# Notes

The following are notes from various courses/talks. They have not been carefully proofread.
• Examples of Relative Duality (html)
An ongoing table of examples of Relative Duality in the program of Ben-Zvi—Sakellaridis—Venkatesh. Please email me if you would like to add an example or if you find a mistake!
• Notes on Intersection Complexes and L-functions (pdf)
Notes from informal lectures on my joint paper with Yiannis Sakellaridis. The notes are intended to be supplementary to the paper and mostly focus on background material for setting up the conjectural relation between certain affine spherical varieties and L-functions.
• 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.

## Exposition

Expository writings of mine. None of the material presented is original.
• Universal local acyclicity (ULA) (pdf)
Note containing proofs of several properties of ULA sheaves that are well-known to experts but hard to find in the literature.
• 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.
• 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.