Research

My research is in set theory, with a particular focus on the application of algebraic and sheaf-theoretic machinery to the subject. Beginning with the work of F. William Lawvere in the 1960s, a vast and rich body of results has established a strong correspondence between the powerful sheaf-theoretic machinery developed by the Grothendieck school and the revolutionary techniques in mathematical logic promulgated by Gödel, Cohen, and others.
The outputs of this project include

• algebraic set theory and the discovery of sheaf-theoretic forcing;
• the field of topos theory and the first-order axiomatization of the category of sets; and
• univalent foundations and the development of homotopy type theory.

Moreover, categorical logic has applications in the development of proof assistants, which seek to formalize all of mathematics and aid in the proof of new results as well. My work aims to better understand the role of sheaf theory in creating and describing set-theoretic phenomena, with a particular focus on two main projects:

1. Higher infinitary combinatorics. The unreasonable effectiveness of Todorcevic's method of minimal walks in capturing the combinatorics of the first uncountable ordinal has led to a series of efforts to extend the theory to higher infinities. In light of Mitchell's theorem on the cohomological dimension of the first omega alephs, and Jeffrey Bergfalk's pursuant notion of higher nontrivial coherence, this can be viewed as part of a broader project to study higher-dimensional analogs of familiar infinitary-combinatorial structures. I am working to bring techniques from homological algebra and sheaf theory to build a more robust theory of such structures.
2. Condensed mathematics and the method of forcing. Condensed mathematics is a program initiated by Dustin Clausen and Peter Scholze to refound and unify disparate theories of geometry. Recent work of Clausen and Scholze and subsequent work of Bergfalk, Lambie-Hanson, and Šaroch has uncovered deep connections between condensed mathematics and the method of forcing. The core hypothesis of this correspondence roughly states that, given an object X and its internalization to an appropriate condensed category, the evaluation of that internalized functor at the Stone space of a boolean algebra can be understood as the object of names for elements of X. With Justin Moore, I am working to develop a theory of condensed sets/groups/etc. which makes this correspondence precise.

My work has been variously supported by teaching assistantships, research assistantships, and (currently) by the NSF Graduate Research Fellowship.

Undergraduate and recreational research

In the summers of 2018, 2019, and 2020, I had the privilege of participating in the University of Chicago REU.
My projects focused on the algebra of arithmetic functions, a novel generalization of the game of Hex, and the model theoretic techniques underlying nonstandard analysis.
I also have a bad habit of thinking far too long about toy problems. You should ask me sometime about one-half morphisms, cardflipping, convoluted monoids, or hops and leaps on graphs.