Woods Hole Mathematics: Perspectives in Mathematics and Physics

Nils Tongring, R. C. Penner9789812560216, 9812560211

These eight papers have been presented by mathematicians and physicists addressing special meetings at the Woods Hole Oceanographic Institution starting in 1998. The papers describe contemporary mathematics and physics, but they also describes such topics of interest to oceanography as the secondary structure of proteins and an overview of arc complexes with proposed applications to macromolecular folding. Topics include quantizing Teichmuller spaces using graphs, indices and relative indices on contact and CR- manifolds, and biologic. They include operads, moduli of surfaces and quantum algebras; fragments of nonlinear Grothendieck- Teichmuller theory, and cell decomposition; and compactification of Riemann’s moduli space in decorated Teichmuller theory. The collection concludes with papers on spatial intermittence in two-dimensional turbulence using a wavelet approach and an elementary definition of Brownian motion in Hilbert space.

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Table of contents :
PREFACE……Page 6
INTRODUCTION……Page 8
CONTENTS……Page 14
1. Introduction……Page 16
2. Classical Teichmüller spaces……Page 17
2.1. Poisson algebra of geodesics and classical skein relations……Page 26
2.2. General Poisson algebras of geodesics……Page 28
3. Quantization……Page 29
3.1. Geodesic length operators……Page 35
3.2. Algebra of quantum geodesics……Page 36
3.3. Quantizing the Nelson–Regge algebras……Page 39
References……Page 40
1. Fredholm operators and Toeplitz operators on the circle……Page 42
2. Contact and CR-manifolds……Page 48
3. CR-functions and a generalization of Toeplitz operators……Page 55
4. Pseudodifferential operators, symbols and radial compactification……Page 56
5. Parabolic compactifications for contact manifolds……Page 63
6. The Heisenberg calculus……Page 66
7. Application of the Heisenberg calculus to several complex variables……Page 72
8. The quantum harmonic oscillator and b……Page 75
9. Fields of harmonic oscillators……Page 78
10. Vector bundle coefficients and the Atiyah-Singer index theorem……Page 84
11. The Boutet de Monvel index formula……Page 89
12. An index theorem for the Heisenberg calculus……Page 92
13. The structure of the higher eigenprojections……Page 94
14. Grauert tubes and the Atiyah-Singer index theorem……Page 100
15. The contact degree and index of FIOs……Page 102
Acknowledgments……Page 106
References……Page 107
1. Introduction……Page 109
2. Replication of DNA……Page 111
3. Logic, Copies and DNA Replication……Page 113
4. Lambda Algebra – Replication Revisited……Page 116
5. Quantum Mechanics……Page 118
5.1. Quantum Formalism and DNA Replication……Page 120
5.2. Quantum Copies are not Possible……Page 121
6. Mathematical Structure and Topology……Page 122
6.1. Projectors and Meanders……Page 131
6.2. Protein Folding and Combinatorial Algebra……Page 134
7. Cellular Automata……Page 138
7.1. Other Forms of Replication……Page 140
8. Epilogue – Logic and Biology……Page 143
References……Page 146
1. Introduction……Page 148
3. Trees……Page 153
3.2. Structures on Rooted Trees……Page 154
3.4. Structures on Planar Trees……Page 155
3.7. Black and White Trees……Page 156
3.8. Notation I……Page 157
3.11. The Map cppin : Tr Tppbp……Page 158
4. Operads……Page 159
4.1. Operads……Page 160
4.2. Induced Operads……Page 161
4.4. The Operad of Functions……Page 162
4.5. Rooted Leaf Labelled Trees……Page 163
4.6. Bordered Surfaces and Corollas……Page 165
4.7. Tree Insertion Operads……Page 166
4.9 Other Tree Insertion Operads and Compatibilities……Page 168
4.10. Variations of Operads……Page 169
4.11. Algebras Over Operads……Page 170
4.13. The Operad for Commutative Algebras……Page 171
4.15. The Operad for Gerstenhaber Algebras……Page 172
4.16. The Operad for Batalin-Vilkovisky (BV) Algebras……Page 173
4.17. The Pre–Lie Operad……Page 174
4.19. Operads of Moduli Spaces of Curves……Page 175
5.1. The space……Page 176
5.3. Several Models for Arcs……Page 178
5.5. Glueing Weighted Arc Families…….Page 180
5.6. A Pictorial Representation of the Glueing……Page 182
5.9. Suboperads and PROPS……Page 183
5.10. Linearity Condition……Page 184
5.12. Relation to Moduli Spaces……Page 185
5.13. Arc Families and their Induced Operations…….Page 187
5.14. The BV Operator……Page 189
5.15. The Associator……Page 190
6.1. Configurations of Loops and their Graphs……Page 194
6.3. Glueing for Cacti……Page 195
6.4. The Chord Diagram and Planar Planted Tree of a Cactus……Page 196
6.6. Gluing for Normalized Cacti……Page 197
6.8. Left, Right and Symmetric Cacti Operads……Page 199
6.10. Framing of a Cactus……Page 200
6.12. The Boundary Circles……Page 201
6.13. The Equivalence Relations Induced by Arcs……Page 202
6.14. From Loops to Arcs……Page 203
6.16. Comments on an Action on Loop Spaces……Page 205
7. Little Discs, Spineless Cacti and the Cellular Chains of Normalized Spineless Cacti……Page 206
7.3. The Perturbed Compositions……Page 207
7.4. The Perturbed Multiplications in Terms of an Action……Page 208
7.5. Cact(i) and the (Framed) Little Discs Operad……Page 209
7.6. A Cell Decomposition for Spineless Cacti……Page 210
7.8. The Differential on Tpp,ntbp……Page 212
7.10. The Action of the Symmetric Group……Page 214
8. Structures on Operads and Meta–Operads……Page 215
8.1. The Universal Concatenations……Page 216
8.2. The pre-Lie Structure of an Operad……Page 217
8.3. The Insertion Operad……Page 218
8.4. Notation……Page 219
8.5. The Hopf Algebra of an Operad……Page 220
9.3. The Gerstenhaber Structure……Page 221
9.5. The Operation of CC*(Cact1) on HomCH……Page 222
9.6. Signs for the Braces……Page 223
9.8. Another Approach to Signs and Actions……Page 224
9.10. Natural Operations on CH* and their Tree Depiction……Page 225
9.12. The Differential……Page 226
9.15. Deligne’s Conjecture……Page 227
10.2. The Top Dimensional Cells of Spineless Cacti and the Pre–Lie Operad……Page 228
10.5. Operad Algebras and a Generalized Deligne Conjecture……Page 230
10.6. Differential on Trees with Tails……Page 231
10.7. A Cyclic Version of Deligne’s Conjecture……Page 232
11.2. A Putative Cell Decomposition……Page 233
11.4. Relations to the Fulton–MacPherson Compactification……Page 234
11.5. Actions of Arc……Page 235
References……Page 236
1. Introduction……Page 240
2. A Short Reminder and A Reading Guide……Page 241
3. Glimpses of the motivic theory……Page 248
4.1. Profinite versus pronilpotent:……Page 252
4.2. Group actions versus linear representations:……Page 256
4.3. Good groups versus rigid ones:……Page 258
4.4. Amalgamation versus extension:……Page 261
4.5. All genera versus genus 0:……Page 266
4.6. Stack inertia versus inertia at infinity:……Page 267
Appendix: Belyi’s theorem and ‘dessins d’enfant’……Page 270
References……Page 274
Introduction……Page 278
1. Definitions and cell decomposition for punctured surfaces……Page 281
2. Coordinates on Teichmüller space for punctured surfaces……Page 284
3. Bordered surfaces……Page 290
4. The arc complex of a bordered surface……Page 293
5. Sphericity……Page 294
6. Punctured surfaces and fatgraphs……Page 304
7. Operads……Page 307
Appendix. Biological Applications……Page 310
Bibliography……Page 315
1. Introduction……Page 317
2. Classical methods for studying intermittency……Page 321
3.1. Orthogonal wavelet transform……Page 323
3.2. Wavelet spectra……Page 324
3.4. Wavelet intermittency measures……Page 325
3.5. Relation to structure functions……Page 326
4.1. Classical statistical analysis……Page 327
4.2. Wavelet statistical analysis……Page 332
4.3. Extended self-similarity……Page 337
5. Conclusion……Page 338
Appendix……Page 339
References……Page 341
1. Introduction……Page 344
References……Page 357
SERIES ON KNOTS AND EVERYTHING……Page 358