Math 885501 Fall 2018

Topics in Geometry & Topology

Kathryn Lindsey

Topics in Geometry & Topology

Kathryn Lindsey

Topic 1: Entropy and pseudo-Anosov maps

- "Expose 10: Some dynamics of pseudo-Anosov diffeomorphisms," (with marked sections) by A. Fathi and M. Shub, from Thurston's work on surfaces, by FLP.

- Thurston's work on surfaces, by Fathi, Laudenbach, Poenaru et al.

- On iterated maps of the interval, by John Milnor and William Thurston.

The core text for our exploration of this topic.

- Entropy in dimension one, by William Thurston.

Classifies numbers which arise as the topological entropy associated to
postcritically finite maps of the unit interval. The manuscript is
full of intriguing ideas and questions, but the exposition is not very
polished and is hard to read. I'll talk about the first few sections;
we may return to subsequent sections later in the course.

- Continuity of core entropy of quadratic polynomials, by Giulio Tiozzo.

Thurston's notion of "core entropy"
extends ideas from Milnor-Thurston kneading theory to the realm of
holomorphic dynamics and, in particular, Julia sets. Tiozzo
proves that core entropy of quadratic polynomials is a continuous
function of external angle in parameter space.

- Math Overflow post: magical powers of the Schwarzian derivative, by William Thurston.

A post by Thurston in response to a question on math overflow. The Schwarzian derivative comes up in On iterated maps of the interval; I'd like to understand this post and how it relates to maps of the unit interval.

Topic 3: Thurston's topological characterization of rational maps

- A proof of Thurston's topological characterization of rational maps, by Adrien Douady and John Hubbard

Our main text for this section.
We broaden our focus from postcritically finite maps of the unit
interval to postcriticallyfinite maps of the Riemann sphere.

- A positive characterization of rational maps, by Dylan Thurston

- Thurston equivalence of topological polynomials, by Laurent Bartholdi and Volodymyr Nekrashevych

Topic 5: Shapes of polyhedra and triangulations of the sphere

- Shapes of polyhedra and triangulations of the sphere, by William Thurston

- Notes on Shapes of Polyhedra, by Rich Schwartz