## October 31, 2024

### Adjoint School 2025

#### Posted by John Baez

Are you interested in using category-theoretic methods to tackle problems in topics like quantum computation, machine learning, numerical analysis or graph theory? Then you might like the Adjoint School! A lot of applied category theorists I know have gotten their start there. It can be a transformative experience, in part thanks to all the people you’ll meet.

You’ll work online on a research project with a mentor and a team of other students for several months. Then you’ll get together for several days at the end of May at the University of Florida, in Gainesville. Then comes the big annual conference on applied category theory, ACT2025.

You can apply here starting November 1st, 2024. The deadline to apply is December 1st.

For more details, including the list of mentors and their research projects, read on.

## October 29, 2024

### Triangulations of the Sphere (Part 2)

#### Posted by John Baez

Thurston’s paper Shapes of polyhedra and triangulations of the sphere is really remarkable. I’m writing about it in my next column for the *Notices of the American Mathematical Society*. Here’s a draft — which is also a much more detailed version of an earlier blog post here.

If you have suggestions or corrections, please let me know: I can try to take them into account before this is due on November 5th. Just don’t ask me to make this *longer*: I have a strict limit.

## October 26, 2024

### Axiomatic Set Theory 6: Gluing

#### Posted by Tom Leinster

*Previously: Part 5. Next: Part 7.*

A category theorist might imagine that a chapter with this title would be about constructing colimits, and they’d be half right.

### Triangulations of the Sphere (Part 1)

#### Posted by John Baez

I’m writing a short column about this paper:

- W. P. Thurston, Shapes of polyhedra and triangulations of the sphere.

Let me describe one of the key ideas as simply as I can.

## October 21, 2024

### Octoberfest 2024

#### Posted by John Baez

The Octoberfest is a noble tradition in category theory: a low-key, friendly conference for researchers to share their work and thoughts. This year it’s on Saturday October 26th and Sunday October 27th.

It’s being run by Rick Blute out of the University of Ottawa. However, the meeting is entirely virtual, so you can join from anywhere using this zoom link. The talks will be recorded, and shared publicly later.

Below you can see a schedule. All times are Eastern Daylight Time (EDT).

I can’t resist mentioning that I’m giving a keynote talk on 2-rigs in topology and representation theory at 2 pm on Saturday. It’s about some work with Joe Moeller and Todd Trimble.

## October 18, 2024

### Axiomatic Set Theory 5: Relations

#### Posted by Tom Leinster

*Previously: Part 4. Next: Part 6*

I called this chapter of my course “Relations”, but I should have called it “Specifying subsets and functions”, because that’s what it’s all about. This week, we saw that it’s possible to define subsets of a set $X$ by expressions like

$\{ x \in X : \text{some property of }\ x\ \text{ holds} \},$

and functions $f$ by expressions like

$f(x) = \text{some formula in }\ x.$

I didn’t want to drown the students in notation, so I didn’t give precise definitions of “property” and “formula”. Instead, I aimed to give them practical tools that would apply to situations they’re likely to meet.

## October 17, 2024

### Associahedra in Quantum Field Theory

#### Posted by John Baez

I haven’t been carefully following quantum field theory these days, but some folks on the Category Theory Community Server asked me what I thought about recent work using the ‘amplitudohedron’ and other polytopes, so I decided to check out these videos:

- Nima Arkani-Hamed, Advanced class on amplitudes.

There are 5, and so far I’ve only finished watching the first. But I have to say: I enjoyed it more than any lecture on physics I’ve seen for a long time!

Arkani-Hamed has the amusing, informal yet clear manner of someone like Feynman or Coleman. And he explains, step by step, how imaginary particle physicists in some other universe could have invented the associahedra just by doing scattering experiments and looking for poles in the S-matrix. That blew my mind.

## October 11, 2024

### Axiomatic Set Theory 4: Subsets

#### Posted by Tom Leinster

*Previously: Part 3. Next: Part 5*

This phase of the course is all about building up the basic apparatus. We’ve stated our axioms, and it might seem like they’re not very powerful. It’s our job now to show that, in fact, they’re powerful enough to do just about everything with sets that mathematicians ever want. We began that job this week, with a chapter on subsets.

## October 10, 2024

### 2-Rigs and the Splitting Principle

#### Posted by John Baez

We’re done!

- John Baez, Joe Moeller and Todd Trimble, 2-rig extensions and the splitting principle.

Our paper categorifies a famous method for studying vector bundles, called the ‘splitting principle’. But it also continues our work on representation theory using categorified rigs, called ‘2-rigs’. We conjecture a splitting principle for 2-rigs, and prove a version of it in the universal example.

But we also do more. I’ll only explain a bit, today.

## October 4, 2024

### Axiomatic Set Theory 3: The Axioms, Part Two

#### Posted by Tom Leinster

*Previously: Part 2. Next: Part 4*

This week, we finished formulating the axioms of the Elementary Theory of the Category of Sets.