My Work
Papers
Extremal Values of Pi (2022), The American Mathematical Monthly, 129:10, 933-951.
Notes
Random Notes: These are notes that I have written on various topics, usually with the goal of understanding details that are elided in some paper or textbook that I am reading.
Connelly’s Proof of Brown’s Collaring Theorem: This is a note that I produced while reading Robert Connelly’s remarkably short paper “A New Proof of Brown’s Collaring Theorem,” to help organize my thoughts while trying to fill in a couple of details that are left to the reader. Although I have not made the connection explicit, it serves as a useful supplement to the generalized Schoenflies theorem, as described in Putman's note on the topic.
The Generalized Schoenflies Theorem: Morton Brown's proof of the generalized Schoenflies theorem has repeatedly been described as "elementary," a rather tenuous descriptor, which in this case refers to the fact that few prerequisites are needed beyond a first course in point-set topology. However, the arguments in the remarkably short paper are still nuanced and various details are implicitly left to the reader. In this note, I do my best to recount Brown's elegant argument, while sacrificing brevity in favor of overly thorough explanation and seeking to restrict the prequisites entirely to a first course in point-set topology (accepting the Jordan-Brouwer separation theorem as a black box).
An Exotic Sphere: These notes were prepared to accompany a final presentation for Inna Zakharevich's course on K-theory and characteristic classes, taken in Fall 2022. I have made an attempt to write out as many details as possible, particularly around places where I faced some initial confusion, which took significant thought or additional computations to resolve. I presuppose some familiarity with quaternions, vector bundles and characteristic classes. Additionally, there are two major black-boxed results: the Hirzebruch Signature Theorem (specifically for 8-manifolds) and the Reeb Sphere Theorem.
The Local Version of Ehresmann's Theorem: In the proof of the Arnold-Liouville Theorem and some related results, we consider the local structure of a smooth map near a compact, regular fiber. This resembles the situation of Ehresmann's theorem, but the global assumption of properness is too strong and a local version is used instead. For my own edification, I have written out the details of the local version in this brief note.
Jordan Normal Form via Exercises: In this set of twenty guided exercises, I outline proofs of the Jordan Normal Form and Jordan-Chevalley Decomposition, in terms of a framework that helped to clarify my own understanding of these results. This is inspired by another note on the topic and makes no claim to originality whatsoever.
The Euler Class and Morse Theory: In this brief note, I described how Morse theory can be used to demonstrate that the Euler class of a closed (oriented) manifold is Poincaré dual to the Euler characteristic (assuming some pre-existing knowledge of all topics involved). This is a very classical argument that is also related to the Poincaré-Hopf index theorem (see §6 of Milnor's Topology from the Differentiable Viewpoint).
Undergraduate Honors Thesis — Morse Theory and the h-Cobordism Theorem: This document provides a lightly illustrated exposition on the topics listed in its title (sans some technical details that space and time precluded). Some brief commentary and errata can be found in this blog post.
Math Circles: In my work for the Berkeley and Stanford Math Circles, I wrote the following sets of notes:
As a warning, these are all from when I had very little experience with teaching or expository writing. I have also delivered sessions on knot theory (*knot colorings, alternating knots, the Kaufmann bracket), graph theory (*Eulerian and Hamiltonian paths, planarity and colorability), the geometry of curves and surfaces (Whitney-Graustein and Gauss-Bonnet theorems), metric geometry (*normed planes and values of pi) and *group theory. For the starred topics, I have written problem sets (some more complete than others); these are not posted publicly, but I am happy to share them with anyone planning to teach a session on one of these topics.
Posters
This work was mostly conducted independently, with helpful advice and inspiration from Cornelia Van Cott. I presented this research at the 2021 MAA Golden Section Meeting and the Math For All in NOLA Conference (March 2021). A full account of these results can be found in my paper "Extremal Values of Pi."
Smooth Resolutions of Gelfand-Zetlin Polytopes and Toric Varieties
This work was conducted in Summer 2019 as part of the Summer Undergraduate Research Fellowship at UC Berkeley, in collaboation with my team-mates and the advisorship of David Nadler. We presented our research the 2019 SURF Conference.