Moduli spaces of Riemann surfaces: [lecture notes from the Graduate Summer School program held in July 2011]
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| Other Authors: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc. [u.a.]
2013
|
| Series: | IAS Park City mathematics series
20 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | X, 356 S. Ill., graph. Darst. |
| ISBN: | 9780821898871 |
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MARC
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| 245 | 1 | 0 | |a Moduli spaces of Riemann surfaces |b [lecture notes from the Graduate Summer School program held in July 2011] |c Benson Farb ... ed. |
| 264 | 1 | |a Providence, RI |b American Math. Soc. [u.a.] |c 2013 | |
| 300 | |a X, 356 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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Record in the Search Index
| _version_ | 1805073843185778688 |
|---|---|
| adam_text |
Contents
Preface
ix
Benson
Farb,
Richard
Hain,
and
Eduard
Looijenga
Introduction
1
Yair
N.
Minsky
A Brief Introduction to Mapping Class Groups
5
1.
Definitions, examples, basic structure
7
2.
Hyperbolic geometry, laminations and foliations
19
3.
The Nielsen-Thurston classification theorem
28
4.
Classification continued, and consequences
35
5.
Further reading and current events
39
Bibliography
41
Ursula
Hamenstädt
Teichmüller
Theory
45
Introduction
47
Lecture
1.
Hyperbolic surfaces
49
Lecture
2.
Quasiconf
ormai
maps
63
Lecture
3.
Complex structures, Jacobians and the Weil
Petersson
form
75
Lecture
4.
The curve graph and the augmented
Teichmüller
space
85
Lecture
5.
Geometry and dynamics of moduli space
93
Bibliography
107
Nathalie
Wahl
The Mumford Conjecture, Madsen-Weiss and Homological Stability
for Mapping Class Groups of Surfaces
109
Introduction 111
Lecture
1.
The Mumford conjecture and the Madsen-Weiss theorem
113
1.
The Mumford conjecture
113
2.
Moduli space, mapping class groups and diffeomorphism groups
113
3.
The Mumford-Morita-Miller classes
115
vi
CONTENTS
4.
Homological
stability
115
5.
The Madsen-Weiss theorem
117
6.
Exercises
118
Lecture
2.
Homological stability: geometric ingredients
119
1.
General strategy of proof
119
2.
The case of the mapping class group of surfaces
119
3.
The ordered arc complex
120
4.
Curve complexes and disc spaces
123
5.
Exercises
124
Lecture
3.
Homological stability: the spectral sequence argument
127
1.
Double complexes associated to actions on simplicial complexes
127
2.
The spectral sequence associated to the horizontal filtration
128
3.
The spectral sequence associated to the vertical filtration
128
4.
The proof of stability for surfaces with boundaries
129
5.
Closing the boundaries
131
6.
Exercises
131
Lecture
4.
Homological stability: the connectivity argument
133
1.
Strategy for computing the connectivity of the ordered arc complex
133
2.
Contractibility of the full arc complex
134
3.
Deducing connectivity of smaller complexes
135
4.
Exercises
136
Bibliography
137
S0ren Galatius
Lectures on the Madsen—Weiss Theorem
139
Lecture
1.
Spaces of submanifolds and the Madsen-Weiss Theorem
143
1.1.
Spaces of manifolds
143
1.2.
Exercises for Lecture
1 147
Lecture
2.
Rational cohomology and outline of proof
149
2.1.
Cohomology of
Ω°°Φ
149
2.2.
Outline of proof
151
2.3.
Exercises for Lecture
2 152
Lecture
3.
Topological monoids and the first part of the proof
153
3.1.
Topological monoids
153
3.2.
Exercises for Lecture
3 159
Lecture
4.
Final step of the proof
161
4.1.
Proof of theorem
4.3 163
4.2.
Exercises for Lecture
4 165
Bibliography
167
CONTENTS
vii
Andrew
Putman
The
Torelli
Group and Congruence Subgroups of the Mapping
Class Group
169
Introduction
171
Lecture
1.
The
Torelli
group
173
Lecture
2.
The Johnson homomorphism
179
Lecture
3.
The abelianization of Modgn(p)
185
Lecture
4.
The second rational homology group of Modg(p)
189
Bibliography
195
Carel
Faber
Tautological Algebras of Moduli Spaces of Curves
197
Introduction
199
Lecture
1.
The tautological ring of Mg
201
Exercises
208
Lecture
2.
The tautological rings of
M
g¡n
and of some natural partial
cornpactifications of Mg>n
211
Exercises
215
Bibliography
217
Scott A. Wolpert
Mirzakhani's Volume Recursion and Approach for the Witten-
Kontsevich Theorem on Moduli Tautological Intersection Numbers
221
Prelude
225
Lecture
1.
The background and overview
231
Lecture
2.
The McShane-Mirzakhani identity
239
Lecture
3.
The
covolume
formula and recursion
243
Lecture
4.
Symplectic reduction, principal S1 bundles and the normal form
249
Lecture
5.
The pattern of intersection numbers and Witten-Kontsevich
255
Questions for the problem sessions
261
Bibliography
265
Martin
Möller
Teichmüller
Curves, Mainly from the Viewpoint of Algebraic
Geometry
267
1.
Introduction
269
viii CONTENTS
2.
Flat surfaces and SL2 (Reaction
270
2.1.
Flat surfaces and translation structures
270
2.2. Affine
groups and the trace field
272
2.3.
Strata of
Ώ,ΛΛ9
and hyperelliptic loci
276
2.4.
Spin structures and connected components of strata
276
2.5.
Stable differentials and Deligne-Mumford compactification
277
3.
Curves and divisors in Aig
277
3.1.
Curves and fibered surfaces
278
3.2.
Picard
groups of moduli spaces
279
3.3.
Special divisors on moduli spaces
279
3.4.
Slopes of divisors and of curves in
Л4д
282
4.
Variation of Hodge structures and real multiplication
284
4.1.
Hubert modular varieties and the locus of real multiplication
288
4.2.
Examples
289
5. Teichmüller
curves
290
5.1.
Square-tiled surfaces and
primitivity
291
5.2.
The
VHS
of
Τ
curves
292
5.3.
Proof of the
VHS
decomposition and real multiplication
294
5.4.
Cusps and sections of
Τ
curves
296
5.5.
The classification problem of
Τ
curves: state of the art
300
6.
Lyapunov exponents
301
6.1.
Motivation: Asymptotic cycles, deviations and the wind-tree model
301
6.2.
Lyapunov exponents
303
6.3.
Lyapunov exponents for
Teichmüller
curves
304
6.4.
Non-
vary ing properties for sums of Lyapunov exponents
309
6.5.
Lyapunov exponents for general curves in
ЛЛд
and in Ag
312
6.6.
Known results and open problems
314
Bibliography
315
Makoto Matsumoto
Introduction to arithmetic mapping class groups
319
Introduction
321
Lecture
1.
Algebraic fundamental groups
325
Lecture
2.
Monodromy representation on fundamental groups
333
Lecture
3.
Arithmetic mapping class groups
335
Lecture
4.
Topology versus arithmetic
337
Lecture
5.
The conjectures of
Oda
and Deligne-Ihara
341
APPENDIX: Algebraic fundamental groups via fiber functors
349
Bibliography
355
Mapping class groups and moduli spaces of Riemann surfaces were the topics of the
Graduate Summer School at the
2011
IAS/Park City Mathematics Institute. This book
presents the nine different lecture series comprising the summer school, covering a
selection of topics of current interest. The introductory courses treat mapping class
groups and
Teichmüller
theory. The more advanced courses cover intersection theory
on moduli spaces, the dynamics of polygonal billiards and moduli spaces, the stable
cohomology of mapping class groups, the structure of
Torelli
groups, and arithmetic
mapping class groups.
The courses consist of a set of intensive short lectures offered by leaders in the field,
designed to introduce students to exciting, current research in mathematics. These
lectures do not duplicate standard courses available elsewhere. The book should be
a valuable resource for graduate students and researchers interested in the topology,
geometry and dynamics of moduli spaces of Riemann surfaces and related topics. |
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| spelling | Moduli spaces of Riemann surfaces [lecture notes from the Graduate Summer School program held in July 2011] Benson Farb ... ed. Providence, RI American Math. Soc. [u.a.] 2013 X, 356 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier IAS Park City mathematics series 20 Riemannsche Fläche (DE-588)4049991-1 gnd rswk-swf Modulraum (DE-588)4183462-8 gnd rswk-swf aModuli theory aRiemann surfaces (DE-588)4143413-4 Aufsatzsammlung gnd-content (DE-588)1071861417 Konferenzschrift gnd-content Riemannsche Fläche (DE-588)4049991-1 s Modulraum (DE-588)4183462-8 s DE-604 Farb, Benson 1967- (DE-588)1020817828 edt Erscheint auch als Online-Ausgabe 978-1-4704-0994-4 IAS Park City mathematics series 20 (DE-604)BV010402400 20 Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026905089&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Regensburg - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=026905089&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Moduli spaces of Riemann surfaces [lecture notes from the Graduate Summer School program held in July 2011] IAS Park City mathematics series Riemannsche Fläche (DE-588)4049991-1 gnd Modulraum (DE-588)4183462-8 gnd |
| subject_GND | (DE-588)4049991-1 (DE-588)4183462-8 (DE-588)4143413-4 (DE-588)1071861417 |
| title | Moduli spaces of Riemann surfaces [lecture notes from the Graduate Summer School program held in July 2011] |
| title_auth | Moduli spaces of Riemann surfaces [lecture notes from the Graduate Summer School program held in July 2011] |
| title_exact_search | Moduli spaces of Riemann surfaces [lecture notes from the Graduate Summer School program held in July 2011] |
| title_full | Moduli spaces of Riemann surfaces [lecture notes from the Graduate Summer School program held in July 2011] Benson Farb ... ed. |
| title_fullStr | Moduli spaces of Riemann surfaces [lecture notes from the Graduate Summer School program held in July 2011] Benson Farb ... ed. |
| title_full_unstemmed | Moduli spaces of Riemann surfaces [lecture notes from the Graduate Summer School program held in July 2011] Benson Farb ... ed. |
| title_short | Moduli spaces of Riemann surfaces |
| title_sort | moduli spaces of riemann surfaces lecture notes from the graduate summer school program held in july 2011 |
| title_sub | [lecture notes from the Graduate Summer School program held in July 2011] |
| topic | Riemannsche Fläche (DE-588)4049991-1 gnd Modulraum (DE-588)4183462-8 gnd |
| topic_facet | Riemannsche Fläche Modulraum Aufsatzsammlung Konferenzschrift |
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