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March 3, 2024

Modular Curves and Monstrous Moonshine

Posted by John Baez

Recently James Dolan and I have been playing around with modular curves — more specifically the curves X 0(n)X_0(n) and X 0 +(n)X^+_0(n), which I’ll explain below. Monstrous Moonshine says that when pp is prime, the curve X 0 +(p)X^+_0(p) has genus zero iff pp divides the order of the Monster group, namely

p=2,3,5,7,11,13,17,19,23,29,31,41,47,59,71 p = 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71

Just for fun we’ve been looking at n=11n = 11, among other cases. We used dessins d’enfant to draw a picture of X 0(11)X_0(11), which seems to have genus 11, so for X 0 +(11)X^+_0(11) to have genus zero it seems we want the picture for X 0(11)X_0(11) to have a visible two-fold symmetry. After all, the torus is a two-fold branched cover of the sphere, as shown by Greg Egan here:

But we’re not seeing that two-fold symmetry. So maybe we’re making some mistake!

Maybe you can help us, or maybe you’d just like a quick explanation of what we’re messing around with.

Posted at 9:43 PM UTC | Permalink | Post a Comment

February 20, 2024

Spans and the Categorified Heisenberg Algebra

Posted by John Baez

I’m giving this talk at the category theory seminar at U. C. Riverside, as a kind of followup to one by Peter Samuelson on the same subject. My talk will not be recorded, but here are the slides:

Abstract. Heisenberg reinvented matrices while discovering quantum mechanics, and the algebra generated by annihilation and creation operators obeying the canonical commutation relations was named after him. It turns out that matrices arise naturally from ‘spans’, where a span between two objects is just a third object with maps to both those two. In terms of spans, the canonical commutation relations have a simple combinatorial interpretation. More recently, Khovanov introduced a ‘categorified’ Heisenberg algebra, where the canonical commutation relations hold only up to isomorphism, and these isomorphisms obey new relations of their own. The meaning of these new relations was initially rather mysterious, at least to me. However, Jeffery Morton and Jamie Vicary have shown that these, too, have a nice interpretation in terms of spans.

Posted at 10:51 PM UTC | Permalink | Followups (4)

February 14, 2024

Cartesian versus Symmetric Monoidal

Posted by John Baez

James Dolan and Chris Grossack and I had a fun conversation on Monday. We came up some ideas loosely connected to things Chris and Todd Trimble have been working on… but also connected to the difference between classical and quantum information.

Posted at 6:46 AM UTC | Permalink | Followups (12)

February 4, 2024

The Atom of Kirnberger

Posted by John Baez

The 12th root of 2 times the 7th root of 5 is

1.333333192495 1.333333192495\dots

And since the numbers 5, 7, and 12 show up in scales, this weird fact has implications for music! It leads to a remarkable meta-meta-glitch in tuning systems. Let’s check it out.

Posted at 8:09 PM UTC | Permalink | Followups (3)

January 29, 2024

Axioms for the Category of Finite-Dimensional Hilbert Spaces and Linear Contractions

Posted by Tom Leinster

Guest post by Matthew di Meglio

Recently, my PhD supervisor Chris Heunen and I uploaded a preprint to arXiv giving an axiomatic characterisation of the category FCon\mathbf{FCon} of finite-dimensional Hilbert spaces and linear contractions. I thought it might be nice to explain here in a less formal setting the story of how this article came to be, including some of the motivation, ideas, and challenges.

Posted at 1:31 PM UTC | Permalink | Followups (6)

January 24, 2024

Summer Research at the Topos Institute

Posted by John Baez

Are you a student wanting to get paid to work on category theory in Berkeley? Then you’ve got just one week left to apply! The application deadline for Research Associate positions at the Topos Institute is February 1st.

Details and instructions on how to apply are here:

Posted at 2:53 AM UTC | Permalink | Post a Comment

December 4, 2023

Magnitude 2023

Posted by Tom Leinster

I’m going to do something old school and live-blog a conference: Magnitude 2023, happening right now in Osaka. This is a successor to Magnitude 2019 in Edinburgh and covers all aspects of magnitude and magnitude homology, as well as hosting some talks on subjects that aren’t magnitude but feel intriguingly magnitude-adjacent.

Slides for the talks are being uploaded here.

What is magnitude? The magnitude of an enriched category is the canonical measure of its size. For instance, the magnitude of a set (as a discrete category) is its cardinality, and the magnitude of an ordinary category is the Euler characteristic of its nerve. For metric spaces, magnitude is something new, but there is a sense in which you can recover from it classical measures of size like volume, surface area and dimension.

What is magnitude homology? It’s the canonical homology theory for enriched categories. The magnitude homology of an ordinary category is the homology of its classifying space. For metric spaces, it’s something new, and has a lot to say about the existence, uniqueness and multiplicity of geodesics.

Let’s go!

Park in Osaka

Posted at 10:13 PM UTC | Permalink | Followups (55)

December 1, 2023

Adjoint School 2024

Posted by John Baez

Are you interested in applying category-theoretic methods to problems outside of pure mathematics? Apply to the Adjoint School!

Apply here. And do it soon.

  • December 31, 2023. Application Due.

  • February - May, 2024. Learning Seminar.

  • June 10 - 14, 2024. In-person Research Week at the University of Oxford, UK.

Posted at 6:01 PM UTC | Permalink | Post a Comment

Seminar on This Week’s Finds

Posted by John Baez

I wrote 300 issues of a column called This Week’s Finds, where I explained math and physics. In the fall of 2022 I gave ten talks based on these columns. I just finished giving eight more! Now I’m done.

Here you can find videos of these talks, and some lecture notes:

Topics include Young diagrams and the representation theory of classical groups, Dynkin diagrams and the classification of simple Lie groups, quaternions and octonions, the threefold way, the periodic table of nn-categories, the 3-strand braid group, combinatorial species, and categorifying the harmonic oscillator.

If my website dies, maybe these lectures will still survive on my YouTube playlist.

Posted at 4:01 PM UTC | Permalink | Post a Comment

November 23, 2023

Classification of Metric Fibrations

Posted by Tom Leinster

Guest post by Yasuhiko Asao

In this blog post, I would like to introduce my recent work on metric fibrations following the preprints Magnitude and magnitude homology of filtered set enriched categories and Classification of metric fibrations.

Posted at 10:28 PM UTC | Permalink | Followups (8)

November 13, 2023

Mathematics for Climate Change

Posted by John Baez

Some news! I’m now helping lead a new Fields Institute program on the mathematics of climate change.

Posted at 8:36 AM UTC | Permalink | Followups (3)

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)