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August 30, 2023

Grothendieck–Galois–Brauer Theory (Part 3)

Posted by John Baez

Last time we saw that Galois theory is secretly all about covering spaces. Among other things, I told you that a field kk gives a funny kind of space called Spec(k)\mathrm{Spec}(k), and a separable commutative algebra over kk gives a covering space of Spec(k)\mathrm{Spec}(k). I gave you a lot of definitions and stated a few big theorems. But I didn’t prove anything, and some big issues were left untouched.

One of these is: why is a separable commutative algebra over kk like a covering space of Spec(k)\mathrm{Spec}(k)? Today I’ll talk about that, and actually prove a few things.

Posted at 10:11 AM UTC | Permalink | Followups (14)

August 28, 2023

Three Postdoc Positions on Quantum Programming in Edinburgh

Posted by Tom Leinster

The University of Edinburgh is looking to recruit three full-time postdoctoral researchers to work on the project Rubber DUQ: flexible Dynamic Universal Quantum programming with Dr. Chris Heunen.

  • Duration: 3 years
  • Salary: £37,099 – £44,263
  • Start: 1 December 2023 or soon thereafter
  • Deadline: 25 September 2023
  • Apply here and here
Posted at 8:50 PM UTC | Permalink | Post a Comment

August 25, 2023

Grothendieck–Galois–Brauer Theory (Part 2)

Posted by John Baez

Last time we took a tiny taste of Grothendieck’s approach to Galois theory. Now let’s dig in a bit deeper!

Posted at 10:29 AM UTC | Permalink | Followups (1)

August 17, 2023

Representation Theory Question

Posted by John Baez

I’m working with Todd Trimble and Joe Moeller on categories and representation theory, and I’ve run into this question:

Suppose kk is a field of characteristic zero. Then any algebraic representation of GL(n,k)\mathrm{GL}(n,k) restricts to give an algebraic representation of the subgroup DGL(n,k)D \subset \mathrm{GL}(n,k) consisting of invertible diagonal matrices. If two algebraic representations of GL(n,k)\mathrm{GL}(n,k) restrict to give equivalent representations of DD, do they have to be equivalent as representations of GL(n,k)\mathrm{GL}(n,k)?

I think the answer is yes, and maybe I can even string together a proof. But for programmatic reasons I’m seeking a proof that avoids the theory of Young diagrams and the theory of roots and weights. I want to only use easy general stuff. I think I see such a proof for k=k = \mathbb{C}. But it doesn’t seem to generalize. Let me explain.

Posted at 8:55 AM UTC | Permalink | Followups (12)

August 15, 2023

8 and 24

Posted by John Baez

On 8/24 I’m giving a talk about the numbers 8 and 24.

Posted at 8:36 AM UTC | Permalink | Followups (5)

August 11, 2023

Agent-Based Models (Part 2)

Posted by John Baez

Some news! Nathaniel Osgood, Evan Patterson, Kris Brown, Xiaoyan Li, Sean Wu, William Waites and I are going to work together at the International Centre for Mathematical Sciences for six weeks starting on May 1st, 2024. We’re going to use category theory to design better software for agent-based models.

Needless to say, we’re getting started on the math now, since 6 weeks is not enough for that.

Here’s a bit more about our plan.

Posted at 2:55 PM UTC | Permalink | Post a Comment

August 4, 2023

Hoàng Xuân Sính’s Thesis

Posted by John Baez

During the Vietnam war, Grothendieck taught math to the Hanoi University mathematics department staff, out in the countryside. Hoàng Xuân Sính took notes and later did a PhD with him — by correspondence, under very difficult conditions!

I wrote about this here a while back, and on that basis got invited to contribute a paper to a volume in honor of her 90th birthday. Here’s a draft:

I’d love corrections or suggestions.

Posted at 5:36 PM UTC | Permalink | Followups (3)

Who Introduced the Term “Categorical Group”?

Posted by John Baez

I’m writing a paper in honor of Hoàng Xuân Sính’s 90th birthday, and I’m running into a lot of questions.

The term “categorical group” is often used to mean a group object in Cat; these days we also call such a thing a strict 2-group. Who first introduced the term “categorical group”, and when? Perhaps it appeared in the French literature under some name like “groupe catégorique”?

Here are some things I know, which don’t answer my question, but might provide clues.

Posted at 9:24 AM UTC | Permalink | Followups (8)