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