# 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.
• 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.
• Seminar on spherical varieties and L-functions The following are handwritten notes from several talks I gave in a seminar at MIT in Spring 2019. I am posting them as they might be helpful as a quick introduction to spherical varieties.
• Talk 1 Introductory talk explaining classical Hecke period integrals in the language of Sakellaridis-Venkatesh for $$X=\mathbb G_m\backslash \mathrm{PGL}_2$$
• Talk 2 Formulation of the global conjecture of Sakellaridis-Venkatesh
• Talk 3 An overview of the combinatorics associated to spherical varieties and the construction of the dual group of $$X$$ following Knop-Schalke
• Talk 4 A brief summary of Knop's construction of the little Weyl group $$W_X$$ using the cotangent bundle of $$X$$.
• Invariant differential operators on spherical varieties (pdf) Notes from a seminar talk at IAS summarizing Knop's paper The asymptotic behavior of invariant collective motion (1994).
• 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.