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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