Research

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I am a dynamicist and geometer! Currently I'm working specifically in marked length spectrum rigidity and thermodynamic formalism.

 

When someone asks me what I do, I usually say that I'm really interested in how knowing the way things move around on an object can give us details about the shape of the object. 

Some research keywords: CAT(0) spaces, differential geometry, dynamical systems, geodesic flow, geodesics, marked length spectrum rigidity, negative curvature, nonpositive curvature, Riemannian geometry, thermodynamic formalism

What is...?

Dynamics

The math of what happens when things move around. Think balls on a pool table, how light travels, or what a soccer ball would do if you kicked it and watched where it went for infinite time. Dynamics, or how things move, are affected by the geometry of the space or object. I like studying how dynamics and geometry affect each other.

Marked Length Spectrum Rigidity

The marked length spectrum of a space keeps track of the lengths of the shortest closed curves (often geodesics)  in a space and the homotopy class that they belong to. Marked length spectrum rigidity asks if the marked length spectrum is enough to tell you everything you need to know about the geometry of the space. Sometimes the answer is yes! This is a survey of rigidity theory by Ralf Spatzier in 2004.

Thermodynamic Formalism

Sounds like physics, right? The part of thermodynamic formalism that I'm the most interested in cares about whether or not systems have equilibrium states and if those equilibrium states are unique. There's pressure and entropy and other physics-sounding words involved, but I'm firmly on the pure math interpretation side of this.

Math Blog Posts

I talk a little bit about some of the settings and tools I use in blog posts.

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Papers and Preprints

Phase transitions for the geodesic flow of a rank one surface with nonpositive curvature
With Keith Burns, Jérôme Buzzi, and Todd Fisher
To appear in Dynamical Systems in 2021

Unique equilibrium states for geodesic flows on flat surfaces with singularities
With Ben Call, David Constantine, Alena Erchenko and Grace Work
Preprint, 2021