Here is a link to my CV (updated January 2020).
I am currently working on understanding some unusual connections between Fourier Restriction Theory and classical Geometric Invariant Theory. The idea here is that Fourier restriction inequalities exhibit natural symmetry under the affine transformations (and not just orthogonal transformations), and this symmetry allows one to identify previously-unknown geometric quantities which measure the "non-flatness" of submanifolds of Euclidean space. The first major results in this direction are available here, which use these ideas to characterize Dan Oberlin's "affine Hausdorff measure."
I am also interested in the relationship between decoupling inequalities and number theory. My AIM SQuaRE collaborators and I have a recent paper in which we show how a well-known principle (namely, that decoupling inequalities imply counting results for solutions of systems of Diophantine equations) can be effectively reversed in some cases.
Here are all my recent papers on the arXiv.
Computational Content Creation
I have written some tools for creating and maintaining large banks of LaTeX-formatted questions. The project is available on GitHub.
project is a direct outgrowth of experiences in the classroom. The underlying observation is that simple randomization (where problems, e.g., in Canvas Quizzes, are generated by plugging in
random values for key variables) often intruduces unwanted algebraic challenges for students. A better approach is to develop and curate massive banks of questions which are highly optimized for computational simplicity. I am happy to provide examples on request.
In recent years, I have been teaching Math 104 in the SAIL (Structured, Active, In-Class Learning) format.
Active learning techniques have been shown to decrease failure rates in STEM classes (from 33.8% to 21.8%).
A core challenge is that students feel less successful despite having learned more.
I have developed a particular emphasis on fostering freshman students' sense of social belonging in the classroom. Here is a link to an article by Aguilar, Walton, and Weiman that has heavily influenced my thinking about teaching. I recently wrote an article about Social Belonging in Introductory Calculus and spoke a little more about these issues in an Omnia article.
I am currently working on a collection of self-contained Ximera activities for Math 104.
Other Online Items
IFS Fractal Archive: Some images that I produced in 2016 in connection with the Penn Summer Math Academy.
CNSF 2013 Animation: An animation that I made in 2013 as part of my presentation at the Coalition for National Science Funding 19th Annual Exhibition.