Here are some preprints of my publications:
Large Salem Sets Avoiding Nonlinear Configurations (2021)
In this paper, 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 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 soon, 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' 2021 Volume, 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:
University of Wisconsin Analysis SEP Problem Solutions (2021-2022)
In Summer 2021 and 2022, together with Max Bacharach, I got the opportunity to prepare incoming / 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.
Here are some notes salvaged from various talks I've given over the years:
The Lax Hormander Parametrix For the Half-Wave Equation (2023)
Heuristics Behind the FIO 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
Salem Sets Avoiding Nonlinear Patterns (2020)
Slides for a talk relating to the paper "Large Salem Sets Avoiding Nonlinear Configurations"
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
A complete and continuous updated set of the lecture and research notes I have taken on various mathematical topics can be found at my github page. Note that, as a continuous work in process, the beginning of these notes tends to be more cohesive than the latter part of the notes. Here are a selection of notes I am particularly proud of: