Today, I share a guest post by Margot Schou, who I had the good fortune of mentoring at the start of her career. With her permission, I inserted footnotes with links to some relevant further readin…| Henri's Math Education Blog
October is here, and with it the crisp fall air here in the northeast US. No matter where you are, you can get a breath of fresh air with some problem solving in your math classes! Here is the 2015…| Reflections and Tangents
The second fact is perhaps not very well known. It may even be hard to understand what it means. Though the octonions are nonassociative, for any nonzero octonion gg the map| golem.ph.utexas.edu
September is here, and so is the school year for my US friends. Start a habit of doing some problem solving in your math classes this year! Here is the 1990 Calendar of Problems from 35 years ago f…| Reflections and Tangents
Wow the summer is flying by, and it is back-to-school time already in many parts of the US. If you’re about to start your academic year, here is the August 2010 Calendar of Problems from 15 y…| Reflections and Tangents
Joram’s seminar 2025 Here is my summary of the recent Joram’s seminar that took place on July 9 and 10 in Jerusalem. Much of the seminar was about the the paper Product Mixing in Compac…| Combinatorics and more
Note: this article| mathesis
Note: this article| mathesis
Note: this article| mathesis
Quanti venti conosci?| wok
The monoid of n×nn \times n matrices has an obvious nn-dimensional representation, and you can get all its representations from this one by operations that you can apply to any representation. So its category of representations is generated by this one obvious representation, in some sense. And it’s almost freely generated: there’s just one special relation. What’s that, you ask? It’s a relation saying the obvious representation is nn-dimensional! | golem.ph.utexas.edu
Frank Cassano and Anya Sturm are math teachers at Marin Academy, an independent high school in San Rafael. They presented on “Integrating Argumentation and Problem-Solving” at the Calif…| Henri's Math Education Blog
Spring is here, and I’m sharing the May Calendar for some problem solving enjoyment in the last stretch of your school year. This is the May 2019 Calendar of Problems from 6 years ago for you…| Reflections and Tangents
On BlueSky, Bryan Meyer asks: Henri, in your experience, what are the pros and cons of using function diagrams with kids (in addition to the more standard Cartesian representation)? My BlueSk…| Henri's Math Education Blog
Update: Let me mention a ninth paper that just appeared on the arXive. IX. … and the optimal sofa for the moving sofa problem is … Gerver’s sofa. Optimality of Gerver’s Sofa…| Combinatorics and more
How can we tackle relational algebra of the sort seen in optimization models using tools from AlgebraicJulia?| blog.algebraicjulia.org
Acsets are great, but what if attributes could be variables?| blog.algebraicjulia.org
A follow-up to Algebraic Geometry for the Working Programmer, this post explains a category-theoretic approach to symbolic open dynamical systems.| blog.algebraicjulia.org
In this series of posts, we investigate the duality between algebra and geometry in order to develop new types of lenses. In this first post, we review some basic ideas about algebraic geometry that will be needed in the coming posts.| blog.algebraicjulia.org
Here’s the thing: I only have one prep this year. The whole year. I have a lot of credentials. I can teach any subject except science. (OK, language and PE as well, but they’re not real…| educationrealist
In early December, I attended the California Math Council Northern Section conference in Asilomar, as I’ve done almost every year since the mid-1980’s. In my last post, I discussed my session on fr…| Henri's Math Education Blog
At some point, maybe thirty years ago, it became fashionable to emphasize functions and their multiple representations in secondary school math. This was in part driven by the newly available elect…| Henri's Math Education Blog
If $latex V$ is a module of a Lie algebra $latex L$, there is one submodule that turns out to be rather interesting: the submodule $latex V^0$ of vectors $latex v\in V$ such that $latex x\cdot v=0$…| The Unapologetic Mathematician
There are a few constructions we can make, starting with the ones from last time and applying them in certain special cases. First off, if $latex V$ and $latex W$ are two finite-dimensional $latex …| The Unapologetic Mathematician
There are a few standard techniques we can use to generate new modules for a Lie algebra $latex L$ from old ones. We’ve seen direct sums already, but here are a few more. One way is to start …| The Unapologetic Mathematician
As might be surmised from irreducible modules, a reducible module $latex M$ for a Lie algebra $latex L$ is one that contains a nontrivial proper submodule — one other than $latex 0$ or $latex…| The Unapologetic Mathematician
Sorry for the delay; it’s getting crowded around here again. Anyway, an irreducible module for a Lie algebra $latex L$ is a pretty straightforward concept: it’s a module $latex M$ such …| The Unapologetic Mathematician
It should be little surprise that we’re interested in concrete actions of Lie algebras on vector spaces, like we were for groups. Given a Lie algebra $latex L$ we define an $latex L$-module t…| The Unapologetic Mathematician
It turns out that all the derivations on a semisimple Lie algebra $latex L$ are inner derivations. That is, they’re all of the form $latex \mathrm{ad}(x)$ for some $latex x\in L$. We know tha…| The Unapologetic Mathematician
We say that a Lie algebra $latex L$ is the direct sum of a collection of ideals $latex L=I_1\oplus\dots\oplus I_n$ if it’s the direct sum as a vector space. In particular, this implies that $…| The Unapologetic Mathematician
Let’s go back to our explicit example of $latex L=\mathfrak{sl}(2,\mathbb{F})$ and look at its Killing form. We first recall our usual basis: $latex \displaystyle\begin{aligned}x&=\begin{…| The Unapologetic Mathematician
The first and most important structural result using the Killing form regards its “radical”. We never really defined this before, but it’s not hard: the radical of a binary form $…| The Unapologetic Mathematician
