A tour through Mirzakhani’s work on moduli spaces of Riemann surfaces

On the Branch Loci of Moduli Spaces of Riemann Surfaces

The spaces of conformally equivalent Riemann surfaces, Mg where g ≥ 1, are not manifolds. However the spaces of the weaker Teichmüller equivalence, Tg are known to be manifolds. The Teichmüller space Tg is the universal covering of Mg and Mg is the quotient space by the action of the modular group. This gives Mg an orbifold structure with a branch locus Bg . The branch loci Bg can be identified...

On Weil-petersson Symmetry of Moduli Spaces of Riemann Surfaces

In this article, we give a perspective on several results, old and new, concerning geometric structures of moduli spaces of Riemann surfaces with respect to the L2 metric (Weil-Petersson metric) on deformations of hyperbolic metrics. In doing so, we aim to demonstrate that the Weil-Petersson metric is suited to account for the geometry of moduli spaces while the topological type, genus in parti...

Recent Results on the Moduli Spaces of Riemann Surfaces

Moduli spaces of isoperiodic forms on Riemann surfaces

This paper describes the global geometry of the moduli space of holomorphic 1-forms (X,ω) with fixed periods on varying Riemann surfaces of genus g. It establishes completeness for general g and then explores several features of the case g = 2, which yields the first detailed, global picture of the transverse dynamics of the Teichmüller geodesic flow.

Geometric Aspects of the Moduli Spaces of Riemann Surfaces