## Expository Writing

There's a lot of mathematics I don't understand. When I do understand something, and am feeling particularly excited about it, I write it up. Usually, the intended audience is me as an undergraduate. Below are a few such write-ups (with more on the way). They are in various stages of (in)completion, and are surely riddled with errors -- so read with a dash of salt! And if you find any errors or have suggestions, please get in touch.

• Arnold's elementary proof of the insolvability of the quintic
In the 1960s, Arnold invented a remarkably elementary proof of the nonexistence of a general quintic formula; among other things, his proof doesn't require any Galois theory (nor any group theory!), and in some ways is more satisfying than the result one obtains from Galois theory. This essay is a brief introduction to Arnold's approach. I'm grateful to Boaz Katz for posting this lovely video which introduced me to Arnold's proof; I strongly urge you to watch it both before and after reading my essay.

• A crash course in Fourier analysis
This is a relateively brief introduction to Fourier analysis, discussing Fourier series, the Fourier transform, and Fourier analysis on finite abelian groups. Proofs are kept to a minimum.

• Are subgroups of prime index normal?
We consider the question of the title, among other things giving an exposition of a lovely result due to Lam.

• Two short proofs that e is irrational
Here are two very short proofs of the irrationality of e. The first is classical, and proceeds by analyzing the Taylor expansion of ex. The second (shown to me recently by Trevor Wooley) relies on a nice characterization of rational numbers in terms of isolated points.

• The easiest (to remember) proof of Quadratic Reciprocity
Quadratic Reciprocity is one of the most important theorems in elementary number theory. Unfortunately, proofs of QR tend to be technical, ad hoc, and difficult to remember. Here I present a little-known proof due to G. Rousseau, which is extremely easy to remember and uses nothing more than the Chinese Remainder Theorem and elementary modular arithmetic. I also show how the supplement to QR can be deduced from QR itself.

• The sum-product problem and Solymosi's theorem
An old conjecture (due to Erdős and Szemerédi) asserts that for any subset A of the integers, either A+A or AA must be exceptionally large. The strongest result towards this conjecture is due to Solymosi; his proof is beautiful and strikingly elementary. In this essay I discuss the problem and give a self-contained exposition of Solymosi's proof.

• Hindman's Theorem via ultrafilters
Hindman's theorem is a beautiful result in additive combinatorics which has a particularly striking proof using ultrafilters. In this paper I survey necessary results on ultrafilters and describe the proof of Hindman's theorem. This material was patiently explained to me by Mike Pawliuk, to whom I am grateful.

• Differentiating under the integral sign
In his autobiography Surely you're joking, Mr. Feynman, Richard Feynman mentions in passing a trick he picked up in high school of differentiating under the integral sign. I'd never seen such a technique, and was curious about it. After running across a short note by Noam Elkies about a homework solution by (then first-year undergraduate) Inna Zakharevich, I realized that this was exactly what Feynman was talking about. Here I illustrate the technique with a few examples.

• The Cantor-Bernstein-Schröder Theorem
The Cantor-Bernstein-Schröder Theorem (first proved by Dedekind in 1887) asserts that if A injects into B and B injects into A, then A and B must be in bijective correspondence. This is trivial to prove for finite sets A and B, but surprisingly tricky for infinite sets. I strongly urge you to try it yourself before reading!

• A short(er) proof of the divergence of the Harmonic series
It is a classical fact that the harmonic series diverges, and for many years the first proof I saw of this fact was my favorite. Recently, I realized that the proof is longer than it needs to be. Here's the short, short version.

• Irrational Algebraic Integers
Here I give two particularly short and striking proofs of the irrationality of the square-root of 2. Both of these proofs have nice generalizations, which I also discuss.

• Legendre, Jacobi, and Kronecker symbols
The Legendre symbol is a staple in courses on elementary number theory. In practice, however, generalizations of this symbol are more useful, the most common being the Jacobi and Kronecker symbols. I kept forgetting exactly how the Kronecker symbol was defined, so one day I decided to write it down. This document was the result. I still forget the definition, but now I know where to look!

• Mertens' theorem
An easy consequence of the Prime Number Theorem is that the sum of the reciprocals of the primes up to x grows like log log x. However, you don't need such a sledgehammer to prove this result, as Mertens demonstrated in 1874. Here I sketch a proof of his result.

• Paley's theorem, revisited
In 1932, Paley proved an omega result on character sums. His proof is a good vehicle for illustrating several standard research tools, including the methods of completing and smoothing a sum, as well as applying the Fejér kernel.

• Furstenberg's topological proof of the infinitude of primes
The infinitude of primes has been proved many times since the classic proof described by Euclid in his Elements. This proof, by Furstenberg, is the most surprising one I know.

• Hölder's inequality as a convexity result
Hölder's inequality has a variety of proofs. Here I describe one which I find particularly appealing; it derives the inequality from a convexity statement. I was introduced to this approach in a course given by Hugh Montgomery.

Here, for lack of a better place to put it, is an essay by Gian-Carlo Rota.

• Ten lessons I wish I had learned before I started teaching differential equations
As a prospective freshman at MIT, I sat in on several lectures. One of them was given by Gian-Carlo Rota. I had never imagined that such a large lecture could feel so intimate, so alive, so electric, and the experience continues to inform my approach to teaching. Unfortunately, I never got the chance to thank Rota -- he died unexpectedly two weeks later.

This essay is a written version of an address Rota gave in 1997. I have never taught differential equations, so I have no strong opinions on much of the content. However, I found it both entertaining and informative, full of the character I remember from that MIT lecture, so I thought I'd share it.