Category Archives: math

Anything mathematical.

A positive characteristic theory for polar representations?

Let $G$ be a split reductive algebraic group over a field $k$ of characteristic zero and $\mathfrak{g}$ it's Lie algebra. If $T\subset G$ is a maximal torus with Lie algebra $\mathfrak{t}$ and Weyl group $W$, then there is a well-known isomorphism of algebras $$k[\gfr]^G\xrightarrow{\sim}k[\mathfrak{t}]^W.$$ This is called the Chevalley restriction theorem. There are many ways […]

Relax, PhDs: applying to 100+ jobs is normal

Applying for jobs after a PhD and my postdoc was one the weirdest things I ever did. I haven't written too much about it before, but because it is so bewildering I thought I'd give out some stats on how my application process went. The most obvious statistic is the number of jobs to which […]

Sums of powers of digits

Take a number written in decimal, like $25$. Take the sum of squares of its digits: $2^2 + 5^2 = 29$. Can you ever get the number you started with? In fact, no positive natural number greater than one is the sum of squares of its decimal digits. However, 75 is pretty close: $7^2 + […]

Automorphic representations: a short list of books

This is a short list of books to get you started on learning automorphic representations. Before I talk about them, I will first define automorphic representation, which will take a few paragraphs. To start, we need an affine algebraic $F$-group scheme $G$ where $F$ is a number field or function field. We let $\A_F$ be […]

Graph: number of primes containing a given digit

You can ask lots of questions about primes. After posting 50 facts about primes, I couldn't resist making another graph. In this one, the x-axis is $n$ and the y-axis is the number of primes up to $n$ that contain a given decimal digit (written in decimal, of course). I've plotted all of these on […]

50 Awesome facts about prime numbers

A prime is a natural number greater than one whose only factors are one and itself. I find primes pretty cool, so I made a list of 50 facts about primes: The first twenty primes are 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, […]