Here’s my second set of lecture notes for a 4 1 2 \frac{1}{2}-hour minicourse at the Summer School on Algebra at the Zografou campus of the National Technical University of Athens. Part 1 is here, and ...

Note: These pages make extensive use of the latest XHTML and CSS Standards. They ought to look great in any standards-compliant modern browser. Unfortunately, they will probably look horrible in older ...

The history This paper got its start in April 2007 when Allen Knutson raised a question about Schur functors here on the n n -Category Café. I conjectured an answer, and later Todd Trimble refined the ...

Maybe people who are all excited about the “field with one element” should start at the beginning and think a bit about the “group with no elements”.

Did you ever have a math teacher who seemed grumpy? At least in America, schools are full of them. Why do they get that way? Maybe it’s because they spent too long controlling rowdy students, or ...

Here in the n -Café we happen to talk about the various notions of generalized smooth spaces every now and then (last time starting here). I was dreaming of having, at one point, a survey of the ...

which you may think of as a Lagrangian depending only on the fields (not on their derivatives) which are elements of M. That complex of vector spaces will extract for us the nature of the critical ...

I’ve just given you a dirty great hint as to what’s behind that provocative first title. For Axler, ‘done right’ means ‘done without determinants’. He claims that determinants should be delayed until ...

Ecologically, I explained some ways to measure the diversity of an ecosystem — but only taking into account abundance (how many of each species there are), not similarity (how much the various species ...

It’s been around a while, so maybe you’ve already seen it… but I just heard about the BBC documentary called Dangerous Knowledge. According to the summary, it’s about “four brilliant mathematicians — ...

With your help, I would like to start amassing a collection of wisdom on gnarly issues in physics. Let’s start with dimensional analysis. I thought I had this pretty much figured out, until Kehrli ...

I’m slowly cooking up a big stew of ideas connecting Grothendieck’s Galois theory to the Brauer 3-group, the tenfold way, the foundations of quantum physics and more. The stew is not ready yet, but ...