Soc. Centennial Pubs - Math into the 21st Century

Half-Title......Page 1
Frontispiece......Page 2
Title......Page 3
Copyright......Page 4
Contents......Page 5
Preface......Page 7
Introduction......Page 8
Symposium Speakers......Page 10
Instantons and their relatives by Karen Uhlenbeck......Page 0
Representations of Finite Groups as Permutation Groups by Michael Aschbacher......Page 18
References......Page 23
Elliptic equations and interior regularity......Page 24
Free boundary problems and harmonic analysis in Lipschitz domains......Page 27
The common setting......Page 29
Introduction to sufficiency......Page 31
Basic results of sufficiency......Page 33
Introduction to exchangeability and equivalence of ensembles......Page 35
Sufficiency and exchangeability......Page 38
Bibliography......Page 40
Atoms and analytic number theory by C. Fefferman......Page 43
References......Page 52
Working and playing with the 2-disk by Michael H. Freedman......Page 53
References......Page 62
The Incompleteness Phenomena by Harvey Friedman......Page 64
1. General incompleteness phenomena......Page 67
2. Independence results via forcing......Page 68
3. The constructible point of view......Page 71
4. The Borel measurable point of view......Page 74
5. Cantor's theorem and the discrete topology......Page 77
6. Borel diagonalization on equivalence relations, linear orders, groups, and graphs......Page 79
7. Strong Borel diagonalization on groups and graphs......Page 82
8. The predicative point of view......Page 85
9. Finite trees and finite graphs......Page 88
10. Finite Ramsey theory......Page 94
Appendix......Page 96
References......Page 98
Elliptic curves and modular forms by Benedict H. Gross......Page 100
Bibliographic Remarks......Page 103
Developments in algebraic geometry by Joe Harris......Page 104
The classical period......Page 107
"Abstract" algebraic geometry......Page 109
Schemes......Page 112
1.......Page 116
1.1......Page 117
1.2......Page 120
1.3......Page 123
1.4......Page 125
1.5......Page 127
Endnotes......Page 130
2.1......Page 133
2.2......Page 136
2.3......Page 137
2.4......Page 140
2.5......Page 142
2.6......Page 145
2.7......Page 146
2.8......Page 147
2.9......Page 150
2.10......Page 151
2.11......Page 155
2.13 Remarks......Page 156
Endnotes......Page 161
3.1 An example: The quantum harmonic oscillator......Page 164
3.2 The orbit method......Page 168
3.3 Nilpotent groups......Page 175
3.4 Solvable groups......Page 178
3.5 Compact groups......Page 184
3.6 Noncompact semisimple groups......Page 219
4.1 Combinatorics......Page 248
4.2 Automorphic forms......Page 261
4.3 Physics, geometry and differential equations......Page 274
Appendix 1. Basic concepts of representation theory......Page 292
Appendix 2. Structure of real semisimple Lie algebras and Lie groups......Page 306
Acknowledgments......Page 314
References......Page 315
From quantum theory to knot theory and back: a von Neumann algebra excursion by V.F.R. Jones......Page 341
Subfactors......Page 344
Braids, knots, and links......Page 346
The Kauffman polynomial......Page 349
Statistical mechanical models......Page 350
Quantum invariant theory......Page 352
Quantum field theory......Page 354
References......Page 355
1. Modular functions......Page 357
2. Infinite-dimensional Lie algebras......Page 358
3. Positive energy representations......Page 359
5. Modular invariant representations of the affine algebra.........Page 360
6. The example of sl_2......Page 361
7. Some applications......Page 362
8. Modular invariant representations of Vir......Page 365
9. The Ising model......Page 367
10. Modular invariance versus unitarity......Page 368
Bibliography......Page 369
Contents......Page 371
Introduction......Page 372
1. The elementary mathematical structure of fluid flows......Page 373
2. Physical phenomena and mathematical theory for nonlinear sound waves..........Page 378
3. Vorticity waves and turbulence......Page 392
4. Examples of the interaction between large scale computing and modern mathematical theory.........Page 395
Bibliography......Page 412
The equations of cardiac fluid dynamics......Page 415
The issue of numerical stability and the computation of the elastic force......Page 420
The equations of Newtonian molecular dynamics......Page 424
The backward-Euler Langevin method......Page 427
The choice of parameters......Page 429
A connection with quantum statistical mechanics......Page 430
Conclusions......Page 433
References......Page 434
Introduction......Page 436
1. Poincare length distortion and smoothness class one plus Zygmund......Page 445
2. The Koebe distortion argument of...........Page 449
3. The a priori real bounds......Page 452
4. Renormalization limits and schlicht mappings--the Epstein class......Page 456
5. Composition of roots and the sector theorem......Page 458
6. The factoring of the sector theorem......Page 461
7. The sector inequality......Page 463
8. The complex quadratic-like mapping produced by renormalization......Page 465
9. The Douady-Hubbard theory and Riemann surface laminations......Page 467
10. The modulus function on external classes......Page 469
11. Thurston equivalences and the pull back conjugacy......Page 470
12. Renormalization of complex quadratic-like mappings......Page 472
13. Teichmuller contraction of renormalization for symmetric complex quadratic-like mappings......Page 473
14. Proof of theorem 2'......Page 475
15. Proof of theorem 2......Page 476
Appendix. Riemann surface laminations and their Teichmuller theory......Page 477
References......Page 482
Introduction and a glimpse of history......Page 486
1. The birth of gauge theory......Page 487
2. What is an instanton......Page 489
3. Abelian vortices......Page 491
4. And if there is a twenty-first century.........Page 493
References......Page 496
1. Introduction......Page 497
2. Physical Hilbert spaces and transition amplitudes......Page 499
3. Axioms of quantum field theory......Page 502
4. Construction of quantum field theories......Page 505
5. Conclusion......Page 507
References......Page 508