Research

I study modern aspects of classical computability theory, with particular focus on its interactions with reverse mathematics and algorithmic randomness. My projects include defining almost-everywhere computability, studying the large-scale structure of the Zoo of reverse-mathematical principles, and investigating fine divisions in reverse mathematics.

Translating that into English: I study objects that can and can't be computed, and computational relationships between them. This turns out to be relevant to the study of which axioms are truly necessary to prove a given theorem, as well as to defining what it means for an object to be random. Specifically, I work on three major projects: defining what it means for an object to be computable "most of the time", trying to find large-scale patterns in the sorts of axioms needed to prove different theorems, and using new tools to distinguish theorems that our previous reverse-math approaches would think are equivalent.

For an in-depth professional summary, please see my Research Statement (updated September 2017).

Journal articles

The determined property of Baire in reverse math.

With Damir Dzhafarov, Antonio Montalbán, Reed Solomon, and Linda Brown Westrick.

Submitted.

Dense computability, upper cones, and minimal pairs.

With Denis Hirschfeldt and Carl G. Jockusch.

*Computability*, vol. 8, no. 2, pp. 155–177, 2019.

The computational content of intrinsic density.

*The Journal of Symbolic Logic*, vol. 83, no. 2, pp. 817–828, 2018.

The uniform content of partial and linear orders.

With Damir Dzhafarov, Reed Solomon, and Jacob Suggs.

*Annals of Pure and Applied Logic*, vol. 168, no. 6, pp. 1153–1171, 2017.

Asymptotic density, immunity, and randomness.

*Computability*, vol. 4, no. 2, pp. 141–158, 2015.

Papers in preparation

The weakness of typicality.

With Laurent Bienvenu, Damir Dzhafarov, Ludovic Patey, Paul Shafer, Reed Solomon, and Linda Brown Westrick.

Unpublished work

Asymptotic density and effective negligibility.

Thesis, University of Chicago, 2015.

Parts of this thesis have appeared in "Asymptotic density, immunity, and randomness", "The computational content of intrinsic density", and "Dense computability, upper cones, and minimal pairs".

The Reverse Mathematics Zoo

The RM Zoo is a program to help organize relations among various reverse mathematical principles, particularly those that fail to be equivalent to any of the big five subsystems of second-order arithmetic. Its goal is to make it easier to see known results and open questions, and thus hopefully to serve as a useful tool to researchers in the field. As such, it includes an annotated bibliography for all papers with referenced results.