Expository Notes
This is a repository for some of the expository math notes I've written over the years (many from early parts of grad school). Read at your own risk for there are likely to be mistakes.Geometry
Messy handwritten note on the Poincaré-Hopf theorem via Morse theory.
Note on 1st Chern class of $\mathbb{CP}^n$ and relative homotopy.
Handwritten note on the simplest example of the Atiyah-Singer Index Theorem I know: $d+d^*$ and the Euler characteristic of genus-$g$ Riemann surfaces.
Note Atiyah's "New Invariants of 3- and 4-Dimensional Manifolds." The original paper is brilliant and outlines some relationships between Floer theory and gauge theory, topology and geometry. You really should read that first.
Handwritten note from a talk on the Jones Polynomial, by Witten.
Notes on Heegaard Floer theory.
Handwritten note from a talk on Hirzebruch's signature theorem.
Handwritten note from a talk on symplectic cohomology.
Note on Milnor fibrations and Picard-Lefschetz theory.
Very brief note on why virtual fundamental cycles appear in symplectic topology.
Note on basic contact geometry, Boothby-Wang bundles, Liouville domains, symplectic (co)homology, and wrapped Lagrangian Floer homology.
Very brief note on complex K3 surfaces.
Brief note on $G_2$ manifolds.
Physics
Note on symplectic geometry and its relationship to classical mechanics.
Note which compares classical and quantum mechanics from a rather mathematical viewpoint.
Notes on quantum field theory.
Some handwritten notes I took based on classical field theory lectures of Charles Torres: Part 1 and Part 2.
Some handwritten notes I took based on Edward Witten's "Supersymmetry and Morse Theory" paper: Part 1, Part 2, Part 3.
Notes from a talk on light rays and black holes by Witten: Part 1 and Part 2.
From an RTG Seminar on Homological Mirror Symmetry
Collected Examples
Qualifying Exam Notes
These are notes I wrote for my major topic in the qualifying exams based on "Morse Theory and Floer Homology" by Michèle Audin and Mihai Damian.These are notes I wrote for my minor topic in the qualifying exams based on "The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds" by John Morgan.Random Notes
Note from a first semester graduate course on real analysis I took. The examples for different modes of convergence are rather useful.
These are condensed notes for SBU's comprehensive math exams. The notes themselves are not comprehensive. For example, they do not cover representation theory.
Finite fields: notes from an abstract algebra class I taught.
Things I learned from 3Blue1Brown.
Irrationality of $\pi$ via Niven.
Short proof that the harmonic series diverges.
Slides for a college presentation on Gödel's Incompleteness Theorems in the context of 20th century analytic philosophy of language.