Notes and Talks

Course notes

These are more polished versions of the notes that I live-TeXed when I took the corresponding courses at MIT. There will inevitably be errors, which are due to me and not the course instructor. Problems that appear in these notes do not correspond to problem sets used in the actual class. Rather, they are problems that I collected from various sources.

Commutative Algebra

Covers standard topics in commutative ring theory that is widely used in algebraic number theory or algebraic geometry. Based on 18.705 in Fall 2023 taught by Prof. Davesh Maulik.

Algebraic Topology

Covers topological spaces, fundamental groups, covering spaces, homology, cohomology, and Poincare duality. Based on 18.901 in Spring 2023 taught by Prof. Anthony Conway and 18.905 in Fall 2024 taught by Prof. Jeremy Hahn.

Expository papers and notes

Here are some of my shorter expository papers or notes.

The Necklace Splitting Problem

Given a necklace containing beads with \(n\) different colors, it is possible to split it evenly into \(k\) pieces using at most \(k(n-1)\) cuts. This article explains a complete proof of this results (due to Alon), which shockingly use a substantial amount of algebraic topology.

Rational Points on Curves

Introduces the reader to some of the main ideas of the proofs of Faltings' Theorem and explains its conditional algorithmic version given by Alpöge and Lawrence's preprint. Written as a final paper in Harvard Math 282Z in Spring 2026, taught by Dr. Max Weinreich.

Primality Testing in Polynomial Time

Introduces the AKS primality testing (an algorithm to determine if a given positive integer is prime in polynomial time) and covers most of its proof. Written in preparation for the talk given at HMMT November 2025 Education.

Cayley's Tree Formula

Proves Cayley's tree formula (that there are \(n^{n-2}\) spanning trees on vertices \(1,2,\dots,n\)) via generating function and Lagrange Inversion Formula. Provides an introduction to complex analysis and residue theorem. Written in preparation for a lecture of MIT 18.A34 (Putnam Seminar) in Fall 2025.

Introduction to Khovanov Homology

Introduces Khovanov homology, a knot invariant obtained by replacing Jones polynomial with appropriate chain complexes. Written as a final paper in 18.904 (Seminar in Topology) Spring 2025 at MIT, taught by Dr. Jonathan Zung.

Power Reciprocity Law

Proves quadratic reciprocity using Gauss sums, discusses the generalization to higher power reciprocity law, and provides a light introduction to class field theory. Written in preparation for the talk given at MOP 2024 Seminar in June 2024.

Desargues Involution Theorem

First written in 2017 and updated in 2020, it is the first ever handout that introduces the Desargues' Involution Theorem and its uses in Olympiad geometry. The pdf file is the same as the AoPS version, except that it is compiled with the new formatting.

Other Presentations

Here are notes or slides from my other presentations. Talks directly related to my papers are linked at my research page.

Secrets of Elliptic Curves

Slides accompanying a 10-minute presentation given at the MIT Math Department Reception during the parents weekend in Fall 2025. Intended for general scientific audience.

Examples of Descent

Notes accompanying the March 5, 2026 talk for MIT STAGE Seminar. It covers examples of descent obstruction and weak Mordell-Weil theorem.