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9783540002031

Graphs on Surfaces and Their Applications

by ; ;
  • ISBN13:

    9783540002031

  • ISBN10:

    3540002030

  • Format: Hardcover
  • Copyright: 2004-02-01
  • Publisher: Springer Verlag
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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.

Author Biography

 Sergei LANDO graduated from the Moscow State University in 1977, got the PhD degree from the Moscow State University in 1986 under the supervision of Prof. V.I. Arnold; worked for the Russian Academy of Sciences in 1986-1990 and since 1996 till now; one of the organizers (1991), and currently professor and vice-president of the Independent University of Moscow. Alexander ZVONKIN graduated from the Moscow State University in 1970, got the PhD degree form the Moscow State University in 1974 under the supervision of Prof. A.N.Shiryaev; worked as associate professor (1973-1975), as industrial researcher (1975-1988), and in the USSR Academy of Sciences (1989-1992); from 1991 till now, professor of computer science at Bordeaux I University, Bordeaux, France.

Table of Contents

0 Introduction: What is This Book About 1(6)
0.1 New Life of an Old Theory
1(1)
0.2 Plan of the Book
2(2)
0.3 What You Will Not Find in this Book
4(3)
1 Constellations, Coverings, and Maps 7(72)
1.1 Constellations
7(6)
1.2 Ramified Coverings of the Sphere
13(13)
1.2.1 First Definitions
13(2)
1.2.2 Coverings and Fundamental Groups
15(3)
1.2.3 Ramified Coverings of the Sphere and Constellations
18(4)
1.2.4 Surfaces
22(4)
1.3 Maps
26(13)
1.3.1 Graphs Versus Maps
26(2)
1.3.2 Maps: Topological Definition
28(5)
1.3.3 Maps: Permutational Model
33(6)
1.4 Cartographic Groups
39(4)
1.5 Hypermaps
43(12)
1.5.1 Hypermaps and Bipartite Maps
43(2)
1.5.2 Trees
45(4)
1.5.3 Appendix: Finite Linear Groups
49(1)
1.5.4 Canonical Triangulation
50(5)
1.6 More Than Three Permutations
55(8)
1.6.1 Preimages of a Star or of a Polygon
56(1)
1.6.2 Cacti
57(4)
1.6.3 Preimages of a Jordan Curve
61(2)
1.7 Further Discussion
63(7)
1.7.1 Coverings of Surfaces of Higher Genera
63(2)
1.7.2 Rift's Theorem
65(3)
1.7.3 Symmetric and Regular Constellations
68(2)
1.8 Review of Riemann Surfaces
70(9)
2 Dessins d'Enfants 79(76)
2.1 Introduction: The Belyi Theorem
79(1)
2.2 Plane Trees and Shabat Polynomials
80(29)
2.2.1 General Theory Applied to Trees
80(8)
2.2.2 Simple Examples
88(6)
2.2.3 Further Discussion
94(7)
2.2.4 More Advanced Examples
101(8)
2.3 Belyi Functions and Belyi Pairs
109(6)
2.4 Galois Action and Its Combinatorial Invariants
115(11)
2.4.1 Preliminaries
115(3)
2.4.2 Galois Invariants
118(5)
2.4.3 Two Theorems on Trees
123(3)
2.5 Several Facets of Belyi Functions
126(20)
2.5.1 A Bound of Davenport-Stothers-Zannier
126(5)
2.5.2 Jacobi Polynomials
131(4)
2.5.3 Fermat Curve
135(2)
2.5.4 The abc Conjecture
137(2)
2.5.5 Julia Sets
139(3)
2.5.6 Pell Equation for Polynomials
142(4)
2.6 Proof of the Belyi Theorem
146(9)
2.6.1 The "Only If" Part of the Belyi Theorem
146(1)
2.6.2 Comments to the Proof of the "Only If" Part
147(3)
2.6.3 The "If", or the "Obvious" Part of the Belyi Theorem
150(5)
3 Introduction to the Matrix Integrals Method 155(68)
3.1 Model Problem: One-Face Maps
155(5)
3.2 Gaussian Integrals
160(19)
3.2.1 The Gaussian Measure on the Line
160(2)
3.2.2 Gaussian Measures in Rk
162(1)
3.2.3 Integrals of Polynomials and the Wick Formula
163(1)
3.2.4 A Gaussian Measure on the Space of Hermitian Matrices
164(3)
3.2.5 Matrix Integrals and Polygon Gluings
167(4)
3.2.6 Computing Gaussian Integrals. Unitary Invariance
171(5)
3.2.7 Computation of the Integral for One Face Gluings
176(3)
3.3 Matrix Integrals for Multi-Faced Maps
179(6)
3.3.1 Feynman Diagrams
179(1)
3.3.2 The Matrix Integral for an Arbitrary Gluing
180(3)
3.3.3 Getting Rid of Disconnected Graphs
183(2)
3.4 Enumeration of Colored Graphs
185(7)
3.4.1 Two-Matrix Integrals and the Ising Model
185(3)
3.4.2 The Gauss Problem
188(2)
3.4.3 Meanders
190(1)
3.4.4 On Enumeration of Meanders
191(1)
3.5 Computation of Matrix Integrals
192(7)
3.5.1 Example: Computing the Volume of the Unitary Group
192(3)
3.5.2 Generalized Hermite Polynomials
195(2)
3.5.3 Planar Approximations
197(2)
3.6 Korteweg-de Vries (KdV) Hierarchy for the Universal One-Matrix Model
199(11)
3.6.1 Singular Behavior of Generating Functions
200(2)
3.6.2 The Operator of Multiplication by a in the Double Scaling Limit
202(2)
3.6.3 The One-Matrix Model and the KdV Hierarchy
204(2)
3.6.4 Constructing Solutions to the KdV Hierarchy from the Sato Grassmanian
206(4)
3.7 Physical Interpretation
210(5)
3.7.1 Mathematical Relations Between Physical Models
211(1)
3.7.2 Feynman Path Integrals and String Theory
211(2)
3.7.3 Quantum Field Theory Models
213(1)
3.7.4 Other Models
214(1)
3.8 Appendix
215(8)
3.8.1 Generating Functions
215(2)
3.8.2 Connected and Disconnected Objects
217(2)
3.8.3 Logarithm of a Power Series and Wick's Formula
219(4)
4 Geometry of Moduli Spaces of Complex Curves 223(46)
4.1 Generalities on Nodal Curves and Orbifolds
223(9)
4.1.1 Differentials and Nodal Curves
223(3)
4.1.2 Quadratic Differentials
226(1)
4.1.3 Orbifolds
227(5)
4.2 Moduli Spaces of Complex Structures
232(2)
4.3 The Deligne-Mumford Compactification
234(3)
4.4 Combinatorial Models of the Moduli Spaces of Curves
237(6)
4.5 Orbifold Euler Characteristic of the Moduli Spaces
243(6)
4.6 Intersection Indices on Moduli Spaces and the String and Dilaton Equations
249(7)
4.7 KdV Hierarchy and Witten's Conjecture
256(1)
4.8 The Kontsevich Model
257(6)
4.9 A Sketch of Kontsevich's Proof of Witten's Conjecture
263(6)
4.9.1 The Generating Function for the Kontsevich Model
263(1)
4.9.2 The Kontsevich Model and Intersection Theory
264(2)
4.9.3 The Kontsevich Model and the KdV Equation
266(3)
5 Meromorphic Functions and Embedded Graphs 269(68)
5.1 The Lyashko-Looijenga Mapping and Rigid Classification of Generic Polynomials
270(7)
5.1.1 The Lyashko-Looijenga Mapping
270(1)
5.1.2 Construction of the LL Mapping on the Space of Generic Polynomials
271(2)
5.1.3 Proof of the Lyashko-Looijenga Theorem
273(4)
5.2 Rigid Classification of Nongeneric Polynomials and the Geometry of the Discriminant
277(11)
5.2.1 The Discriminant in the Space of Polynomials and Its Stratification
277(2)
5.2.2 Statement of the Enumeration Theorem
279(1)
5.2.3 Primitive Strata
280(2)
5.2.4 Proof of the Enumeration Theorem
282(6)
5.3 Rigid Classification of Generic Meromorphic Functions and Geometry of Moduli Spaces of Curves
288(16)
5.3.1 Statement of the Enumeration Theorem
288(1)
5.3.2 Calculations: Genus 0 and Genus 1
289(3)
5.3.3 Cones and Their Segre Classes
292(2)
5.3.4 Cones of Principal Parts
294(3)
5.3.5 Hurwitz Spaces
297(2)
5.3.6 Completed Hurwitz Spaces and Stable Mappings
299(1)
5.3.7 Extending the LL Mapping to Completed Hurwitz Spaces
300(2)
5.3.8 Computing the Top Segre Class; End of the Proof
302(2)
5.4 The Braid Group Action
304(23)
5.4.1 Braid Groups
304(5)
5.4.2 Braid Group Action on Cacti: Generalities
309(3)
5.4.3 Experimental Study
312(6)
5.4.4 Primitive and Imprimitive Monodromy Groups
318(7)
5.4.5 Perspectives
325(2)
5.5 Megamaps
327(10)
5.5.1 Hurwitz Spaces of Coverings with Four Ramification Points
328(1)
5.5.2 Representation of H as a Dessin d'Enfant
329(2)
5.5.3 Examples
331(6)
6 Algebraic Structures Associated with Embedded Graphs 337(62)
6.1 The Bialgebra of Chord Diagrams
337(13)
6.1.1 Chord Diagrams and Arc Diagrams
337(2)
6.1.2 The 4-Term Relation
339(3)
6.1.3 Multiplying Chord Diagrams
342(1)
6.1.4 A Bialgebra Structure
343(3)
6.1.5 Structure Theorem for the Bialgebra M
346(1)
6.1.6 Primitive Elements of the Bialgebra of Chord Diagrams
347(3)
6.2 Knot Invariants and Origins of Chord Diagrams
350(9)
6.2.1 Knot Invariants and their Extension to Singular Knots
350(3)
6.2.2 Invariants of Finite Order
353(2)
6.2.3 Deducing 1-Term and 4-Term Relations for Invariants
355(2)
6.2.4 Chord Diagrams of Singular Links
357(2)
6.3 Weight Systems
359(8)
6.3.1 A Bialgebra Structure on the Module V of Vassiliev Knot Invariants
359(1)
6.3.2 Renormalization
360(2)
6.3.3 Weight Systems
362(2)
6.3.4 Vassiliev Knot Invariants and Other Knot Invariants
364(3)
6.4 Constructing Weight Systems via Intersection Graphs
367(17)
6.4.1 The Intersection Graph of a Chord Diagram
367(1)
6.4.2 Tutte Functions for Graphs
368(1)
6.4.3 The 4-Bialgebra of Graphs
369(10)
6.4.4 The Bialgebra of Weighted Graphs
379(4)
6.4.5 Constructing Vassiliev Invariants from 4-Invariants
383(1)
6.5 Constructing Weight Systems via Lie Algebras
384(9)
6.5.1 Free Associative Algebras
385(2)
6.5.2 Universal Enveloping Algebras of Lie Algebras
387(3)
6.5.3 Examples
390(3)
6.6 Some Other Algebras of Embedded Graphs
393(6)
6.6.1 Circle Diagrams and Open Diagrams
393(2)
6.6.2 The Algebra of 3-Graphs
395(1)
6.6.3 The Temperley-Lieb Algebra
395(4)
A Applications of the Representation Theory of Finite Groups (by Don Zagier) 399(30)
A.1 Representation Theory of Finite Groups
399(9)
A.1.1 Irreducible Representations and Characters
399(4)
A.1.2 Examples
403(3)
A.1.3 Frobenius's Formula
406(2)
A.2 Applications
408(21)
A.2.1 Representations of Sn and Canonical Polynomials Associated to Partitions
409(6)
A.2.2 Examples
415(1)
A.2.3 First Application: Enumeration of Polygon Gluings
416(2)
A.2.4 Second Application: the Goulden-Jackson Formula
418(5)
A.2.5 Third Application: "Mirror Symmetry" in Dimension One
423(6)
References 429(16)
Index 445

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