduction 
204 
2 Definitions and Main Results on Geometric Crystals 
207 
3 Unipotent Crystals 
214 
4 Examples of Unipotent Crystals 
221 
5 Proofs 
228 
6 Projections of Bruhat Cells to the Parabolic Subgroups 
241 
7 Appendix: Combinatorial Pre-crystals and W -crystals 
248 
References 
250 
Y-FUNCTIONS OF REPRESENTATIONS AND LIFTING 252
1 Introduction and Statement of the Results 
253 
2 Lifting 
259 
3 Gamma-functions and Bernstein's Center 
261 
4 The Case of Split Tori 
265 
5 Local Spaces of Functions 
269 
6 The Poisson Summation Formula 
273 
7 ''Algebraic'' Integrals over Local Fields 
274 
8 The Case of G(m, n) and its Applications 
278 
9 The Case of Finite Fields 
284 
10 Appendix (by V . Vologodsky) 
290 
References 
292 
PDE AS A UNIFIED SUBJECT 
294 
Introduction 
294 
Between Mathematics and Physics 
300 
The Main Equations 
307 
The Problem of Breakdown 
314 
The Problem of Well-posedness for Nonlinear Epuations 
319 
References 
327 
LESSONS FOR TURBULENCE 
331 
1 Navier-Stokes Equations and Turbulence 
331 
2 The Initial Value Problem 
333 
3 Statistical Hydrodynamics 
334 
4 Universality 
335 
5 Random Advection 
340 
References 
347 
THE MATHEMATICS OF THE SECOND LAW OF THERMODYNAMICS 
349 
1 A Guide to Entropy and the Second Law of Thermodynamics [LiY2] 
350 
2 Some Speculations and Open Problems 
369 
References 
372 
DISCRETE AND CONTINUOUS: TWO SIDES OF THE SAME? 
374 
1 Introduction 
374 
2 Discrete in Continuous and Continuous in Discrete 
377 
3 Optimization Discrete, Linear, Semidefinite
380 
4 Discrepancy Theory 
385 
5 The Laplacian 
387 
References 
394 
PYTHAGOREAN AND PLATONIC CONCEPTIONS IN XXTH CENTURY PHYSICS 
398 
1 Introduction: Why Start in Greece? 
398 
2 Greek ''Meta-science'' as Distinguished from Greek Science 
400 
3 Pythagoreanisms, as induced by Quantum Mechanics 
402 
4 The First Geometrization (Relativity): Global (1905) and Local (1915) Phases 
405 
5 Extending the Geometry so as to Include Local ''Internal'' Symmetries (Creation of the Means, in an Early Attempt at Unification) 
406 
6 The Second Geometrization (Gauge Theories): Global and Quasi-local Phase (1955-1966) 
408 
7 The Second Geometrization's Local Phase: (a) QCD (1973) 
410 
8 The Second Geometrization's Local Phase: (b) the Electroweak Theory (1967-1982) 
412 
9 The Standard Model and Beyond: Towards a Third Geometrization? 
414 
References 
417 
1. CLASSICAL AND MODERN TOPOLOGY 2. TOPOLOGICAL PHENOMENA IN REAL WORLD PHYSICS 
421 
1 Classical and Modern Topology 
421 
2 Topological Phenomena in Real World Physics 
427 
References 
438 
SOME PROBLEMS IN THE THEORY OF DYNAMICAL SYSTEMS AND MATHEMATICAL PHYSICS 
440 
1 Generically Quasi-periodic and Hyperbolic Dynamics are the Only Structurally Stable Types of Dynamics 
440 
2 Qualitative Theory of PDE and Infinite-Dimensional Systems 
444 
3 Quantum Chaos and Anderson Localization 
445 
References 
447 
SOME GEOMETRICAL CONCEPTS ARISING IN HARMONIC ANALYSIS 
449 
1 ''Classical Theory'' 
450 
2 Extensions The Setting
452 
3 Control Metric and Associated Singular Intergrals 
454 
4 The Conflict of Metrics First Example
456 
5 The Conflict of Metrics Second Example
458 
6 A New Viewpoint 
460 
7 A New Viewpoint, II 
462 
8 Some Aspects of a General Theory 
464 
References 
466