Here are some preprints of my publications:
The Maximum Codimension of a Salem Submanifold (2026)
In this paper, we obtain a complete classification of the parameters $m$ and $k$ such that an $m$-dimensional manifold can be embedded in $\mathbb{R}^{m+k}$ as a Salem Set, that is, supporting a probability measure $\mu$ with Fourier decay $|\widehat{\mu}(\xi)| \lesssim |\xi|^{-m/2}$ for all $|\xi| \geq 1$. The classification has a perhaps surprising connection to the Radon-Hurwitz numbers, which describe the number of linearly independent vector fields one may choose on the sphere.
Multipliers of Spherical Harmonic Expansions (2025)
In my PhD thesis, I provide a more expository look at work on bounds for multipliers of the Laplace-Beltrami operators on compact manifolds described in the paper Multipliers for Spherical Harmonic Expansions, as well as discussing new methods for combining estimates for such multipliers at different frequency scales via a theory of atomic decompositions in $L^p(M)$ on a compact manifold $M$ for $1 < p < 2$, planned for publication in a separate paper.
Multipliers for Spherical Harmonic Expansions (2024)
In this paper, accepted for publication in the Indiana University Mathematics Journal, using methods of Fourier Integral Operators, with relations to Finsler geometry, we obtain a simple and effective condition which is necessary and sufficient for an operator on the sphere $S^n$ commuting with rotations to be bounded on $L^p(S^n)$. As a consequence, we obtain the first non-Euclidean transference principle, which takes $L^p$ estimates for Fourier multipliers on Euclidean space and giving $L^p$ estimates for associated operators on the sphere.
Large Salem Sets Avoiding Nonlinear Configurations (2021)
In this paper, published in the Proceedings of the Royal Society of Edinburgh, robust concentration inequalities are used to improve upon results previously only applied to the construction of sets avoiding linear patterns, which support measures with Fourier decay, to the construction of such sets avoiding nonlinear patterns.
Cartesian Products Avoiding Rough Patterns (2019)
In my MSc. thesis, I provide a more expository look at the work completed with Malabika Pramanik and Joshua Zahl described in the paper Large Sets Avoiding Rough Patterns, as well as new work planned for publication in a separate paper, on the topic of the Fourier decay of measures supported on sets avoiding configurations, and on the topic of low rank rough configurations.
Large Sets Avoiding Rough Patterns (2019)
In this paper, published in the Springer Series Harmonic Analysis and Applications, Malabika Pramanik, Joshua Zahl, and I used a novel dyadic discretization strategy combined with a simple application of the pidgeonhole principle to construct fractals avoiding a 'rough' family of patterns.
Proofs in Three Bits or Less (2018)
In this expository article, published as the front-page article in the Spring 2018 edition of the Canadian Mathematical Society's student journal Notes From the Margin, I describe the fundamentals ideas behind the PCP theorem in computational complexity theory, geared towards a general mathematical audience.
Here are some teaching resources I have used to help math students in past courses I've taught:
Linear Algebra and Optimization Course Notes (2025)
In the Summer of 2026, I was hired by Kaitlyn Phillipson and Sebastien Roch as a contributing author to develop an online textbook for Math 345: Linear Algebra and Optimization, which aims to teach Linear Algebra and Multivariate Calculus with a view towards Data Science. Our main goal was to introduce applications of the theory to data science (such as the analysis of time series, the method of gradient descent, and clustering analysis) as early as possible in order to motivate material, including various programming activities using Python and Numpy. All course material was edited in PreTeXt, a language based on LaTeX which provides a much greater support for people with visual impairment via conversion of material into various formats, including web formats and Nemeth Braille (both electronically and in embossable formats). Supplementary programming exercises were also written to accompany the textbook.
Discrete Mathematics Discussion Materials (2025)
In the Spring of 2026, Mitch Kellar and I edited the discussion material for the Math 240 course at the University of Wisconsin, Madison, which is the introductory course in Discrete Mathematics. Our main goal was to improve the accessibility of the contents by editing the material using PreTeXt, a language based on LaTeX which provides a much greater support for people with visual impairment via conversion of material into various formats, including web formats and Nemeth Braille (both electronically and in embossable formats).
University of Wisconsin Analysis SEP Course Notes (2021, 2022, 2025)
In Summer 2021, 2022, and 2025, together with Max Bacharach, I got the opportunity to prepare first-year graduate students at the University of Madison, Wisconsin for the analysis qualifying exam in that department. To help students review for the exam, we produced a list of solutions to various problems for the analysis SEP. Our goal with writing these solutions was to try and isolate the general heuristics / principles which can often be applied to problems on different problems on the exam, so students could conciously think of these principles when applying them to the new questions that would occur on their upcoming qualifying exam.
A continuously updated set of notes I have taken on various mathematical topics can be found on my github page. Most notes are a work in progress, with the latter parts of these notes much less cohesive than the latter part of the notes. Here are a selection of notes that are the most developed:
Here are some notes salvaged from various talks I've given over the years:
On Trilinear Oscillatory Integral Inequalities And Related Topics (2024)
after a paper by Michael Christ
Detangling Entangled Paraproducts (2024)
after a paper by Polona Durcik
The Lax Hormander Parametrix For the Half-Wave Equation (2023)
Heuristics Behind the Lax-Hormander Parametrix for variable coefficient Half-Wave Equation via High-Frequency Asymptotic Solutions
Anticoncentration and Polynomial Decompositions (2022)
after a paper by Daniel Kane
Logarithmic Improvements To Spectral Bound Projection in the Presence of Negative Curvature (2022)
after a paper by Christopher D. Sogge and Matthew D. Blair
Algorithmic Aspects of Brascamp Lieb Inequalities (2021)
after papers by Ankit Garg, Leonid Gurvits, Rafael Oliveira, and Avi Wigderson
Incidence Theorems in Arbitrary Characteristic (2019)
the polynomial method, flecnodes, and arithmetic genus
laplacians, riemannian manifolds, and cohomology
modular forms, sums of squares, and forbidden Eisenstein series
Radon Transforms and Exceptional Projections (2018)
singular directions, mixed norm estimates, and interpolation of besov spaces
characters, fourier inversion, and poisson summation
The Fourier-Stieltjes Transform (2016)
weak-$*$ density and physical measurement
operators, closed subspaces, and abstract harmonic analysis
Lift and Project Techniques (2016)
combinatorial optimization, vertex cover, and non-linear linear programming
On Molecular Gases and the Natural Numbers (2016)
an introduction to ergodic theory
The Brouwer Fixed Point Theorem (2016)
homology, vector Fields, and hex
The Jordan Separation Theorem (2016)
curves, intuition, and the van kampen theorem
Abstract Nonsense in Computing Science (2015)
categories, the curry howard isomorphism, and cartesian closure