Gluing opposite sides of a parallelogram together gives a torus. A translation surface is a surface obtained by identifying sides of more complicated polygons by translations. They arise naturally when exploring both billiards trajectories and geodesics in the moduli space of curves. I will give an introduction to these objects and explain these connections. I will then present an important system of coordinates on the space parameterising translation surfaces, the period coordinates, and discuss results characterising the periods that can arise.

Abstract:

In this talk, I focus on regularity of so-called very weak solutions to the Poisson  Problem for the linear fractional Laplacian on a bounded domain and equipped with homogeneous Dirichlet boundary conditions. I will recall known results and show you how we could improve these results recently by using classical symmetrization techniques.

 

This is a joint work with  Barbara Brandolini (University of Palermo, Italy) and Vincenzo Ferone (university of Naples, Federico II, Italy)