Mathematics Research

I am an applied algebraic topologist, which means that neither pure nor applied mathematicians will count me among their ranks. My work uses the tools of algebraic topology and category theory to study concrete problems, especially those over networks and other stratified structures.

My thesis work extends the theory of weighted cellular sheaves to the setting of infinite-dimensional Hilbert spaces and unbounded operators. In a few words, my work provides diffusion-based tools for understanding and solving consensus and coordination problems in infinite-dimensional spaces. Suppose a network of agents each have an estimated distribution for the value of the S&P 500 at the end of the year. Each agent communicates exclusively with her neighbors. Moreover, each agent may share different information with each neighbor; perhaps Alice shares her expected value with Bob, her full distributional estimate with Charlie, and lies to Dan. My work provides a mathematical model for how the distribution estimate of each agent changes through time. Natural applications of this work include opinion dynamics and sheaf neural networks.

Other research interests include applications of lattice theory in finance, categorical frameworks for linear programming, and a burning desire to solve the edge reconstruction conjecture in graph theory.

Writing

My first book, I Wish They'd Taught Me That, is a collection of conversational-yet-nearly-rigorous essays on math topics that aren't part of the standard undergraduate curriculum, but should be. Topics such as stochastic calculus, transfinite induction, approximation theory, and ultrafilters were selected due to their beauty, low barriers to entry, and lack of presence in the "shared toolkit" of mathematicians. While the book is primarily aimed at undergraduates and early graduate students, this book will have new ideas for all but the most polymathematical mathematicians. This work was co-authored with Robin Pemantle and is to be published by CRC Press in November 2025.

I also write stand-alone essays about philosophy, politics, and whatever else interests me at the moment. You can read some of them here.

I formerly wrote for Machine Appreciation, a mathematics group blog. Some of the posts here are excellent, some are fun and silly, some are a little too ambitious, and some are largely unfinished. Several of my blog posts evolved into more polished essays on this site.

Teaching

I count my teaching ability as among my greatest assets. Through extensive teaching experience inside and outside the classroom, I have become extremely opinionated about good and bad pedagogical practices. I have been disseminating these opinions through my role as a Master TA in the math department, organizing and leading the annual teacher training for the last three years.

I have been a teaching assistant for a variety of courses, including calculus, real analysis, optimization, cryptography, and linear algebra. Most recently, I was the head TA for a new "Linear Algebra with Applications to Engineering and AI" course in the Electrical and Systems Engineering department at Penn. I have also tutored privately and mentored six undergraduate students through Penn's Directed Reading Program on topics including topological data analysis for prediction markets, financial mathematics, mathematical models of international diplomacy, category theory, and all manner of mathematical logic. For my efforts, I have received both the departmental Moez Alimohamed Graduate Student Award for Distinguished Teaching in Mathematics and the college-wide Dean's Awards for Distinguished Teaching by Graduate Students.

In the background, I am (very slowly) working on a linear algebra textbook that reflects my beliefs about how to teach mathematics. The book teaches a standard linear algebra curriculum through a series of guided exercises and minimal connecting exposition. I personally used a subset of these problems to teach linear algebra to an advanced high school student.

Elsewhere

When not doing my work, I enjoy cooking, picking up heavy things, trading on prediction markets, and arguing with my father about politics. I count myself as good at three of these hobbies, and merely alright at the other; I won't tell you which is which. I am also an avid reader of all manner of books and essays.