Third Year Major Oral: Squiggly Circles

Abstract: Given a self-homemorphism $\varphi$ of a surface $\Sigma$, we can form the mapping torus, $M_\varphi$, a 3-manifold fibering over the circle. In this talk, I will begin by covering the Nielsen–Thurston classification of surface automorphisms---discussing the topological properties of periodic, reducible, and pseudo-Anosov maps with an additional focus on the structure of the mapping torus. I will then discuss the work of Cannon and Thurston studying the geometric structure of the universal cover of hyperbolic surface bundles. Along the way, we will encounter a variety of twisted and distorted copies of the circle---including the infamous "space filling curves," which encode geometric and dynamical properties of the 3-manifolds used to construct them. I will end by discussing my own research into $S^2$-filling Peano curves and decompositions of the circle which might shed light on the structure of these Peano curves.

Advisor: Steven Frankel