Visions in Mathematics: GAFA 2000 Special Volume, Part II pp. 455-983

Cover......Page 1
Visions in Mathematics, Part II......Page 4
ISBN 9783034604246......Page 5
Table of Contents......Page 7
Foreword
......Page 9
1 Introduction......Page 12
2.1 Combinatorial Nullstellensatz......Page 13
2.2 The dimension argument......Page 18
3 Probabilistic Methods......Page 20
4 The Algorithmic Aspects......Page 24
References......Page 25
CHALLENGES IN ANALYSIS......Page 28
1 Digital Transcriptions of Functions, Libraries of Waveforms......Page 29
2 Transcribing Dense Matrices for Efficient Computations......Page 33
References......Page 37
NONCOMMUTATIVE GEOMETRY YEAR 2000......Page 38
2 Geometry......Page 39
3 Quantum Mechanics......Page 42
4 Noncommutative Geometry......Page 45
5 A Basic Example......Page 47
6 Topology......Page 53
7 Differential Topology......Page 55
8 Calculus and Infinitesimals......Page 62
9 Spectral Triples......Page 67
10 Noncommutative 4-manifolds and the Instanton Algebra......Page 77
11 Noncommutative Spectral Manifolds......Page 80
12 Test with Space-time......Page 84
13 Operator Theoretic Index Formula......Page 86
14 Deffeomorphism Invariant Geometry......Page 88
15 Characteristic Classes for Actions of Hopf Algebras......Page 90
16 Hopf Algebras, Renormalization and the Riemann-Hilbert Problem......Page 92
17 Number Theory......Page 102
18 Appendix, the Cyclic Category......Page 110
References......Page 111
INTRODUCTION TO SYMPLECTIC FIELD THEORY......Page 117
1.1 Contact preliminaries......Page 120
1.2 Dynamics of Reeb vector fields......Page 123
1.3 Splitting of a symplectic manifold along a contact hypersurface......Page 124
1.4 Compatible almost complex structures......Page 126
1.5 Holomorphic curves in symplectic cobordisms......Page 127
1.6 Compactification of the moduli spaces MAg,r(r-,r+)......Page 129
1.7 Dimension of the moduli spaces MAg,r(r-,r+)......Page 135
1.8.1 Determinants......Page 139
1.8.2 Cauchy-Riemann type operators on closed surfaces......Page 140
1.8.3 A special class of Cauchy-Riemann type operators on punctured Riemann surfaces......Page 144
1.8.4 Remark about the coherent orientation for asymptotic operators with symmetries......Page 148
1.8.5 Coherent orientations of moduli spaces......Page 149
1.9.1 Recollection of finite-dimensional Floer theory......Page 150
1.9.2 Floer homology for the action functional......Page 155
1.9.3 Examples......Page 159
1.9.4  Relative contact homology and contact non-squeezing theorems......Page 161
2.1 Informal introduction......Page 164
2.2.1 Evaluation maps......Page 170
2.2.2 Correlators......Page 171
2.2.3 Three differential algebras......Page 172
2.3.1 Evaluation maps and correlators......Page 179
2.3.2 Potentials of symplectic cobordisms......Page 181
2.4 Chain homotopy......Page 186
2.5 Composition of cobordisms......Page 192
2.6 Invariants of contact manifolds......Page 198
2.7 A differential equation for potentials of symplectic cobordisms......Page 200
2.8 Invariants of Legendrian knots......Page 204
2.9.1 Dealing with torsion elements in H1......Page 208
2.9.2 Morse-Bott formalism......Page 209
2.9.3 Computing rational Gromov-Witten invariants of CPn......Page 218
2.9.4 Satellites......Page 222
References......Page 226
1 A Relationship Between Certain Vector Fields and a Holomorphic Curve Theory......Page 231
2 The Behavior of a Finite Energy Map Near a Puncture......Page 238
3 The Conley-Zehnder Index......Page 242
4 Holomorphic Curves and More Dynamics......Page 246
5 Finite Energy Foliations and Dynamics......Page 251
6 About Possible Generalizations to Other Manifolds......Page 253
References......Page 258
1 Introduction and Background......Page 262
2 Fundamental Conjectures......Page 269
3 Function Field Analogues......Page 274
4 Dirichlet L-Functions GL(1)/Q......Page 276
5 Special Values......Page 278
6 Subconvexity and Equidistribution......Page 281
7 GL(2) Tools......Page 286
8 Symmetry and Attacks on GRH......Page 291
References......Page 293
Introduction......Page 299
1.1 Radon's theorem and order types (oriented matroids)......Page 300
1.2 Tverberg's theorem......Page 301
1.4 The dimension of Tverberg's points......Page 302
1.5.2 Point configurations from graphs......Page 303
1.5.3 The four color theorem......Page 304
1.6.2 Eckhoff's partitions conjecture......Page 305
1.7 Some links and references......Page 306
2.1.2 The upper bound theorem......Page 307
2.2 Stanley-Reisner rings and their generic initial ideals (algebraic shifting)......Page 308
2.3.1 Witt spaces......Page 309
2.4.1 The Dehn-Sommerville relations......Page 310
2.4.3 Partial unimodality and the Braden-MacPherson theorem......Page 311
2.4.4 Other duality relations......Page 312
2.5.2 Triangulations of manifolds......Page 313
2.5.3 Neighborly embedded manifolds......Page 314
2.7 Some links and references......Page 315
3.1.1 The discrete cube......Page 316
3.1.2 Influence of variables......Page 317
3.1.5 Noise sensitivity......Page 319
3.2.1 Discrete isoperimetric inequalities......Page 320
3.3 Advanced theorems on influences......Page 321
3.5.2 Majority of majorities, tribes, runs......Page 322
3.5.6 General graph properties......Page 323
3.5.9 First passage percolation......Page 324
3.5.13 Linear objective functions......Page 325
4.1 Kirchhoff, Cayley, Kasteleyn and Tutte......Page 326
4.3 Random spanning trees II......Page 327
4.4 Random spanning trees III......Page 328
4.5 Higher dimensions......Page 329
4.7  Some links and references......Page 330
5.2 The combinatorics of linear programming......Page 331
5.3 Some classes of pivot rules......Page 333
5.4 Can geometry help?......Page 334
5.5 Can geometry help? II: How to distinguish geometric objective functions......Page 335
5.6 Some links and references......Page 336
References......Page 337
1 About the Subject......Page 349
2.1 Entropy and volume behavior in high dimension......Page 352
2.2 "Isomorphic" geometry......Page 353
2.2.1 Remarks......Page 355
2.3.1 Example of phase transition; Local form......Page 356
2.3.3......Page 357
2.3.4......Page 358
2.4 Approximation; what we expected from our old intuition and reality of the new one......Page 359
3.1 The standard form......Page 360
3.2 Metric G-spaces (X, p)......Page 364
3.4 Functional point of view......Page 365
4.1......Page 366
5 Some Open Problems of Asymptotic Geometic Analysis......Page 367
References......Page 370
1 Introduction......Page 373
2 Shannon Theory......Page 374
3 Quantum Mechanics......Page 375
4 Von Neumann Entropy......Page 380
5 Source Coding......Page 381
6 Accessible Information......Page 383
7 The Classical Capacity of a Quantum Channel......Page 386
8 Quantum Teleportation and Superdense Coding......Page 389
9 Other Results from Quantum Information Theory......Page 391
References......Page 394
1 Introduction......Page 396
2 Random Walk and Self-avoiding Walk......Page 398
3 Ising Model......Page 400
4 Lattice Field Models and Anharmonic Oscillators......Page 406
5 Random Schrodinger, Random Matrices and Supersymmetry......Page 408
References......Page 412
1 Introduction......Page 416
2 n-wave Equations and n-orthogonal Coordinate Systems......Page 419
3 Theory of Surfaces as a Chapter of Theory of Solitons......Page 425
4 Long-time Asymptotics in the Hamiltonian PDE Equation......Page 430
5 Briefly on Collapses......Page 434
References......Page 436
Introdution......Page 438
DISCUSSION on MATHEMATICAL PHYSICS......Page 440
DISCUSSION on GEOMETRY......Page 452
DISCUSSION on MATHEMATICS in the REAL WORLD......Page 472
DISCUSSION on COMPUTER SCIENCE and DISCRETE MATHEMATICS......Page 507
REFLECTIONS ON THE DEVELOPMENT OF MATHEMATICS IN THE 20TH CENTURY......Page 535