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Lando S.K., Zvonkin A.K. — Graphs on Surfaces and Their Applications
Lando S.K., Zvonkin A.K. — Graphs on Surfaces and Their Applications



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Название: Graphs on Surfaces and Their Applications

Авторы: Lando S.K., Zvonkin A.K.

Аннотация:

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.


Язык: en

Рубрика: Математика/Алгебра/Комбинаторика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 2003

Количество страниц: 459

Добавлена в каталог: 30.04.2005

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
Framed graph      367
Framed knot      348
Framing      348
Frobenins's formula      9 396
Frobenins's formula, generalized      397
Frobenins, automorphism of      48
Function, Airy matrix      259
Function, Baker      203
Function, Belyi      78
Function, Belyi, pure      108
Function, generating      209
Function, meromorphic      70
Function, meromorphic, on a nodal curve, stable      290
Fundamental group      15
Galois group      114
Galois group, absolute      113
Galois group, universal      113
Galois orbit      89
Galois, covering      17
Galois, extension      114
Gaussian measure in $\mathbb{R}^k$      158
Gaussian measure on the line      156
Gaussian measure on the space of Hermitian matrices      160
Gaussian measure on the space of Symmetric matrices      163
Gaussian measure, standard      159
Gegenbauer polynomial      130
General linear group      47
Generalized Chebyshev polynomial      80
Generalized constellation      63
Generalized Hermit© polynomial      190
Generating function      209
Generating function, exponential      210
Genus of a constellation      22
Genus of a map      27
Genus of a nodal curve      218
Goulden — Jackson formula      58 271 40?
Goulden — Jackson formula, generalized      408
Graph      26
Graph with rotation      1
Graph, bicolored      41
Graph, bipartite      41
Graph, complete      31
Graph, complete, tripartite      133
Graph, complexity      359
Graph, connected      26
Graph, embedded      1
Graph, embedded, marked      231
Graph, fat      1
Graph, framed      367
Graph, intersection      358
Graph, invariant      360
Graph, modular      229
Graph, planar      31
Graph, ribbon      1
Graph, simple      360
Graph, star      17 22
Graph, weighted      370
Group, affine linear      47
Group, algebra      392
Group, braid      12 296
Group, braid, diagonal      306
Group, braid, Hurwitz      297
Group, braid, pure      297
Group, braid, sphere      297
Group, braid, surface      297
Group, cartographic      7
Group, fundamental      15
Group, Galois      114
Group, Galois, universal (or absolute)      113
Group, general linear      47
Group, imprimitive      7
Group, Klein modular      222
Group, mapping class      28
Group, Mathieu, $M_{11}$      121
Group, Mathieu, $M_{12}$      41 121
Group, Mathieu, $M_{23}$      11 46 120
Group, Mathieu, $M_{24}$      10 38 41
Group, monodromy      7 16
Group, orthogonal      171
Group, primitive      7
Group, protective linear      47
Group, representation of      389
Group, representation of, left      389
Group, representation of, right      389
Group, special      7
Group, special, linear      48
Group, special, projective      48
Group, unitary      167
Half-edge      32
Hermite polynomial      187
Hermite polynomial, generalized      190
Hermitian matrix      161
Hierarchy, KdV      200 249
Hierarchy, Korteweg — de Vries      200 249
Hodge, bundle      280
Hodge, integral      282
Holomorphic 1-form      217
Homfly polynomial      357
Homotopic loops      15
Hopf algebra      338
Hurwitz braid group      297
Hurwitz formula      281
Hurwitz numbers      279
Hurwitz problem      261
Hurwitz space      279 320
Hurwitz space, completed      289
Hyperdual      53
Hyperdual hypermap      53
Hyperdual, reciprocal      52
Hyperdual, rooted      44
HyperMAP      41 42
Hypermap, dual      52
Hypermap, elementary      43
icosahedron      111
IHX-relation      385
Imprimitiye group      7
Incidence relation      26
Integral, elliptic      140
Integral, Hodge      282
Integral, quasi-elliptic      140
Internal vertex      384
Intersection graph of a chord diagram      355 358
Invariant of finite order      344
Invariant, Tutte      359
Invariant, Vassiliev      344
Inverse enumeration problem      96
Irreducible character      9 390
Irreducible components      218
Irreducible representation      390
Irreducible representation, character of      390
Isomorphic cacti      56
Isomorphic constellations      7
Isomorphic coverings      13
Isomorphic Riemann surfaces      69
Isthmus      29
Jacobi polynomial      130
Jenkins — Strebel quadratic differential      232
Jenkins — Strebel quadratic differential, canonical      233
Julia set      137
k-constellation      6
KdV hierarchy      200 249
Klein modular group      222
Knot      341
Knot, framed      348
Knot, invariant      341
Knot, invariant, of order $\le n$      344
Knot, singular      342
Knots, concatenation of      350
Knots, connected sum of      350
Kontsevich model      250
Kontsevich theorem      249
Korteweg — de Vries hierarchy      200 249
Leading term      287
Legendre polynomial      130
Leila's flower      103
Lemma, Schur’s      390
Length of constellation      6
Lickerish bilinear form      387
Lifting      16
Loop      15 26
Loops, homotopic      15
Loops, product of      15
Lyashko — Looijenga mapping      263
Mandelbrot set      137
MAP      27
Map, bicolored      41
Map, combinatorial      36
Map, dual      51
Map, face of      27
Map, genus of      27
Map, plane      31
Map, self-dual      53
Map, special      39
Map, topological      27
Mapping class group      28
Mapping, finite      264
Mapping, Lyashko — Looijenga      263
Mapping, quasihomogeneous      266
Marked embedded graph      231
Mass-formula      46
Matching polynomial      365
Mathieu group, $M_{11}$      121
Mathieu group, $M_{22}$      41 121
Mathieu group, $M_{23}$      11 46 120
Mathieu group, $M_{24}$      10 38 41
Matrix, Airy, equation      259
Matrix, Airy, function      259
Matrix, covariance      159
Matrix, hermitian      161
Matrix, orthogonal      171
Matrix, symmetric      163
Matrix, unitary      167
Maxwell stratum      271
Mean value of a function      158
Meander      185 387
Megamap      321
Meromorphic function      70
Meromorphic function on a nodal curve      218
Meromorphic function, pole of      70
Meromorphic function, stable      290
Meromorphic function, zero of      70
Meromorphic, 1-form      216
Meromorphic, 1-form, residue of      217
Meromorphic, differential      216
Meromorphic, differential, quadratic      219
Midpoint of an edge      36 108
Mil Implication of chord diagrams      333
Mil Implication of knot invariants      350
Mirror Symmetry      412
Modified chromatic polynomial      361
Modular curve      222
Modular form      415
Modular graph      229
Module of chord diagrams      332
Moduli space, coarse      226
Moduli space, decorated      230
Moduli space, fine      226
monodromy      16
Monodromy group      7 16
Mueller theorem      309
Multiple edges      26
Multiplicative 4-invariant      363
Multiplicative weight system      353
Multiplicity of a critical point      71 79
Negative resolution of a singular point      342
Negative triangle      50
Nodal curve      217
Nodal curve, genus of      218
Nodal curve, normalization of      218
Node of a complex curve      217
Non-rooted cactus      57
Normal extension      114
Normal family of functions      136
Normalization of a nodal curve      218
Normalized double point      218
Number field      113
Numbers, Bernoulli      236 241 282
Numbers, Catalan      154 174 186 193 210 387
Numbers, Hurwitz      279
Open diagram      385
Operator, chord separating      352
Operator, renormalization      352
Orbifold      221
Orbifold, Euler characteristic      223
Orbit of braid group      25 299
Orbit of diagonal braid group      306
Orbit, Galois      89
Order of a critical point      71 79
Order of a pole      217
Order of symmetry      57
Orthogonal group      171
Orthogonal matrix      171
Orthogonality relation, first      390
Orthogonality relation, second      393
Painleve I equation      196
Painleve I equation, discrete      192
Pair, Belyi      78
Pair, stable      225
Parasitic solution      90
Partition      8
Partition, canonical polynomials associated to      398
Partition, degeneracy of      270
Partition, degenerate      270
Passport      79
Passport of constellation      8
Passport of constellation, refined      8
Passport of type $A_n$      316
Passport of type $S_n$      316
Passport, polynomial      58
Passport, valuable, cactus      58
Passport, valuable, tree      44
Path      26
Pell equation      139
Pell — Abel equation      139
Perfect matching      365
Periodic point      137
Permutation, cycle structure of      8
Planar graph      31
Plane map      31
Plane tree      43
Plane tree, canonical geometric form of      85
Platonic solid      111
Point, critical      14 20 71 79
Point, critical, degree of      71 79
Point, critical, multiplicity of      14 71 79
Point, critical, order of      71 79
Point, double      217
Point, double, normalized      218
Point, eventually periodic      137
Point, periodic      137
Point, ramification      14 20
Point, strictly preperiodic      137
Pole of a meromorphic function      70
Pole, order of      217
Polynomial passport      58
Polynomial, bialgebra      336
Polynomial, Chebyshev      80
Polynomial, Chebyshev, generalized      80
Polynomial, Chebyshev, of the first kind      130
Polynomial, Chebyshev, of the second kind      130
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