Mathematics Department Colloquium
Fall 2005

Date: Thursday, December 8 Time: 4:00 p.m. Location: 114 McAllister Building Name: Scott Wolpert Affiliation: University of Maryland Title: Integration and symplectic reduction for Teichmueller space Abstract: The Teichmüller space $\mathcal T(S)$ is the space of homotopy conformal structures for a surface $S$. Recent progress on the Weil-Petersson (WP) geometry of $\mathcal T$ includes understanding: the $CAT(0)$ WP geometry, harmonic maps to $\mathcal T$, the quasi isometric equivalence of $\mathcal T$ to the Thurston-Hatcher pants complex, the existence of designer metrics, and the work of Maryam Mirzakhani on integration and symplectic reduction. We will present an overview of Mirzakhani's integration scheme for $\mathcal T$ and connections to applications. The considerations include McShane's identity, and for the moduli space of Riemann surfaces recursion formulas for WP volumes and intersection numbers of tautological bundles. The recursion formulas satisfy the string equation and the dilaton equation. Gregg McShane found for the lengths $\{\ell_{\gamma}\}$ of simple closed geodesics on a once punctured torus with hyperbolic metric the universal identity $$\sum_{scg's\ \gamma}\frac{1}{1+e^{\ell_{\gamma}}}=\frac12.$$ Mirzakhani generalized the identity and developed a scheme for calculating WP volumes. An example is provided by the volume of the moduli space of hyperbolic metric tori with geodesic boundaries of prescribed lengths $L_1,\ L_2$ $$V_{1,2}=\frac{1}{192}(4\pi^2+L_1^2+L_2^2)(12\pi^2+L_1^2+L_2^2).$$ For more see http://www.math.harvard.edu/$\sim$ mirzak/ http://www.math.umd.edu/$\sim$ saw/