Expository Writing

I often write mathematical exposition for my own consumption – once in a while, these things are polished enough that I am willing to share them here.

Recent stuff

\(G/H\) embeds in \(G\) using character theory

Naomi Jochnowitz frequently teaches MATH 436 (first course in graduate algebra, covering groups, rings, and modules) at UofR, including when I took it. Naomi has a reputation for having very long and hard homework sets (but also a complementary reputation for giving pizza to those who come to her problem solving sessions and work on these sets).

The problem I found hardest was the following: Let \(G\) be a finite Abelian group, and \(H \leq G\) be a subgroup. Show that \(G\) contains a subgroup isomorphic to \(G/H\).

This problem seems easy, but it's subtler than it looks, even if one assumes the classification of finite Abelian groups. Naomi's solution for this problem was quite complicated, and I was never quite able to wrap my head around all of it. However, about a year or two after the course was over, I found a different solution to the problem while reading one of Keith Conrad's blurbs. This solution doesn't use the fundamental theorem of finite Abelian groups at all – instead, it uses some basic character theory (i.e., Fourier analysis) on finite Abelian groups.

Notes for Haessig's topic course

These are some notes I wrote up when I was auditing Doug Haessig's topics course on \(p\)-adic analysis and \(p\)-adic cohomology – it covers some extremely basic facts about non-Archimedean fields.

Undergrad stuff

I wrote a moderately large number of theses, projects reports, and whatnot as an undergraduate, partly because the continuation of my scholarship required me to do scholarly writing every year. Some of these are pretty rough, and may have mistakes.

Master's Thesis

My thesis supervisor was Somnath Jha in the Dept. of Mathematics and Statistics, IITK. There were two parts to this thesis, each written in a different semester:

Part I: The Vinogradov Theorem (July 2015 — November 2015)

This is an exposition of an old theorem (due to Hardy and Littlewood) that the Generalized Riemann Hypothesis for Dirichlet \(L\)-functions implies that there are only finitely many exceptions to the ternary Goldbach conjecture. I.M. Vinogradov famously removed the assumption of GRH here, which is where the name of the report comes from.

Part II: Prime Numbers and Arithmetic Progressions (January 2016 — April 2016)

This is an exposition on the prime number theorem in arithmetic progressions.

Undergraduate Projects (UGPs)

The IITK math department let us do up to four project based courses as electives, a fact I took full advantage of. All four of my projects were supervised by Shobha Madan de jure; she was de facto superviser in the first two, while in the latter two, I also had external supervisers from the CSE department.

Dirichlet's Theorem (July 2013 — November 2013)

This is an exposition on the basics of Fourier analysis on finite Abelian groups as well as a proof of Dirichlet's theorem on the infinitude of primes in arithmetic progressions.

Arithmetic Progressions in Sets of Integers (January 2014 — April 2014)

This continued the trend of Fourier analysis, this times using Fourier analysis on \(\mathbb{Z}/n\mathbb{Z}\) to prove Roth's theorem on arithmetic progression.

Additive Combinatorics and Incidence Geometry: The Kakeya Problem (July 2014 — November 2014)

This was supervised by Nitin Saxena from CSE, and done jointly with Vijay Keswani. Some basic stuff about the Kakeya problem in finite fields, and in \(\mathbb R\). The slides from the final talk are here.

Additive Combinatorics and Szémeredi’s Regularity Lemma (January 2015 — April 2015)

This was supervised by Rajat Mittal from CSE, and also done jointly with Vijay. Some basic stuff about Szémeredi’s regularity lemma. The slides from the final talk are here.

Miscellaneous

The Hidden Subgroup Problem (January 2016 — April 2016)

This one was written for a course project, specifically a course on quantum computing taught by Rajat Mittal.

The Bombieri-Vinogradov Theorem (May 2013 — July 2013)

This note was the outcome of a summer research fellowship, which gave me funding to stay at IMSc, Chennai for a few months. It was supervised R. Balasubramanian and Sanoli Gun.

Partitions and Rademacher's Exact Formula (June 2012 — July 2012)

This was my first piece of writing for the continuation of the KVPY fellowship. It was supervised by Amitabha Tripathi at IIT Delhi, and written jointly with Rijul Saini.