I’m reading through Primes of the Form $x^2+ny^2$, by David Cox (link; it’s good!). Here are the high-level notes I took on the first chapter, which is about the theory of quadratic forms.

(Meta point re blog: I’m probably going to start posting more and more of these more high-level notes/sketches on this blog on topics that I’ve been just learning. Up til now I’ve been mostly only posting things that I understand well and for which I have a very polished exposition. But the perfect is the enemy of the good here; given that I’m taking these notes for my own sake, I may as well share them to help others.)

1. Overview

Definition 1. For us a quadratic form is a polynomial $Q = Q(x …$

In this post we’ll make sense of a holomorphic square root and logarithm. Wrote this up because I was surprised how hard it was to find a decent complete explanation.

Let $f \colon U \rightarrow \mathbb C$ be a holomorphic function. A holomorphic $n$-th root of $f$ is a function $g \colon U \rightarrow \mathbb C$ such that $f(z) = g(z)^n$ for all $z \in U$. A logarithm of $f$ is a function $g \colon U \rightarrow \mathbb C$ such that $f(z) = e^{g(z)}$ for all $z \in U$.

The main question …

In Spring 2016 I was taking 18.757 Representations of Lie Algebras. Since I knew next to nothing about either Lie groups or algebras, I was forced to quickly learn about their basic facts and properties. These are the notes that I wrote up accordingly. Proofs of most of these facts can be found in standard textbooks, for example Kirillov.

1. Lie groups

Let $K = \mathbb R$ or $K = \mathbb C$, depending on taste.

Definition 1. A Lie group is a group $G$ which is also a $K$-manifold; the multiplication maps $G \times G \rightarrow G$ (by $(g_1, g_2) \mapsto g_1g_2$) and the inversion map $G \rightarrow G$ (by $g\mapsto {\mathrm{g\; \dots }}^{}$

More than six months late, but here are notes from the combinatorial nullsetllensatz talk I gave at the student colloquium at MIT. This was also my term paper for 18.434, “Seminar in Theoretical Computer Science”.

1. Introducing the choice number

One of the most fundamental problems in graph theory is that of a graph coloring, in which one assigns a color to every vertex of a graph so that no two adjacent vertices have the same color. The most basic invariant related to the graph coloring is the chromatic number:

Definition 1. A simple graph $G$ is $k$-colorable if it’s possible to properly color its vertices with $k$ colors. The smallest such $k$ is the chromatic number $\chi(G)$.

In this exposition we study a more general notion in which the set of permitted colors …

This will be old news to anyone who does algebraic topology, but oddly enough I can’t seem to find it all written in one place anywhere, and in particular I can’t find the bit about $\mathsf{hPairTop}$ at all.

In algebraic topology you (for example) associate every topological space $X$ with a group, like $\pi_1(X, x_0)$ or $H_5(X)$. All of these operations turn out to be functors. This isn’t surprising, because as far as I’m concerned the definition of a functor is “any time you take one type of object and naturally make another object”.

The surprise is that these objects also respect homotopy in a nice way; proving this is a fair amount of the “setup …

These are the notes of my last lecture in the 18.099 discrete analysis seminar. It is a very high-level overview of the Green-Tao theorem. It is a subset of this paper.

1. Synopsis

This post as in overview of the proof of:

Theorem 1 (Green-Tao)

The prime numbers contain arbitrarily long arithmetic progressions.

Here, Szemerédi’s theorem isn’t strong enough, because the primes have density approaching zero. Instead, one can instead try to prove the following “relativity” result.

Theorem (Relative Szemerédi)

Let $S$ be a sparse “pseudorandom” set of integers. Then subsets of $A$ with positive density in $S$ have arbitrarily long arithmetic progressions.

In order to do this, we have to accomplish the following.

• Make precise the notion of “pseudorandom”.
• Prove the Relative Szemerédi theorem, and then
• Exhibit a “pseudorandom” set $S$ which subsumes the prime numbers.

This post …

Epistemic status: highly dubious. I found almost no literature doing anything quite like what follows, which unsettles me because it makes it likely that I’m overcomplicating things significantly.

1. Synopsis

Recently I was working on an elegant problem which was the original problem 6 for the 2015 International Math Olympiad, which reads as follows:

Problem [IMO Shortlist 2015 Problem C6]

Let $S$ be a nonempty set of positive integers. We say that a positive integer $n$ is clean if it has a unique representation as a sum of an odd number of distinct elements from $S$. Prove that there exist infinitely many positive integers that are not clean.

Proceeding by contradiction, one can prove (try it!) that in fact all sufficiently large integers have exactly one representation as a sum of an even subset of $S$. Then, the problem reduces to …

This lecture, which I gave for my 18.434 seminar, focuses on the MAX-E3LIN problem. We prove that approximating it is NP-hard by a reduction from LABEL-COVER.

1. Introducing MAX-E3LIN

In the MAX-E3LIN problem, our input is a series of linear equations $\pmod 2$ in $n$ binary variables, each with three terms. Equivalently, one can think of this as $\pm 1$ variables and ternary products. The objective is to maximize the fraction of satisfied equations.

Example 1 (Example of MAX-E3LIN instance)

$$\begin{aligned} x_1 + x_3 + x_4 &\equiv 1 \pmod 2 \\ x_1 + x_2 + x_4 &\equiv 0 \pmod …$$

This was my final paper for 18.099, seminar in discrete analysis, jointly with Sammy Luo and Ryan Alweiss.

We prove that every sufficiently large odd integer can be written as the sum of three primes, conditioned on a strong form of the prime number theorem.

1. Introduction

In this paper, we prove the following result:

Theorem 1 (Vinogradov)

Every sufficiently large odd integer $N$ is the sum of three prime numbers.

In fact, the following result is also true, called the “weak Goldbach conjecture”.

Theorem 2 (Weak Goldbach conjecture)

Every odd integer $N \ge 7$ is the sum of three prime numbers.

The proof of Vinogradov’s theorem becomes significantly simpler if one assumes the generalized Riemann hypothesis; this allows one to use a strong form of the prime number theorem (Theorem 9). This conditional proof was given by Hardy and Littlewood in …

In this post I’ll describe the structure theorem over PID’s which generalizes the following results:

• Finite dimensional vector fields over $k$ are all of the form $k^{\oplus n}$,
• The classification theorem for finitely generated abelian groups,
• The Frobenius normal form of a matrix,
• The Jordan decomposition of a matrix.

1. Some ring theory prerequisites

(Prototypical example for this section: $R = \mathbb Z$.)

Before I can state the main theorem, I need to define a few terms for UFD’s, which behave much like $\mathbb Z$: Our intuition from the case $R = \mathbb Z$ basically carries over verbatim. We don’t even need to deal with prime ideals and can factor elements instead.

Definition 1. If $R$ is a UFD, then $p \in R$ is a …