The “what” and the “why” of framed mapping class groups

Abstract: Given a family of Riemann surfaces, the monodromy representation, valued in the mapping class group of the fiber, is a key invariant that encodes a great deal of information about the topological and algebraic structure of the family. Many natural families, including families of translation surfaces, smooth sections of line bundles on surfaces (e.g. plane curves), and the family of Milnor fibers of a plane curve singularity, are equipped with the additional data of a preferred section of a line bundle (e.g. a holomorphic 1-form). In such circumstances, the monodromy group is valued in a special subgroup known as the framed mapping class group. I will discuss some new tools to understand framed mapping class groups, and the sorts of insight they can bring to the study of the families listed above. This encompasses joint work with Aaron Calderon and Pablo Portilla Cuadrado.

algebraic topologydifferential geometrygeometric topologymetric geometry

Audience: researchers in the topic

University of Toronto Geometry & Topology seminar