Research

Metric Space Geometry

My research is primarily focused on the geometry of metric spaces. In particular, I have spent a lot of time studying metric spaces that are highly symmetric in the sense that they admit transitive actions by collections of metrically controlled self-mappings (e.g. isometries, bi-Lipschitz mappings, quasi-symmetric mappings, or something along those lines). More recently, I've been investigating the Lipschitz geometry of quasiconformal arcs and trees in connection with Lipschitz free spaces and Lipschitz dimension. Here is a list of my publications and preprints in these areas:

Continued Fractions

I've also done a bit of work with continued fractions exhibiting a generalized form of palindromic symmetry. In particular, I generalized a transcendence criteria of Adamczewski and Bugeaud. Here's a link:

Math + Art

My recent exploration of the space between math and art began with some experiments involving epicycloid curves and continued fractions. These experiments led to the following paper and a few images in the Bridges Conference art exhibit:

After that, I became interested in aperiodic tilings of the plane. After learning a bit of Python code (via the help of my friend Noah Weaver) I was able to reproduce various tilings appearing in the literature. Here are a few I produced for inclusion in the Bridges exhibit:

As a result of working a bit with tilings, I was asked by a colleague to be interviewed for an article on the Penrose tiling. Here's a link to the article (in French), published by the online magazine of the University of Fribourg:

Math + Philosophy

I've become quite interested in questions regarding the ontology of abstract mathematical objects. I'm particularly interested in possible theological implications of such questions, and vice-versa (along the lines of those examined in William Lane Craig's book God and Abstract Objects). This interest led to the following article: