Framed mapping class groups, or the topology of families of flat surfaces

Abstract:

Families of surfaces are everywhere in mathematics, not just in topology, but in algebraic geometry, complex analysis, dynamics, and even number theory. The topology of a family of surfaces is governed by a “monodromy representation” that is valued in the mapping class group. I’m interested in (a) developing tools within the mapping class group to better understand monodromy and (b) applying these tools to problems involving families of surfaces, inside and out of topology proper. The thrust of my work over the past few years has been to understand the monodromy of families of surfaces endowed with certain tangential structures (e.g. a framing, a holomorphic 1-form with prescribed zeroes, an “r-spin structure”, etc.) and to apply this to study the topology of the spaces supporting such families (strata of abelian differentials, linear systems in certain algebraic surfaces, versal deformation spaces of plane curve singularities). This represents joint work with Aaron Calderon and Pablo Portilla Cuadrado.

This talk is hosted by Virtual Trends in LDT. Link to the talk site.