| Contents | 第5-7页 |
| 0.1 Abstract | 第7-9页 |
| 0.2 Acknowledgements | 第10-11页 |
| 0.3 Appendix: Bernoulli numbers | 第11-13页 |
| 1 Moduli space | 第13-21页 |
| 1.1 Yoneda lemma and representable functors | 第13-15页 |
| 1.2 Moduli problem and moduli space | 第15-17页 |
| 1.3 An example: cross ratios | 第17-21页 |
| 2 The moduli space of Curves | 第21-39页 |
| 2.1 The moduli space of stable n-pointed curves with genus zero | 第21-27页 |
| 2.1.1 Stable n-pointed rational curves | 第22-23页 |
| 2.1.2 Stabilization and contraction | 第23-25页 |
| 2.1.3 The boundary | 第25-27页 |
| 2.2 The moduli space of stable curves with genus g | 第27-33页 |
| 2.2.1 Stable curves with genusg | 第28-29页 |
| 2.2.2 Tautological classes on M_g | 第29-30页 |
| 2.2.3 Faber conjecture | 第30-33页 |
| 2.3 The moduli space of stable n-pointed curves with genus g | 第33-39页 |
| 2.3.1 Tautological classes on (?)_(g,n) | 第34页 |
| 2.3.2 String and dilation equations | 第34-36页 |
| 2.3.3 Tautological rings on (?)_(g,n) | 第36-39页 |
| 3 The moduli space of stable maps and Virasoro constraints | 第39-45页 |
| 3.1 The moduli space of stable maps | 第39-40页 |
| 3.2 Virasoro constraints | 第40-41页 |
| 3.3 The degree zero gravitational descendents of X | 第41-43页 |
| 3.4 Witten conjecture and Kontsevich theorem | 第43-45页 |
| 4 Some Results of the Mari(?)o-Vafa Formula | 第45-63页 |
| 4.1 Partitions | 第45-46页 |
| 4.2 Symmetric polynomials and Chern character of Hodge bundle | 第46-48页 |
| 4.3 Mari(?)o-Vafa formula | 第48-50页 |
| 4.4 Mumford's relations | 第50页 |
| 4.5 Mari(?)o-Vafa formula for one marked point | 第50-55页 |
| 4.6 Another Simple Proof of λ_g Conjecture | 第55-57页 |
| 4.7 A recursion Formula of the λ_(g-1) Integral | 第57-63页 |
| 5 Localization, Hurwitz Numbers and the Witten Conjecture | 第63-71页 |
| 5.1 Localization and the Hurwitz Numbers | 第63-65页 |
| 5.2 Symmetrization Operator and Rooted Tree Series | 第65-68页 |
| 5.3 Proof of the DVV Conjecture | 第68-71页 |
| Bibliography | 第71-76页 |
