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.
• Roots of unity in an arbitrary group
  Gauss famously proved the existence of a primitive root mod p, or in other words, that the multiplicative group of the field of p elements is cyclic. We examine his proof and generalize it to other settings, in particular proving that a finite group is cyclic iff it doesn't have too many roots of unity. We also describe some results of Frobenius connected to the roots of unity in an arbitrary finite group.


• How to discover a proof of Quadratic Reciprocity*
  If you happen to be familiar with the basic properties of Gauss sums, and decide to try to prove Quadratic Reciprocity using these... here's one approach. Of all the proofs of QR that I've seen, this is one of two that I can reconstruct from memory on the fly. In this essay I try to convey how one might be led to discover the proof.


• Computing quadratic gauss sums
  Gauss discovered a simple formula for the value of quadratic gauss sums. We present Dirichlet's proof of Gauss' formula.


• The Wigdersons' novel approach to uncertainty principles
  The Heisenberg uncertainty principle, originally discovered in the context of physics, quickly inspired a general theorem about fourier transforms (that it's impossible for both a function and its fourier transform to be too localized). Since then, a number of variants on this uncertainty principle have been discovered, each with its own ad hoc proof. Very recently, Avi and Yuval Wigderson discovered a unified framework that recovers many of these variants, and also proves new versions and generalizations. In particular, their approach only requires a few basic properties of the fourier transform, and as a result, applies to a large class of linear operators. In this short note I describe their approach and illustrate it by proving the original Heisenberg uncertainty principle. There's nothing original in this essay--everything proved is a special case of what appears in the Wigdersons' manuscript--and my hope is that the relative brevity and the concreteness of my aims will serve as motivation to read the Wigersons' original paper, which covers a lot more ground and is beautifully written.


• A brief introduction to the Cauchy-Schwarz and Holder inequalities
  The Cauchy-Schwarz inequality (and its most famous generalization, Holder's inequality) are fundamental to many areas of mathematics, physics, statistics, and computer science. The primary goal of this essay is to motivate Cauchy-Schwarz, prove it, and provide a few examples of applications. I also share a mnemonic that helped me remember the inequality; as a bonus, the mnemonic inspires a version of Holder's inequality that I find much more natural than the usual formulation.


• A motivated proof of Chebyshev's theorem
  In the 1850s, Chebyshev proved several fundamental results about the distribution of primes, including Bertrand's Postulate and a slightly weaker version of the Prime Number Theorem. In this essay I give self-contained proofs of both of these.


• A motivated introduction to normal subgroups and quotient groups (aka factor groups)
  Normal subgroups and quotient groups are fundamental concepts taught in every first course on abstract algebra, but in my experience, most students never fully internalize these ideas. The purpose of this essay is to try to motivate these concepts.


• A short, non-inductive proof of Euler's formula for graphs
  A fundamental result in graph theory, originally discovered by Euler, is that the Euler characteristic of any connected planar graph is 2. The goal of this essay is to present a lovely, non-inductive proof of this, discovered jointly by Stephanie Mathew (then a second-year undergraduate) and Red Burton (a computer program written by Siemion Fajtlowicz). The proof relies on Euler's famous characterization of graphs admitting an Euler tour; for completeness, we sketch a proof of this as well. I'm grateful to David Eppstein for posting this proof on his delightful collection of proofs of Euler's formula; I encourage the reader to check out the many other awesome expository articles he's posted.


• 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 e^x. The second (shown to me 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.

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