On a smoothness characterization for good moduli spaces, joint with Dan Edidin and Matt Satriano, published in Advances in Mathematics in April 2024.

A preprint is available for free on the arXiv. I was responsible for writing the second section of this paper (pp. 5 - 20 in the preprint).

In mathematics, we are often interested in the spaces that result when taking a large space and 'identifying' some points (making a new, smaller space by declaring some points to be indistinguishable from each other). A typical example would be folding a piece of paper in half: so the top left corner is identified with the bottom left corner, and so on. Identifications that arise from a procedure like folding a sheet of paper leave a crease in the paper, points that are not identified to any other point. Mathematicians call such points singular (meaning 'exceptional/rare'), as the crease line is a one-dimensional object, while the paper itself is two-dimensional. The opposite of singular is smooth.

We considered the general problem of identifying which paper-folding rules were singular, and which were smooth (in mathematical jargon: which complex representations of a group have smooth GIT quotients?) The answer for a list of finitely many rules has been known since the 1950s, due to Shephard and Todd—the space is smooth only when the rules are generated by those rotating one dimension and leaving fixed a hyperplane. In our work, we allow for infinitely many rules, and recovered in some cases a natural generalization of the theorem of Shephard and Todd: we prove roughly that smoothness follows when the rules are generated by a rotation in one dimension leaving fixed a hypersurface.

Generalized Pythagorean Lutes, published in the Proceedings of Bridges 2022 in August 2022.

The paper is available for free from the proceedings here (PDF, 8MB).

In this paper I investigated the construction of a geometric pattern (attributed to Pythagoras, but probably much more recent) called the "lute of Pythagoras". The pattern features an appealing self-symmetry between the pentagon and its stellation, the pentagram, demonstrating how they fit 'within' each other in an infinite chain, descending in scale exponentially with the golden ratio. In this short paper, I formalized the geometric structure of the lute of Pythagoras and generalized it to higher dimensions. I proved that such structures can exist only in dimensions 2, 3, and 4, and that the lute of Pythagoras itself is a two-dimensional shadow of a larger 'generalized lute of Pythagoras' existing in four dimensions, based on an exceptional four-dimensional shape called the dodecaplex.

Integrality theorems for symmetric instantons, M.Math thesis (supervised by Benoit Charbonneau), August 2022.

This thesis is available online through UWSpace.

In this thesis, I studied objects from mathematical physics called 'instantons': particular solutions to the Yang—Mills equations that appear in various places in geometry, topology, knot theory, and high-energy particle physics. The instanton equations are very complicated, and thus difficult to solve directly. One promising idea is to look just for solutions that have some prescribed symmetry; these symmetries impose additional equations that can in some cases be used to directly construct interesting examples of instantons.

The original methodology for constructing instantons by symmetries was set out in a series of papers by Singer, Sutcliffe, et al. from the late 1990s and early 2000s. In my thesis, I generalized their framework and provided a formal proof of correctness. Then, I used the generalized framework to resolve a series of questions that had been left open in the original papers, constructing two new examples of instantons with the symmetries of the dodecahedron, and using a computer to prove that any instanton with the symmetries of a dodecaplex must satisfy a certain lower bound on its charge, resolving a 2013 question of Allen—Sutcliffe.

Studying Wythoff and Zometool constructions using Maple, joint with Benoit Charbonneau, published in Maple in Mathematics Education and Research in February 2020.

A preprint is available for free on the arXiv. I was responsible for the initial writing of the paper and most of the software development.

In this paper, we concern ourselves with the mathematical educational toy Zometool. We had experience working in large-scale projects with Zometool previously (see the Great Omnitruncated Dodecaplex Barn Raising Project, University of Waterloo 2019), in which we built certain four-dimensional geometric objects using Zometool.

Any time one builds a four-dimensional object in three-dimensions, it is necessary to "project a dimension away": for example, by placing the skeleton of a three-dimensional cube in front of a light source, one can project it to a two-dimensional image on a wall. Not all projections are equal: some (like the stereographic projection distort lengths, while others (such as the orthogonal projection, where we imagine an infinitely strong light source placed infinitely far away) may cause edges to cross each other. In this project, we created a software package in Maple to determine what projections could actually be constructed in the Zometool building system, and to generate plans for building complicated geometric projections in Zometool. We focused especially on a class of shapes called Wythoff constructions, which are the result of looking at images of a point in a system of mirrors called a kaleidoscope.