Skip to the Main Content

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 browsers, like Netscape 4.x and IE 4.x. Moreover, many posts use MathML, which is, currently only supported in Mozilla. My best suggestion (and you will thank me when surfing an ever-increasing number of sites on the web which have been crafted to use the new standards) is to upgrade to the latest version of your browser. If that's not possible, consider moving to the Standards-compliant and open-source Mozilla browser.

October 27, 2023

Grothendieck–Galois–Brauer Theory (Part 6)

Posted by John Baez

I’ve been talking about Grothendieck’s approach to Galois theory, but I haven’t yet touched on Brauer theory. Both of these involve separable algebras, but of different kinds. For Galois theory we need commutative separable algebras, which are morally like covering spaces. For Brauer theory we’ll need separable algebras that are as noncommutative as possible, which are morally like bundles of matrix algebras. One of my ultimate goals is to unify these theories — or, just as likely, learn how someone has already done it, and explain what they did.

Both subjects are very general and conceptual. But to make sure I understand the basics, my posts so far have focused on the most classical case: separable algebras over fields. I’ve explained a few different viewpoints on them. It’s about time to move on. But before I do, I should at least classify separable algebras over fields.

Posted at 9:00 AM UTC | Permalink | Followups (2)

October 19, 2023

The Flora Philip Fellowship

Posted by Tom Leinster

The School of Mathematics at the University of Edinburgh is pleased to invite applications for the 2023 Flora Philip Fellowship. This four-year Fellowship is specifically aimed at promising early-career postdoctoral researchers from backgrounds that are under-represented in the mathematical sciences academic community (e.g. gender, minority ethnicity, disability, disadvantaged circumstances, etc.). The Fellowship aims to provide a supportive and collegial environment for early-career researchers to develop their research and prepare themselves, with support from an academic mentor, for future independent roles in academia and beyond.

The closing date is 24 November and the job ad is here.

Posted at 10:30 PM UTC | Permalink | Post a Comment

October 12, 2023

Grothendieck–Galois–Brauer Theory (Part 5)

Posted by John Baez

Lately I’ve been talking about ‘separable commutative algebras’, writing serious articles with actual proofs in them. Now it’s time to relax and reap the rewards! So this time I’ll come out and finally explain the geometrical meaning of separable commutative algebras.

Just so you don’t miss it, I’ll put it in boldface. And in case that’s not good enough, I’ll also say it here! Any commutative algebra AA gives an affine scheme XX called its spectrum, and AA is separable iff X×XX \times X can be separated into the diagonal and the rest!

I’ll explain this better in the article.

Posted at 8:48 AM UTC | Permalink | Followups (11)

October 1, 2023

The Free 2-Rig on One Object

Posted by John Baez

These are notes for the talk I’m giving at the Edinburgh Category Theory Seminar this Wednesday, based on work with Joe Moeller and Todd Trimble.

(No, the talk will not be recorded.)

Schur Functors

The representation theory of the symmetric groups is clarified by thinking of all representations of all these groups as objects of a single category: the category of Schur functors. These play a universal role in representation theory, since Schur functors act on the category of representations of any group. We can understand this as an example of categorification. A ‘rig’ is a ‘ring without negatives’, and the free rig on one generator is [x]\mathbb{N}[x], the rig of polynomials with natural number coefficients. Categorifying the concept of commutative rig we obtain the concept of ‘symmetric 2-rig’, and it turns out that the category of Schur functors is the free symmetric 2-rig on one generator. Thus, in a certain sense, Schur functors are the next step after polynomials.

Posted at 3:00 PM UTC | Permalink | Followups (15)