Exposition
Expository writings of mine. None of the material presented is original.

Introduction to \(D\)modules and representation theory (pdf)
Cambridge Part III Essay. This document attempts to provide a succinct yet thorough introduction to
some basic properties of algebraic \(D\)modules. I hope to modify/update this essay as my knowledge of the
subject improves.

The moduli stack of \(G\)bundles (pdf) (arXiv)
Harvard University Senior Thesis. This paper provides an expository account of the geometric properties of the moduli stack of \(G\)bundles.

Functor of points description of the flag variety (pdf)
Personal note on the connection between \(G\)equivariant line bundles on the flag variety \(G/B\) and the functor of points of \(G/B\) using Plücker relations.

Existence of the quotient scheme \(G/H\) (pdf)
Proof of the representability of the fppf sheaf \(G/H\) by a scheme for (not necessarily reduced) algebraic groups \(H \subset G\) over an arbitrary field. Also proves that the quotient is a geometric quotient in the case \(H\) and \(G\) are smooth.

Local algebra in algebraic geometry
(pdf)
An overview of some facts from local algebra and how they relate to algebraic geometry. Based on
the course Math 233B. Theory of Schemes, taught by Dennis Gaitsgory at Harvard, Spring 2010.

Theorem on formal functions, Stein factorization, and
Zariski's Main Theorem (pdf)
Discussion and proofs of the theorem on formal functions, Stein factorization, and
the various forms and applications of Zariski's Main Theorem. Everything is proved
for proper morphisms (as opposed to only projective morphisms).

Higher direct images of coherent sheaves under a proper morphism
(pdf)
A proof that for a proper map of noetherian schemes, higher direct images of a
coherent sheaf remain coherent. Includes a short introduction to derived functors.
Notes
The following are notes from various courses/talks. They have not been carefully proofread.

Topics in calculus and algebra
(html)
Taught by Ian Grojnowski at University of Cambridge, Lent 2012.

Moduli stacks of vector bundles
(pdf)
Notes from my talk for the Part III Algebraic Geometry Seminar, Lent 2012.

Abstract algebra
(pdf)
Math 55A lecture notes, taught by Dennis Gaitsgory at Harvard, Fall 2007.

Cryptography
(pdf)
Computer Science 220R lecture notes, taught by Michael O. Rabin at Harvard, Fall 2009.