These are a collection of notes on mathematical topics.

Notes on the homotopy groups of topological modular forms at the prime 3 (for a talk at Juvitop).

Notes on the Browder-Novikov theorem, giving necessary and sufficient conditions for a simply connected 4k-dimensional space to have the homotopy type of a manifold.

Some notes on a theorem of Snaith that constructs complex K-theory.

Notes on simplicial commutative rings (written for a talk at a workshop).

The fundamental group of a smooth projective curve in nonzero characteristic (in particular, the fact that it admits 2g topological generators). .pdf

Notes on Kodaira vanishing (and a few other things, like Chern-Weil theory). .pdf

Verdier duality .pdf

The Dold-Kan correspondence between simplicial abelian groups and chain complexes. .pdf

(Unfinished) notes on the semicontinuity theorem (on the cohomology of a flat family) in algebraic geometry. .pdf

Notes on chapter 1 of "Higher Topos Theory," for a talk. .pdf

Interactive proofs, notes from a talk I gave at an undergraduate mathematics colloquium. .pdf

Zariski's Main Theorem and some of its applications (following mostly EGA and Raynaud's Anneaux locaux henseliens). .pdf

Notes from a talk I gave at HMMT on Sperner's lemma and some of its applications. .pdf

A short expository paper on Arrow's theorem. .pdf

An exposition of the Synge-Weinstein theorem in Riemannian geometry. .pdf

Some notes on divisors and line bundles..pdf