Higher spin mapping class groups and their applications

A higher spin structure is a structure on a Riemann surface that assigns a ``winding number’’ to each isotopy class of simple closed curve. Higher spin structures appear in many problems involving families of Riemann surfaces, such as the study of smooth plane curves or strata of Abelian differentials. In this talk I will discuss the basic theory of higher spin structures, the subgroup of the mapping class group preserving such a structure, and some recent advances in the theory that lead to developments in algebraic geometry and the theory of translation surfaces. Portions of this work are joint with Aaron Calderon.