Abstract

In her article, M. Mirzakhani generalized the notion of moduli space of marked curves to the moduli space of bordered Riemann surfaces for studying Weil-Petersson Symplectic form on it. She gave a method for integrating geometric functions over this space and for the first time, she gave a recursive formula for calculating the volume of this moduli space of bordered Riemann surfaces. She shows that this volume function is a polynomial in the lengths of the border components. The constant term in this polynomial is the classic Weil-Petersson Volume of moduli space of curves.

In the first talk we review the basic definitions, and the main theorem. We apply the new method for the case of the moduli space of curves of genus 1 with 1 marked point using McShane identity. Then we give a vast generalization of the McShane identity that is used for the general case. In the second talk, we give the statement of the recursive formula for the volume of and prove the polynomial behavior of the volume function. Finally in the third talk, we investigate the Weil-Petersson symplectic structure and give a method to calculate the geometric functions on it. Then using the generalized McShane identity we prove the main theorem. .



Information: