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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
$\tau$-function      201
1-Factor      365
1-flag      32
1-form, holomorphic      217
1-form, meromorphic      216
1-form, meromorphic, residue of      217
1-term relations      346
2-term relations      354
2-term relations for graphs      366
3-term element, weighted      370
4-bialgebra of graphs      362
4-invariant      363
4-invariant of a graph      361
4-invariant, multiplicative      363
4-term element      331
4-term relations      346
4-term relations for graphs      362
abc conjecture      134
Absolute Galois group      113
Adjacency matrix of a graph      366
Adjoint action      167
Afiine linear group      47
Airy function matrix      259
Algebra of 3-graphs      386
Algebra, Hopf      338
Algebra, Temperley — Lieb      386
Algebra, universal enveloping      377
Algebraic numbers, conjugate      113
Antipode      338
Antisymmetry relation      385
Arc diagram      330
Arc diagrams, concatenation      333
Archimedean solid      111
Automorphism group of a constellation      66
Automorphism group of a map      36
Automorphism of Frobenins      48
Average value of a function      158
Bachelor      95 122
Baker function      203
Base hypermap      63
Base star      18
Belyi function      78
Belyi function, dynamical      136
Belyi function, pure      108
Belyi pair      78
Belyi theorem      77
Bernoulli numbers      236 241 282
Bialgebra      335
Bialgebra of 4-invariants      363
Bialgebra of finite order knot invariants      350
Bialgebra of weighted graphs      370
Bialgebra, polynomial      336
Bialgebra, primitive element of      337
Bicolored graph      41
Bicolored map      41
Bicolored plane tree      43
Bipartite graph      41
Block of imprimitivity      7
Bound of Davenport — Stothers — Zannier      124
Boundary of a moduli space      228
Braid      296
Braid group      12 296
Braid group, action on cacti      300
Braid group, action on constellations      12
Braid group, diagonal      306
Braid group, Hurwita      297
Braid group, orbit of      299
Braid group, pure      297
Braid group, sphere      297
Braid group, surface      297
Bundle, Hodge      280
Cacti, family of      301
Cacti, isomorphic      56
Cactus      55
Cactus, non-rooted      57
Cactus, rooted      57
Cactus, special      309
Cactus, symmetric      57
Canonical geometric form of a plane tree      85
Canonical Jenkins — Strebel quadratic differential      233
Canonical triangulation      50
Canonically labelled tree      44
Cartographic group of a constellation      7
Cartographic group of a generalized constellation      63
Catalan numbers      154 174 186 193 210 387
Caustic      272
Center of a face      36 78
Character, irreducible      9 390
Chebyshev polynomial      80
Chebyshev polynomial of the first kind      130
Chebyshev polynomial of the second kind      130
Chebyshev polynomial, generalized      80
Chern class of a line bundle first      242
Chinese character      384
Chord diagram      329
Chord diagram of a singular knot      345
Chord diagram, decomposable      339
Chord diagram, intersection graph of      355 358
Chord diagrams, comuImplication      334
Chord diagrams, multiplication      333
Chord separating operator      352
Chromatic polynomial      361
Chromatic polynomial, modified      361
Circle diagram      384
Class, Chern first      242
Class, Segre      284
Class, Segre, top      284
Classification of coverings, flexible      12 295
Classification of coverings, rigid      23 262 269
Coarse moduli space      226
Combinatorial map      36
Complete graph      31
Complete graph, genus of      31
Complete tripartite graph      133
Complete weight system      381
Completed Hurwitz space      289
complex curve      69
Complex curve, node of      217
Complex equivalence of coverings      74
Complex protective line      69
Complex structure      69
Composition type      312
Comultiplic ation of chord diagrams      334
Comultiplic ation of knot invariants      350
Concaten ation of arc diagrams      333
Concaten ation of knots      350
Cone      283
Cone over a variety      284
Conjecture, $A_n$      316
Conjecture, $S_n$      316
Conjecture, Witten      249
Conjugate algebraic numbers      113
Conjugate constellations      7
Conjugate trees      89
Connected graph      26
Connected sum of knots      350
Constellation      6
Constellation, degree of      6
Constellation, generalized      63
Constellation, genus of      22
Constellation, length of      6
Constellation, passport of      8
Constellation, quotient      67
Constellation, regular      68
Constellations, conjugate      7
Constellations, isomorphic      7
Contraction of an edge in a weighted graph      370
Contribution of a coupling      165
Convolution product in a bialgebra      338
Conway polynomial      356
Counit      335
Covariance matrix      159
Covering      12
Covering, decomposable      64
Covering, degree of      13
Covering, finite-sheeted      13
Covering, Galois      17
Covering, n-sheeted      13
Covering, ramified      14
Covering, regular      17
Covering, universal      17
Covering, unramified      12
Coverings, complex equivalence of      74
Coverings, flexible classification of      295
Coverings, isomorphic      13
Coverings, rigid classification of      269
Critical point      14 20 71 79
Critical point, degree of      71 79
Critical point, multiplicity of      14 71 79
Critical point, order of      71 79
Critical value      14 20 71 79
Critical value, simple      301
Cubic field, cyclic      102
Cubic field, purely      103
Curve, complex      69
Curve, elliptic      104 108
Curve, Format      132
Curve, modular      222
Curve, nodal      217
Curve, nodal, genus of      218
Curve, stable      227
Curve, unicursal      183
Curve, universal      227
CYCLE      26
Cycle structure of permutation      8
Cyclic cubic field      102
Dart      32
Dart, incident to a face      33
Dart, incident to a vertex      32
Davenport — Stothers bound      124
Decomposable chord diagram      339
Decomposable element of a graded bialgebra      339
Decomposable polynomial      311
Decomposable, covering      64
Decorated moduli space      230
Defect      302
Defined over      70 77 89
Defining polynomial      89 114
Degeneracy of a partition      270
Degenerate partition      270
Degree of a critical point      71 79
Degree of a face      29
Degree of a vertex      26
Degree of constellation      6
Degree of covering      13
Deletion of an edge in a weighted graph      370
Deligne — Mumford compactification of the moduli space      228
Delm twist      28
Density of a Gaussian measure      158
Dessin d'enfant      78
Dessins d'enfants, theory of      75
Diagonal braid group      306
Diagram, arc      330
Diagram, chord      329
Diagram, circle      384
Diagram, open      85
Diameter of a tree      98
Differential, meromorphic      216
Differential, quadratic      219
Dilaton equation      248
Discrete, action of a group      221
Discrete, Painleve I equation      192
Discriminant      116 269
Discriminant, locus      269
Double flower      129
Double point      217
Double point, normalized      218
Double, scaling limit      196
Dual hypermap      52
Dual map      51
Dynamical Belyi function      136
EDGE      26
Edge, midpoint of      36
Edge, polygon in a graph      364
Edge, quadrangle in a graph      364
Elementary hypermap      43
Elementary transformation      343
Elementary triangulation      51
Elimination of quantifiers      115
Elliptic curve      104 108
Elliptic integral      140
Embedded graph      1
Embedded graph, marked      231
Enler characteristic      29
Enler characteristic, orbifold      223
Enler characteristic, virtual      223
Enumeration problem, inverse      96
Enveloping algebra, universal      377
Equation, dilaton      248
Equation, discrete      192
Equation, matrix Airy      259
Equation, Painleve I      196
Equation, Pell      139
Equation, Pell — Abel      139
Equation, string      246
Equivalence of coverings, flexible      23
Equivalence of coverings, rigid      23
Equivalent Shabat polynomials      82
Eventually periodic point      137
Exponential generating function      210
Extension, Galois      114
Extension, normal      114
External vertex      384
Face of a map      27
Face, center of      36 78
Face, degree of      29
Face, valency of      29
Family of cacti      301
Family of functions, normal      136
Family of trees      92
Fat graph      1
Fatou set      137
Feit — Jones theorem      318
Fermat curve      132
Fiber      13
Field of definition      148
Field of moduli      77 94
Field of moduli of a dessin      115
Field of moduli of an orbit      115
Field, cubic cyclic      102
Field, purely cubic      103
Field, splitting      114
Fine moduli space      226
Finite mapping      264
Finite order invariant      344
First Chern class of a line bundle      242
First Orthogonality Relation      390
Flexible classification of coverings      12 295
Flexible equivalence of coverings      23
Flower, double      129
Flower, Leila's      103
Form, modular      415
Form, quasimodular      415
Formula, Frobenins's      9 396
Formula, Frobenins's, generalized      397
Formula, Goulden — Jackson      58 271 408
Formula, Goulden — Jackson, generalized      408
Formula, Hurwitz      281
Formula, Riemann — Hurwitz      21
Formula, Whitney      285
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