Synthesis Report Of The Curve Moduli Spaces And Gromov,-witten Theory

This article is about the intersection theory on Moduli Space of curves and Gromov-Witten Theory. In the first chapter,we introduce the basic knowlodge of Moduli Space of curves and moduli space of stable maps,on which the intersection theory and Gromov-Witten theory is defined. In the second chapter, we introduce Witten conjecture and one of the proof of this conjecture which is given by localization method,and then we describe Virasoro conjucture. In the third theory, we introduce some other theorems and conjectures about Moduli Space of curves and intersection theory, including ELSV theorem, Marino-Vafa theorem and Faber conjecture. This theory play an important role in the Mathematical Physics in these years. There are many conjectures which has not been proved.This article is a simple summary about this theory.I described the necessary knowledge to help one who first steps into this field.