Teaching

SEP materials

Notes on Algebra

From 2022-2025, I was the instructor of the Algebra SEP (Summer Enhancement Program) at UW—Madison. You can read more about the SEP here. In Summer 2024, as part of my SEP materials, I wrote a series of expository notes on various topics in graduate-level algebra. The notes are not intended to be read front-to-back, but rather with the intention that if the reader is facing an unfamiliar topic for the first time (say, tensor products), they can navigate to that section and find a short, conversational introduction, along with example problems, many of which are drawn from or oriented towards problems that have appeared on past Algebra Qualifying Exams at UW.

• Basic Linear Algebra
• Advanced Linear Algebra
• Ring Theory 1
• Ring Theory 2
• Module Theory 1
• Module Theory 2
• Module Theory 3
• Group Theory 1
• Group Theory 2: Group Actions
• Group Theory 3
• Fields and Galois Theory

Number theory notes

Number theory notes

Recently, I have been relearning my field. As I go, I've been writing myself notes in a novel format: rather than write down whatever proofs I come up with, I write down only the questions I asked myself (or would have asked myself in an ideal problem-solving process) that brought me to those proofs.

• Algebraic Number Theory. Notes about the two fundamental theorems of algebraic number theory: finite generation of the unit group and finiteness of the class group. I have ambitions to someday expand this to include other topics from algebraic number theory, such as an introduction to number fields and number rings, an introduction to Minkowski's theorem, a discussion of Frobenius elements, etc.
• Classifying cyclic cubic extensions. Inspired by a problem I got from Brain Lawrence when I took Class Field Theory with him. I have found that every hour I've spent thinking about this problem has repaid me tenfold in learning something else in number theory.
• A solution to a Silverman exercise I wrote up because a friend of mine got stuck on it.

Other math notes

Short expositions

I have written several short expositions of a more recreational nature:

• A bit about why e is transcendental.
• My reconstruction of how Galois might have thought about the Galois theory of cubic extensions.
• A short note about Inclusion-Exclusion which I wrote for my own reference.
• Always Use Pascal: a lesson about binomial coefficients and thinking mathematically. I wrote this in undergrad, and it is aimed towards undergraduates taking discrete math or combinatorics classes.