I was staring at a bonfire on a beach the other day and realized that I didn’t understand anything about fire and how it works. (For example: what determines its color?) So I looked up some s…| Annoying Precision
In Part I we discussed some conceptual proofs of the Sylow theorems. Two of those proofs involve reducing the existence of Sylow subgroups to the existence of Sylow subgroups of $latex S_n$ and $la…| Annoying Precision
As an undergraduate the proofs I saw of the Sylow theorems seemed very complicated and I was totally unable to remember them. The goal of this post is to explain proofs of the Sylow theorems which …| Annoying Precision
This is a post I wanted to write some time ago; I’ve forgotten why, but it was short and cute enough to finish. Our starting point is the following observation: Theorem 1: Universal lossless …| Annoying Precision
Note: this is a repost of a Facebook status I wrote off the cuff about a year ago, lightly edited. As such it has a different style from my other posts, but I still wanted to put it somewhere where…| Annoying Precision
In this post we’ll describe the representation theory of the additive group scheme $latex \mathbb{G}_a$ over a field $latex k$. The answer turns out to depend dramatically on whether or not $…| Annoying Precision
As a warm-up to the subject of this blog post, consider the problem of how to classify $latex n \times m$ matrices $latex M \in \mathbb{R}^{n \times m}$ up to change of basis in both the source ($l…| Annoying Precision
Let $latex k$ be a commutative ring. A popular thing to do on this blog is to think about the Morita 2-category $latex \text{Mor}(k)$ of algebras, bimodules, and bimodule homomorphisms over $latex …| Annoying Precision
In the previous post we described a fairly straightforward argument, using generating functions and the saddle-point bound, for giving an upper bound $latex \displaystyle p(n) \le \exp \left( \pi \…| Annoying Precision
(Part I of this post is here) Let $latex p(n)$ denote the partition function, which describes the number of ways to write $latex n$ as a sum of positive integers, ignoring order. In 1918 Hardy…| Annoying Precision