- Ropes, fractions, and moduli spaces (an exposition of Conway's tangle trick)
Abstract:
This is an exposition of John H. Conway’s tangle trick. We discuss what the trick is, how to perform it, why it works
mathematically, and finally offer a conceptual explanation for why a trick like this should exist in the first place. The mathematical centerpiece is
the relationship between braids on three strands and elliptic curves, and we a draw a line from the tangle trick back to work of Weierstrass, Abel, and
Jacobi in the 19th century. For the most part we assume only a familiarity with the language of group actions, but some prior exposure to the fundamental
group is beneficial in places.
- Higher spin mapping class groups in algebraic and flat geometry (course notes for the Cuernavaca winter school, January 2020)
Abstract:
An r-spin structure is a choice of rth root of the canonical bundle of a Riemann surface. Such structures arise in a variety of settings in geometry;
in this lecture series, we will focus on their role in two places at the interface of algebraic geometry and topology: linear systems on algebraic surfaces
(especially toric surfaces), and translation surfaces (also known as abelian differentials). In both these settings, there are “topological monodromy
groups” valued in the mapping class group that encode important information about these families of Riemann surfaces and their degenerations, and the
presence of r-spin structures is reflected in the underlying group theory. We will outline some recent developments in the theory of these “higher spin
mapping class groups” that allow us to understand monodromy in the above problems, and ultimately to gain new insights into the behavior of these families.
No specialized knowledge of topology or the mapping class group will be assumed. Portions of this work are joint with Aaron Calderon.
- Surface bundles in topology, algebraic geometry, and group theory, with Bena Tshishiku. Notices of the AMS (February 2020)
Abstract:
This is a survey article discussing some of the places that surface bundles appear in mathematics. The aim of the article is to highlight the unusual
diversity of mathematical themes that come together in the subject.