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March 26, 2025
The McGee Group
Posted by John Baez
This is a bit of a shaggy dog story, but I think it’s fun. There’s also a moral about the nature of mathematical research.
Once I was interested in the McGee graph, nicely animated here by Mamouka Jibladze:
This is the unique (3,7)-cage, meaning a graph such that each vertex has 3 neighbors and the shortest cycle has length 7. Since it has a very symmetrical appearance, I hoped it would be connected to some interesting algebraic structures. But which?
March 20, 2025
Visual Insights (Part 2)
Posted by John Baez
From August 2013 to January 2017 I ran a blog called Visual Insight, which was a place to share striking images that help explain topics in mathematics. Here’s the video of a talk I gave last week about some of those images:
It was fun showing people the great images created by Refurio Anachro, Greg Egan, Roice Nelson, Gerard Westendorp and many other folks. For more info on the images I talked about, read on….
March 12, 2025
Category Theory 2025
Posted by Tom Leinster
Guest post by John Bourke.
The next International Category Theory Conference CT2025 will take place at Masaryk University (Brno, Czech Republic) from Sunday, July 13 and will end on Saturday, July 19, 2025.
Brno is a beautiful city surrounded by nature with a long tradition in category theory. If you are interested in attending, please read on!
Important dates
- April 2: talk submission
- April 18: early registration deadline
- May 7: notification of speakers
- May 23: registration deadline
- July 13-19: conference
In addition to 25 minute contributed talks, there will be speed talks replacing poster sessions, and we hope to accommodate as many talks as possible.
The invited speakers are:
- Clark Barwick (University of Edinburgh)
- Maria Manuel Clementino (University of Coimbra)
- Simon Henry (University of Ottawa)
- Jean-Simon Lemay (Macquarie University)
- Wendy Lowen (University of Antwerp)
- Maru Sarazola (University of Minnesota)
March 7, 2025
Visual Insights (Part 1)
Posted by John Baez
I’m giving a talk next Friday, March 14th, at 9 am Pacific Daylight time here in California. You’re all invited!
(Note that Daylight Savings Time starts March 9th, so do your calculations carefully if you do them before then.)
Title: Visual Insights
Abstract: For several years I ran a blog called Visual Insight, which was a place to share striking images that help explain topics in mathematics. In this talk I’d like to show you some of those images and explain some of the mathematics they illustrate.
Zoom link: https://virginia.zoom.us/j/97786599157?pwd=jr0dvbolVZ6zrHZhjOSeE2aFvbl6Ix.1
Recording: This talk will be recorded, and eventually a video will appear here: https://www.youtube.com/@IllustratingMathSeminar
March 4, 2025
How Good are Permutation Represesentations?
Posted by John Baez
Any action of a finite group on a finite set gives a linear representation of on the vector space with basis . This is called a ‘permutation representation’. And this raises a natural question: how many representations of finite groups are permutation representations?
Most representations are not permutation representations, since every permutation representation has a vector fixed by all elements of , namely the vector that’s the sum of all elements of . In other words, every permutation representation has a 1-dimensional trivial rep sitting inside it.
But what if we could ‘subtract off’ this trivial representation?
There are different levels of subtlety with which we can do this. For example, we can decategorify, and let:
the Burnside ring of be the ring of formal differences of isomorphism classes of actions of on finite sets;
the representation ring of be the ring of formal differences of isomorphism classes of finite-dimensional representations of .
In either of these rings, we can subtract.
There’s an obvious map , since any action of on a finite set gives a permutation representation of on the vector space with basis .
So I asked on MathOverflow: is typically surjective, or typically not surjective?
February 26, 2025
Potential Functions and the Magnitude of Functors 2
Posted by Tom Leinster
Despite the “2” in the title, you can follow this post without having read part 1. The whole point is to sneak up on the metricky, analysisy stuff about potential functions from a categorical angle, by considering constructions that are categorically reasonable in their own right. Let’s go!
February 23, 2025
Potential Functions and the Magnitude of Functors 1
Posted by Tom Leinster
Next: Part 2
In the beginning, there were hardly any spaces whose magnitude we knew. Line segments were about the best we could do. Then Mark Meckes introduced the technique of potential functions for calculating magnitude, which was shown to be very powerful. For instance, Juan Antonio Barceló and Tony Carbery used it to compute the magnitude of odd-dimensional Euclidean balls, which turn out to be rational functions of the radius. Using potential functions allows you to tap into the vast repository of knowledge of PDEs.
In this post and the next, I’ll explain this technique from a categorical viewpoint, saying almost nothing about the analytic details. This is category theory as an organizational tool, used to help us understand how the various ideas fit together. Specifically, I’ll explain potential functions in terms of the magnitude of functors, which I wrote about here a few weeks ago.
February 17, 2025
Category Theorists in AI
Posted by John Baez
Applied category theorists are flocking to AI, because that’s where the money is. I avoid working on it, both because I have an instinctive dislike of ‘hot topics’, and because at present AI is mainly being used to make rich and powerful people richer and more powerful.
However, I like to pay some attention to how category theorists are getting jobs connected to AI, and what they’re doing. Many of these people are my friends, so I wonder what they will do for AI, and the world at large — and what working on AI will do to them.
Let me list a bit of what’s going on. I’ll start with a cautionary tale, and then turn to the most important program linking AI and category theory today.
February 4, 2025
A Characterization of Standard Borel Spaces
Posted by John Baez
People in measure theory find it best to work with, not arbitrary measurable spaces, but certain nice ones called standard Borel spaces. I’ve used them myself.
The usual definition of these looks kind of clunky: a standard Borel space is a set equipped with a -algebra for which there exists a complete metric on such that is the -algebra of Borel sets. But the results are good. For example, every standard Borel space is isomorphic to one of these:
- a finite or countably infinite set with its -algebra of all subsets,
- the real line with its sigma-algebra of Borel subsets.
So standard Borel spaces are a good candidate for Tom Leinster’s program of revealing the mathematical inevitability of certain traditionally popular concepts. I forget exactly how he put it, but it’s a great program and I remember some examples: he found nice category-theoretic characterizations of Lebesgue integration, entropy, and the nerve of a category.
Now someone has done this for standard Borel spaces!
February 2, 2025
Backing Up US Federal Databases
Posted by John Baez
I hope you’ve read the news:
- Ethan Singer, Thousands of U.S. government web pages have been taken down since Friday, New York Times, 2 Feburary 2025.
Many of the pages taken down mention DEI, but they also include research papers on optics, chemistry, medicine and much more. They may reappear, but at this time nobody knows.
If you want to help save US federal web pages and databases, here are some things to do.
January 29, 2025
Comagnitude 2
Posted by Tom Leinster
Previously: Part 1
Last time, I talked about the magnitude of a set-valued functor. Today, I’ll introduce the comagnitude of a set-valued functor.
I don’t know how much there is to the comagnitude idea. Let’s see! I’ll tell you all the interesting things I know about it.
Along the way, I’ll also ask an elementary question about group actions that I hope someone knows how to answer.
January 22, 2025
Comagnitude 1
Posted by Tom Leinster
Next: Part 2
In this post and the next, I want to try out a new idea and see where it leads. It goes back to where magnitude began, which was the desire to unify elementary counting formulas like the inclusion-exclusion principle and the simple formula for the number of orbits in a free action of a group on a finite set.
To prepare the ground for comagnitude, I need to present magnitude itself in a slightly different way from usual. I won’t assume you know anything about magnitude, but if you do, watch out for something new: a connection between magnitude and entropy (ordinary, relative and conditional) that I don’t think has quite been articulated before.
January 15, 2025
The Dual Concept of Injection
Posted by Tom Leinster
We’re brought up to say that the dual concept of injection is surjection, and of course there’s a perfectly good reason for this. The monics in the category of sets are the injections, the epics are the surjections, and monics and epics are dual concepts in the usual categorical sense.
But there’s another way of looking at things, which gives a different answer to the question “what is the dual concept of injection?”
December 15, 2024
Random Permutations (Part 14)
Posted by John Baez
I want to go back over something from Part 11, but in a more systematic and self-contained way.
Namely, I want to prove a wonderful known fact about random permutations, the Cycle Length Lemma, using a bit of category theory. The idea here is that the number of -cycles in a random permutation of things is a random variable. Then comes a surprise: in the limit as , this random variable approaches a Poisson distribution with mean . And even better, for different choices of these random variables become independent in the limit.
I’m stating these facts roughly now, to not get bogged down. But I’ll state them precisely, prove them, and categorify them. That is, I’ll state equations involving random variables — but I’ll prove that these equations come from equivalences of groupoids!
December 10, 2024
Martianus Capella
Posted by John Baez
I’ve been blogging a bit about medieval math, physics and astronomy over on Azimuth. I’ve been writing about medieval attempts to improve Aristotle’s theory that velocity is proportional to force, understand objects moving at constant acceleration, and predict the conjunctions of Jupiter and Saturn. A lot of interesting stuff was happening back then!
As a digression from our usual fare on the -Café, here’s one of my favorites, about an early theory of the Solar System, neither geocentric nor heliocentric, that became popular thanks to a quirk of history around the time of Charlemagne. The more I researched this, the more I wanted to know.