Lectures on Orbifolds and Group Cohomology&Alejandro Adem and Michele Klaus 1
1 Introduction 1
2 Classical orbifolds 2
3 Examples of orbifolds 3
4 Orbifolds and manifolds 5
5 Orbifolds and groupoids 6
6 The orbifold Euler characteristic and K-theory 10
7 Stringy products in K-theory 13
8 Twisted version 15
References 18
Lectures on the Mapping Class Group of a Surface&Thomas Kwok-Keung Au,Feng Luo and Tian Yang 21
Introduction 21
1 Mapping class group 22
2 Dehn-Lickorish Theorem 31
3 Hyperbolic plane and hyperbolic surfaces 37
4 Quasi-isometry and large scale geometry 48
5 Dehn-Nielsen Theorem 54
References 60
Lectures on Orbifolds and Reflection Groups&Michael W.Davis 63
1 Transformation groups and orbifolds 63
2 2-dimensional orbifolds 71
3 Reflection groups 76
4 3-dimensional hyperbolic reflection groups 83
5 Aspherical orbifolds 87
References 93
Lectures on Moduli Spaces of Elliptic Curves&Richard Hain 95
1 Introduction to elliptic curves and the moduli problem 96
2 Families of elliptic curves and the universal curve 104
3 The orbifold M1,1 110
4 The orbifold ？1,1 and modular forms 120
5 Cubic curves and the universal curve？→？1,1 127
6 The Picard groups of M1,1 and ？1,1 141
7 The algebraic topology of ？1.1 148
8 Concluding remarks 151
Appendix A Background on Riemann surfaces 156
Appendix B A very brief introduction to stacks 163
References 166
An Invitation to the Local Structures of Moduli of Genus One Stable Maps&Yi Hu 167
1 Introduction 167
2 The structures of the direct image sheaf 170
3 Extensions of sections on the central fiber 188
References 193
Lectures on the ELSV Formula&Chiu-Chu Melissa Liu 195
1 Introduction 195
2 Hurwitz numbers and Hodge integrals 197
3 Equivariant cohomology and localization 201
4 Proof of the ELSV formula by virtual localization 207
References 214
Formulae of One-partition and Two-partition Hodge Integrals&Chiu-Chu Melissa Liu 217
1 Introduction 217
2 The Mari？o-Vafa formula of one-partition Hodge integrals 219
3 Applications of the Mari？o-Vafa formula 222
4 Three approaches to the Mari？o-Vafa formula 224
5 Proof of Proposition 4.3 227
6 Generalization to the two-partition case 231
References 235
Lectures on Elements of Transformation Groups and Orbifolds&Zhi Lü 239
1 Topological groups and Lie groups 239
2 G-actions(or transformation groups)on topological spaces 241
3 Orbifolds 249
4 Homogeneous spaces and orbit types 251
5 Twisted product and slice 253
6 Equivariant cohomology 255
7 Davis-Januszkiewicz theory 265
References 275
The Action of the Mapping Class Group on Representation Varieties&Richard A.Wentworth 277
1 Introduction 277
2 Action of Out(π) on representation varieties 279
3 Action on the cohomology of the space of flat unitary connections 286
4 Action on the cohomology of the SL(2,C)character variety 291
References 296