Let $F$ be a field and let $n$ be a positive integer. Does there exist an integer $k$ such that $A^k = 0$ whenever $A\in M_n(F)$ is nilpotent? Here, $M_n(F)$ denotes the ring of $n\times n$ square matrices with entries in $F$. Such an integer exists. For any nilpotent matrix $A$, we have $A^p = […]
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 […]
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 […]
Consider $n$ observed data points $(x_1,y_1),\dots, (x_n, y_n)$. We think they might satisfy a linear model $y = ax + b$. Finding the coefficients $a$ and $b$ is called linear regression, and the most typical way to find them is the method of least squares: that is, we find $a$ and $b$ that minimize the […]
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 […]
I've talked a lot about von Neumann regular rings on this blog, so I thought I'd write an informal short survey on them, collecting some facts we've already seen and many new ones. It should give you an idea of what von Neumann regular rings are. Most of the facts that I did not explicitly […]
We say that an associative ring $A$ is von Neumann regular if for every $a\in A$ there exists a $x\in A$ such that $axa = a$. That is a rather strange condition, isn't it? But, you can think of $x$ as a pseudoinverse to $a$. This weakening of inverses has a homological counterpart: if every […]
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 […]
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 […]
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 + […]