I am a Professor of Mathematics at UPenn. I am also affiliated with the AMCS program. My research interests lie at the intersection of harmonic analysis and geometry, including the study of geometric averaging operators (generalizing the Radon transform), oscillatory integral operators, sublevel set estimates, Fourier restriction, and related objects. I have also worked on applications of harmonic analysis to PDEs including the Boltzmann equation and the Gross-Pitaevskii Hierarchy.

Research

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.

Talk Slides

• Swarthmore College Math-Stat Colloquium: Kakeya Needle Problem Here is a YouTube animation that I produced for the talk.
• AMS September 2019 Section Meeting: Multilinear Oscillatory Integral Operators
• Madison Lectures in Fourier Analysis 2019

Important Links

Department of Mathematics Bridge to PhD Program: I have been a co-advisor with Ryan Hynd since shortly after he created the program. See this Penn Today article for more information.

The Mathematics Major

The Calculus Homepages

Mathematics Department Calendar

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.

Inclusive Teaching

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.

Worksheet Templates

Here is a sample worksheet produced using XeLaTeX. The source is available here. Compilation requires downloading and installing the font Fira Sans.

Ximera

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.

Animated Spherical Harmonics

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.

A Continuous Nowhere-Differentiable Function