Abstract:

A flat surface can be easily obtained by gluing Euclidean polygons together. Local deformation of flat surfaces via deforming polygons is classical and well-understood. In this talk the focus will be placed on a family of flat surfaces called half-translation surfaces. The space of quadratic differentials, interpreted as a cotangent space of the moduli space of hyperbolic surfaces, can be used to describe optimal quasiconformal maps between hyperbolic surfaces. The goal of the talk is to introduce deformations of half-translation surfaces called flat grafting to develop a better geometric understanding of the space of quadratic differentials. Every fixed topological surface admits a large family of flat structures and deformations of flat surfaces are useful in proving local and boundary properties. One particularly interesting aspect of deformations is the geodesic length spectrum.