Graphs on Surfaces and Their Applications

by Sergei K. Lando; Alexander K. Zvonkin

• ISBN13: 9783540383611
• ISBN10: 3540383611
• Format: Nonspecific Binding
• Copyright: 2013-04-17
• Publisher: Springer Nature

Summary

Graphs drawn on two-dimensional surfaces have always attracted researchers by their beauty and by the variety of difficult questions to which they give rise. The theory of such embedded graphs, which long seemed rather isolated, has witnessed the appearance of entirely unexpected new applications in recent decades, ranging from Galois theory to quantum gravity models, and has become a kind of a focus of a vast field of research. The book provides an accessible introduction to this new domain, including such topics as coverings of Riemann surfaces, the Galois group action on embedded graphs (Grothendieck's theory of "dessins d'enfants"), the matrix integral method, moduli spaces of curves, the topology of meromorphic functions, and combinatorial aspects of Vassiliev's knot invariants and, in an appendix by Don Zagier, the use of finite group representation theory. The presentation is concrete throughout, with numerous figures, examples (including computer calculations) and exercises, and should appeal to both graduate students and researchers.

Table of Contents

Introduction: What is This Book About	p. 1
New Life of an Old Theory	p. 1
Plan of the Book	p. 2
What You Will Not Find in this Book	p. 4
Constellations, Coverings, and Maps	p. 7
Constellations	p. 7
Ramified Coverings of the Sphere	p. 13
First Definitions	p. 13
Coverings and Fundamental Groups	p. 15
Ramified Coverings of the Sphere and Constellations	p. 18
Surfaces	p. 22
Maps	p. 26
Graphs Versus Maps	p. 26
Maps: Topological Definition	p. 28
Maps: Permutational Model	p. 33
Cartographic Groups	p. 39
Hypermaps	p. 43
Hypermaps and Bipartite Maps	p. 43
Trees	p. 45
Appendix: Finite Linear Groups	p. 49
Canonical Triangulation	p. 50
More Than Three Permutations	p. 55
Preimages of a Star or of a Polygon	p. 56
Cacti	p. 57
Preimages of a Jordan Curve	p. 61
Further Discussion	p. 63
Coverings of Surfaces of Higher Genera	p. 63
Ritt's Theorem	p. 65
Symmetric and Regular Constellations	p. 68
Review of Riemann Surfaces	p. 70
Dessins d'Enfants	p. 79
Introduction: The Belyi Theorem	p. 79
Plane Trees and Shabat Polynomials	p. 80
General Theory Applied to Trees	p. 80
Simple Examples	p. 88
Further Discussion	p. 94
More Advanced Examples	p. 101
Belyi Functions and Belyi Pairs	p. 109
Galois Action and Its Combinatorial Invariants	p. 115
Preliminaries	p. 115
Galois Invariants	p. 118
Two Theorems on Trees	p. 123
Several Facets of Belyi Functions	p. 126
A Bound of Davenport-Stothers-Zannier	p. 126
Jacobi Polynomials	p. 131
Fermat Curve	p. 135
The abc Conjecture	p. 137
Julia Sets	p. 139
Pell Equation for Polynomials	p. 142
Proof of the Belyi Theorem	p. 146
The "Only If" Part of the Belyi Theorem	p. 146
Comments to the Proof of the "Only If" Part	p. 147
The "If", or the "Obvious" Part of the Belyi Theorem	p. 150
Introduction to the Matrix Integrals Method	p. 155
Model Problem: One-Face Maps	p. 155
Gaussian Integrals	p. 160
The Gaussian Measure on the Line	p. 160
Gaussian Measures in <$>{\op R}^k<$>	p. 162
Integrals of Polynomials and the Wick Formula	p. 163
A Gaussian Measure on the Space of Hermitian Matrices	p. 164
Matrix Integrals and Polygon Gluings	p. 167
Computing Gaussian Integrals. Unitary Invariance	p. 171
Computation of the Integral for One Face Gluings	p. 176
Matrix Integrals for Multi-Faced Maps	p. 179
Feynman Diagrams	p. 179
The Matrix Integral for an Arbitrary Gluing	p. 180
Getting Rid of Disconnected Graphs	p. 183
Enumeration of Colored Graphs	p. 185
Two-Matrix Integrals and the Ising Model	p. 185
The Gauss Problem	p. 188
Meanders	p. 190
On Enumeration of Meanders	p. 191
Computation of Matrix Integrals	p. 192
Example: Computing the Volume of the Unitary Group	p. 192
Generalized Hermite Polynomials	p. 195
Planar Approximations	p. 197
Korteweg-de Vries (KdV) Hierarchy for the Universal One-Matrix Model	p. 199
Singular Behavior of Generating Functions	p. 200
The Operator of Multiplication by ¿ in the Double Scaling Limit	p. 202
The One-Matrix Model and the KdV Hierarchy	p. 204
Constructing Solutions to the KdV Hierarchy from the Sato Grassmanian	p. 206
Physical Interpretation	p. 210
Mathematical Relations Between Physical Models	p. 211
Feynman Path Integrals and String Theory	p. 211
Quantum Field Theory Models	p. 213
Other Models	p. 214
Appendix	p. 215
Generating Functions	p. 215
Connected and Disconnected Objects	p. 217
Logarithm of a Power Series and Wick's Formula	p. 219
Geometry of Moduli Spaces of Complex Curves	p. 223
Generalities on Nodal Curves and Orbifolds	p. 223
Differentials and Nodal Curves	p. 223
Quadratic Differentials	p. 226
Orbifolds	p. 227
Moduli Spaces of Complex Structures	p. 232
The Deligne-Mumford Compactification	p. 234
Combinatorial Models of the Moduli Spaces of Curves	p. 237
Orbifold Euler Characteristic of the Moduli Spaces	p. 243
Intersection Indices on Moduli Spaces and the String and Dilaton Equations	p. 249
KdV Hierarchy and Witten's Conjecture	p. 256
The Kontsevich Model	p. 257
A Sketch of Kontsevich's Proof of Witten's Conjecture	p. 263
The Generating Function for the Kontsevich Model	p. 263
The Kontsevich Model and Intersection Theory	p. 264
The Kontsevich Model and the KdV Equation	p. 266
Meromorphic Functions and Embedded Graphs	p. 269
The Lyashko-Looijenga Mapping and Rigid Classification of Generic Polynomials	p. 270
The Lyashko-Looijenga Mapping	p. 270
Construction of the LL Mapping on the Space of Generic Polynomials	p. 271
Proof of the Lyashko-Looijenga Theorem	p. 273
Rigid Classification of Nongeneric Polynomials and the Geometry of the Discriminant	p. 277
The Discriminant in the Space of Polynomials and Its Stratification	p. 277
Statement of the Enumeration Theorem	p. 279
Primitive Strata	p. 280
Proof of the Enumeration Theorem	p. 282
Rigid Classification of Generic Meromorphic Functions and Geometry of Moduli Spaces of Curves	p. 288
Statement of the Enumeration Theorem	p. 288
Calculations: Genus 0 and Genus 1	p. 289
Cones and Their Segre Classes	p. 292
Cones of Principal Parts	p. 294
Hurwitz Spaces	p. 297
Completed Hurwitz Spaces and Stable Mappings	p. 299
Extending the LL Mapping to Completed Hurwitz Spaces	p. 300
Computing the Top Segre Class; End of the Proof	p. 302
The Braid Group Action	p. 304
Braid Groups	p. 304
Braid Group Action on Cacti: Generalities	p. 309
Experimental Study	p. 312
Primitive and Imprimitive Monodromy Groups	p. 318
Perspectives	p. 325
Megamaps	p. 327
Hurwitz Spaces of Coverings with Four Ramification Points	p. 328
Representation of <$>\overline {H}<$> as a Dessin d'Enfant	p. 329
Examples	p. 331
Algebraic Structures Associated with Embedded Graphs	p. 337
The Bialgebra of Chord Diagrams	p. 337
Chord Diagrams and Arc Diagrams	p. 337
The 4-Term Relation	p. 339
Multiplying Chord Diagrams	p. 342
A Bialgebra Structure	p. 343
Structure Theorem for the Bialgebra <$>{\cal M}<$>	p. 346
Primitive Elements of the Bialgebra of Chord Diagrams	p. 347
Knot Invariants and Origins of Chord Diagrams	p. 350
Knot Invariants and their Extension to Singular Knots	p. 350
Invariants of Finite Order	p. 353
Deducing 1-Term and 4-Term Relations for Invariants	p. 355
Chord Diagrams of Singular Links	p. 357
Weight Systems	p. 359
A Bialgebra Structure on the Module <$>{\cal V}<$> of Vassiliev Knot Invariants	p. 359
Renormalization	p. 360
Weight Systems	p. 362
Vassiliev Knot Invariants and Other Knot Invariants	p. 364
Constructing Weight Systems via Intersection Graphs	p. 367
The Intersection Graph of a Chord Diagram	p. 367
Tutte Functions for Graphs	p. 368
The 4-Bialgebra of Graphs	p. 369
The Bialgebra of Weighted Graphs	p. 379
Constructing Vassiliev Invariants from 4-Invariants	p. 383
Constructing Weight Systems via Lie Algebras	p. 384
Free Associative Algebras	p. 385
Universal Enveloping Algebras of Lie Algebras	p. 387
Examples	p. 390
Some Other Algebras of Embedded Graphs	p. 393
Circle Diagrams and Open Diagrams	p. 393
The Algebra of 3-Graphs	p. 395
The Temperley-Lieb Algebra	p. 395
Applications of the Representation Theory of Finite Groups	p. 399
Representation Theory of Finite Groups	p. 399
Irreducible Representations and Characters	p. 399
Examples	p. 403
Frobenius's Formula	p. 406
Applications	p. 408
Representations of Sn and Canonical Polynomials Associated to Partitions	p. 409
Examples	p. 415
First Application: Enumeration of Polygon Gluings	p. 416
Second Application: the Goulden-Jackson Formula	p. 418
Third Application: "Mirror Symmetry" in Dimension One	p. 423
References	p. 429
Index	p. 445
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