Math

Research

Broadly speaking, I enjoy thinking about problems of a combinatorial flavor that use tools from other areas of mathematics. Right now, I am mostly working in discrete geometry and descriptive set theory.

Current projects

Integer points in dilates

Given a d-dimensional simplex P with vertices in [−1, 1]d, how many integer points lie in the dilate k ⋅ P? When k is much larger than the dimension, of course, the asymptotic bound given by the Ehrhart polynomial is the best we can hope for. My goal is to find tight bounds for small values of k that depend only on the original polytope, not the dimension.

Measurable circle--squaring

Marks and Unger provided the first Borel solution to Tarski's circle--squaring problem by studying the graphs generated by algebraically independent translations. My goal is to optimize the complexity of this equidecomposition with more advanced number-theoretic and graph-theoretic methods.

Sparse polynomial factors

If a multivariate polynomial is sparse, i.e. only has a few monomials, must its factors also be sparse? This is a major open problem in algebraic complexity theory. My work focuses on the special case when one of the factors is known to be sparse.

Papers and preprints

Year Title Link
2023 Colouring bottomless rectangles and arborescences, with Cardinal, Knauer, Micek, Pálvölgyi, and Ueckerdt ComGeo
2023 Explicit bounds for the layer number of the grid, with Dillon arXiv
2020 Pattern problems related to the Arithmetic Kakeya conjecture, with Cowen-Breen, Karangozishvilli, and Wang arXiv

Talks

Year Title Location
2021 Orthogonal projections for quantum channels and operator systems Developments in Computer Science, Budapest
2020 Colouring bottomless rectangles and arborescences 36th European Workshop on Computational Geometry, 2020

Notes

As a fierce advocate for high-quality mathematical writing, I aim to practice what I preach. Please let me know if you find any errors in my notes.