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 (pdf)
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 I (Coming Soon!)
Covers singular homology, cohomology, and Poincaré Duality. Based on 18.905 in Fall 2024 taught by Prof. Jeremy Hahn.
Expository papers and notes
- Primality Testing in Polynomial Time (pdf)
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 (pdf)
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 (pdf)
Introduces Khovanov homology, a knot invariant obtained by replacing Jones polynomial with appropriate chain complexes. This was written as a final paper in 18.904 (Seminar in Topology) Spring 2025 at MIT, taught by Dr. Jonathan Zung. - Power Reciprocity Law (pdf)
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 (pdf)
(AoPS)
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
- Secrets of Elliptic Curves (slides)
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.