Riemann surfaces, dynamics and geometry

Riemann surfaces , dynamics and geometry Course

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.

The work of Maryam Mirzakhani 18

Maryam Mirzakhani has been awarded the Fields Medal for her outstanding work on the dynamics and geometry of Riemann surfaces and their moduli spaces.

Riemann Surfaces and Differential

We present a new approach to the diierential geometry of surfaces in R 3 and R 4 that treats this theory as a \quaternioniied" version of the complex analysis and algebraic geometry of Riemann surfaces.

The Conformal Geometry of Billiards

This article provides an introduction to some recent results in billiard dynamics. We present results that follow from a study of compact Riemann surfaces (equipped with a holomorphic 1-form) and an SL2R action on the moduli spaces of these surfaces. We concentrate on the progress toward classification of “optimal” billiard tables, those with the simplest trajectory structure.