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Lectures on Orbifolds and Group Cohomology&Alejandro Adem and Michele Klaus1

1 Introduction1

2 Classical orbifolds2

3 Examples of orbifolds3

4 Orbifolds and manifolds5

5 Orbifolds and groupoids6

6 The orbifold Euler characteristic and K-theory10

7 Stringy products in K-theory13

8 Twisted version15

References18

Lectures on the Mapping Class Group of a Surface&Thomas Kwok-Keung Au,Feng Luo and Tian Yang21

Introduction21

1 Mapping class group22

2 Dehn-Lickorish Theorem31

3 Hyperbolic plane and hyperbolic surfaces37

4 Quasi-isometry and large scale geometry48

5 Dehn-Nielsen Theorem54

References60

Lectures on Orbifolds and Reflection Groups&Michael W.Davis63

1 Transformation groups and orbifolds63

2 2-dimensional orbifolds71

3 Reflection groups76

4 3-dimensional hyperbolic reflection groups83

5 Aspherical orbifolds87

References93

Lectures on Moduli Spaces of Elliptic Curves&Richard Hain95

1 Introduction to elliptic curves and the moduli problem96

2 Families of elliptic curves and the universal curve104

3 The orbifold M1,1110

4 The orbifold ？1,1 and modular forms120

5 Cubic curves and the universal curve？→？1,1127

6 The Picard groups of M1,1 and ？1,1141

7 The algebraic topology of ？1.1148

8 Concluding remarks151

Appendix A Background on Riemann surfaces156

Appendix B A very brief introduction to stacks163

References166

An Invitation to the Local Structures of Moduli of Genus One Stable Maps&Yi Hu167

1 Introduction167

2 The structures of the direct image sheaf170

3 Extensions of sections on the central fiber188

References193

Lectures on the ELSV Formula&Chiu-Chu Melissa Liu195

1 Introduction195

2 Hurwitz numbers and Hodge integrals197

3 Equivariant cohomology and localization201

4 Proof of the ELSV formula by virtual localization207

References214

Formulae of One-partition and Two-partition Hodge Integrals&Chiu-Chu Melissa Liu217

1 Introduction217

2 The Mari？o-Vafa formula of one-partition Hodge integrals219

3 Applications of the Mari？o-Vafa formula222

4 Three approaches to the Mari？o-Vafa formula224

5 Proof of Proposition 4.3227

6 Generalization to the two-partition case231

References235

Lectures on Elements of Transformation Groups and Orbifolds&Zhi Lü239

1 Topological groups and Lie groups239

2 G-actions(or transformation groups)on topological spaces241

3 Orbifolds249

4 Homogeneous spaces and orbit types251

5 Twisted product and slice253

6 Equivariant cohomology255

7 Davis-Januszkiewicz theory265

References275

The Action of the Mapping Class Group on Representation Varieties&Richard A.Wentworth277

1 Introduction277

2 Action of Out(π) on representation varieties279

3 Action on the cohomology of the space of flat unitary connections286

4 Action on the cohomology of the SL(2,C)character variety291

References296