Real and Complex Singularities

by Edited by M. Manoel , M. C. Romero Fuster , C. T. C. Wall

• ISBN13: 9780521169691
• ISBN10: 0521169690
• Format: Paperback
• Copyright: 2010-11-08
• Publisher: Cambridge University Press

Summary

The biennial meetings at São Carlos have helped create a worldwide community of experts and young researchers working on singularity theory, with a special focus on applications to a wide variety of topics in both pure and applied mathematics. The tenth meeting, celebrating the 60th birthdays of Terence Gaffney and Maria Aparecida Soares Ruas, was a special occasion attracting the best known names in the area. This volume contains contributions by the attendees, including three articles written or co-authored by Gaffney himself, and survey articles on the existence of Milnor fibrations, global classifications and graphs, pairs of foliations on surfaces, and Gaffney's work on equisingularity.

Table of Contents

Preface	p. xiii
On a conjecture	p. 1
Introduction	p. 1
Invariants of singularities	p. 2
Superisolated surface singularities	p. 5
Zariski pairs	p. 11
References	p. 15
On normal embedding of complex algebraic surfaces	p. 17
Introduction	p. 17
Proof	p. 18
References	p. 21
Local Euler obstruction, old and new, II	p. 23
Introduction	p. 23
Definition of the Euler obstruction	p. 26
Local Euler obstruction for a frame	p. 29
Euler obstruction and hyperplane sections	p. 30
The local Euler obstruction of a function	p. 32
The Euler obstruction and the Euler defect	p. 33
The Euler defect at general points	p. 34
The Euler obstruction via Morse theory	p. 34
Generalizations of Milnor number	p. 35
Bruce and Roberts' Milnor Number	p. 36
Local Euler obstruction of an analytic map	p. 38
Local Euler obstruction of 1-forms	p. 39
Local Chern obstruction of collections of 1-forms and special points	p. 41
References	p. 43
Branching of periodic orbits in reversible Hamiltonian systems	p. 46
Introduction	p. 46
Preliminaries	p. 49
Belitskii normal form	p. 55
Liapunov-Schmidt reduction	p. 55
Birkhoff normal form	p. 59
Two degrees of freedom	p. 60
Three degrees of freedom	p. 65
Examples	p. 68
References	p. 69
Topological invariance of the index of a binary differential equation	p. 71
Introduction	p. 71
Index of a binary differential equation	p. 72
Topological invariance of I(E, 0)	p. 75
Examples	p. 79
References	p. 80
About the existence of Milnor fibrations	p. 82
Introduction	p. 82
Milnor fibrations for complex maps	p. 86
Milnor fibrations for real maps	p. 91
References	p. 101
Counting hypersurfaces invariant by one-dimensional complex foliations	p. 104
Introduction	p. 104
Statement of results	p. 106
Examples	p. 109
Proofs	p. 110
References	p. 113
A note on topological contact equivalence	p. 114
Introduction	p. 114
Topological contact equivalence	p. 115
Properties and invariants	p. 115
References	p. 123
Bi-Lipschitz equivalence, integral closure and invariants	p. 125
Introduction	p. 125
The Lipschitz equisingularity of families of curves	p. 126
Bi-Lipschitz equisingularity of families of curves	p. 132
References	p. 136
Solutions to PDEs and stratification conditions	p. 138
Introduction	p. 138
The method of characteristics and W stratifications	p. 138
References	p. 145
Real integral closure and Milnor fibrations	p. 146
Introduction	p. 146
Notations and setup	p. 149
Some results	p. 150
References	p. 156
Surfaces around closed principal curvature lines, an inverse problem	p. 158
Introduction	p. 158
Preliminary results	p. 160
Hyperbolic principal cycles	p. 162
Concluding remarks	p. 165
References	p. 166
Euler characteristics and atypical values	p. 167
Cohomological results	p. 168
Absence of vanishing cycles	p. 174
Topological triviality	p. 178
References	p. 183
Answer to a question of Zariski	p. 185
Introduction	p. 185
Classification of the equisingularity class (7, 8)	p. 185
The question of Zariski	p. 186
References	p. 189
Projections of timelike surfaces in the de Sitter space	p. 190
Introduction	p. 190
Preliminaries	p. 191
The family of projections along geodesies in Sn1	p. 195
Projections of timelike surfaces in S31	p. 198
Duality	p. 205
Appendix	p. 208
References	p. 209
Spacelike submanifolds of codimension at most two in de Sitter space	p. 211
Introduction	p. 211
Lightcone Gauss images of spacelike hypersurfaces	p. 212
Lightcone height functions	p. 215
Generic properties and de Sitter hyperhorospheres	p. 217
Spacelike surfaces in de Sitter 3-space	p. 220
Spacelike submanifolds of codimension 2	p. 221
Lightlike hypersurfaces and contact with lightcomes	p. 223
Lightcone Gauss maps and lightcone height functions	p. 224
Classification in de Sitter 4-space	p. 226
References	p. 227
The geometry of Hopf and saddle-node bifurcations for waves of Hodgkin-Huxley type	p. 229
Introduction - equations of Hodgkin-Huxley type	p. 230
Linearisation	p. 233
The exchange of stability set	p. 235
The exchange of stability set in (HHW)	p. 238
Example: single channel Hodgkin-Huxley equations	p. 241
References	p. 244
Global classifications and graphs	p. 246
Introduction	p. 246
Surfaces and graphs	p. 247
3-Manifolds and graphs	p. 262
References	p. 264
Real analytic Milnor fibrations and a strong Lojasiewicz inequality	p. 268
Introduction	p. 268
Singular values of matrices	p. 271
L-maps and L-weights	p. 274
Milnor's conditions (a) and (b)	p. 282
The main theorem	p. 286
Comments and questions	p. 290
References	p. 291
An estimate of the degree of L-determinacy by the degree of A-determinacy for curve germs	p. 293
Introduction	p. 293
Proof of Theorem 1.1	p. 294
References	p. 297
Regularity of the transverse intersection of two regular stratifications	p. 298
Introduction	p. 298
Regular stratifications	p. 299
References	p. 303
Pairs of foliations on surfaces	p. 305
Introduction	p. 305
Examples from differential geometry	p. 307
Classification	p. 311
Invariants	p. 325
Bifurcations	p. 330
Appendix	p. 331
References	p. 334
Bi-Lipschitz equisingularity	p. 338
An oft-heard slogan	p. 338
Existence of Whitney and Verdier stratifications	p. 339
O-minimal structures	p. 340
Local triviality along strata	p. 341
Mostowski's Lipschitz stratifications	p. 342
Consequences of Valette's triangulation theorem	p. 344
Appendix. Definition of Lipschitz stratifications	p. 345
References	p. 347
Gaffney's work on equisingularity	p. 350
Introduction	p. 350
Early results on equisingularity	p. 351
Notations	p. 356
Integral closures	p. 357
Invariants	p. 362
Criteria for equisingularity	p. 366
Recent work	p. 369
Bibliography of Terry Gaffney's papers	p. 372
References	p. 374
Singularities in algebraic data acquisition	p. 378
Introduction	p. 378
Examples of moment inversion	p. 381
Sinularities in reconstruction of linear combinations of ¿-functions	p. 389
Noisy measurements	p. 393
References	p. 394
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