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The man who knew partition asymptotics
(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...
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More on partition asymptotics
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...
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What’s a fire, and why does it – what’s the word – burn?
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...
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Higher linear algebra
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...
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Singular value decomposition
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...
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The representation theory of the additive group scheme
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...
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Gradient descent
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...
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Compression and Kolmogorov complexity
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...
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Meditation on the Sylow theorems I
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...
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Meditation on the Sylow theorems II
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...
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