Author Archives: Jason Polak

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 […]

Using Cartopy to draw maps and plot points

Let's do something simple but useful for geographical data science: drawing a map in Python. I will assume you are using the Anaconda distribution. Unfortunately, there are many mapping libraries for different programming ecosystems and it's a little hard to find out which you should use. For a long time in Python, Basemap was the […]

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 […]

University of California: Goodbye Elsevier

On July 10, 2019, the University of California gave up its access to Elsevier journals. According to Elsevier, The contract ended in December 2018. Since then, while working to find a solution, we have continued to provide access without payment to University of California campuses. Unfortunately, we've been unable to come to an agreement. The […]

Some thoughts on proof assistants

A proof assistant is a computer program that takes as input a proof in a formal language and outputs true if and only if the proof is a valid proof in a formal system. An example would be first-order logic with its inference rules. Proof assistants are supposed to tell you whether your proof is […]

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 + […]