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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.
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.
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 gives an affine scheme called its spectrum, and is separable iff can be separated into the diagonal and the rest!
I’ll explain this better in the article.
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 , 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.