"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." - Paul Halmos
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 and respectively. The goal of this post is to understand the Sylow -subgroups of in more detail and see what we can learn from them […]
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 I am actually able to remember, several of which use our old friend The -group fixed point theorem (PGFPT): If […]
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 compression is impossible. That is, there is no function which takes as input finite strings (over some fixed alphabet) and always […]
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 it’d be easier to find and share than Facebook. Gradient descent, in its simplest where […]
In this post we’ll describe the representation theory of the additive group scheme over a field . The answer turns out to depend dramatically on whether or not has characteristic zero. Preliminaries over an arbitrary ring (All rings and algebras are commutative unless otherwise stated.) The additive group scheme over a base ring has functor of […]
As a warm-up to the subject of this blog post, consider the problem of how to classify matrices up to change of basis in both the source () and the target (). In other words, the problem is to describe the equivalence classes of the equivalence relation on matrices given by . It turns out that […]
Let be a commutative ring. A popular thing to do on this blog is to think about the Morita 2-category of algebras, bimodules, and bimodule homomorphisms over , but it might be unclear exactly what we’re doing when we do this. What are we studying when we study the Morita 2-category? The answer is that […]
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 stuff, and here’s what I learned. Fire Fire is a sustained chain reaction involving combustion, which is an exothermic reaction in […]
In the previous post we described a fairly straightforward argument, using generating functions and the saddle-point bound, for giving an upper bound on the partition function . In this post I’d like to record an elementary argument, making no use of generating functions, giving a lower bound of the form for some , which might help explain intuitively […]
(Part I of this post is here) Let denote the partition function, which describes the number of ways to write as a sum of positive integers, ignoring order. In 1918 Hardy and Ramanujan proved that is given asymptotically by . This is a major plot point in the new Ramanujan movie, where Ramanujan conjectures this result […]