Abstract:

The earthquake map was developed by Thurston and Kerckhoff to solve the Nielsen realization problem in the 80's. Earthquake is a generalization of applying a Dehn twist, a very simple surgery that played a central role in the study of the mapping class group. On the other hand, the map can be seen as a flow on a bundle over the moduli space. Mirzakhani used the dynamics viewpoint and made multiple breakthrough in the field. In my talk I will survey some of Mirzakhani's contributions before introducing my result regarding the systole of hyperbolic surfaces. The theme of the talk will be the interplay between geometry, group theory, and dynamical systems. There are also explicit examples where we can use continued fractions to characterize the systole behavior under the earthquake flow.