Abstract:
An r-spin structure on a surface can be thought of as a gadget for measuring “Z/rZ-valued winding numbers” of curves on the surface. There is an evident action of the mapping class group on the set of such objects; an r-spin mapping class group is the associated stabilizer. r-spin structures appear in a wide variety of contexts at the interface of topology and algebraic geometry: singularity theory, translation surfaces/Abelian differentials, linear systems on algebraic surfaces. I will explain what is now known about r-spin mapping class groups and the uses to which the theory can be put. This encompasses various projects with my collaborators Aaron Calderon and Pablo Portilla Cuadrado.