I hadn’t expanded on my thoughts about category theory in AI because it’s a very emotional subject for me and I didn’t want to contaminate my reporting of a potentially momentous event with too much of my own perspective. But okay:| golem.ph.utexas.edu
Comparing two mathematical notions of diversity.| The n-Category Café
I hear that Jack Morava died on August 1, 2025.| The n-Category Café
Can the Duflo isomorphism explain the extra 1/2 in the Hamiltonian for the quantum harmonic oscillator?| The n-Category Café
Announcing the Clowder Project: a wiki and reference work for category theory built using the same general infrastructure and tag system of the Stacks Project.| The n-Category Café
Kevin Coulembier has come out with a paper claiming to prove some conjectures on 2-rigs that Todd Trimble, Joe Moeller and I made.| The n-Category Café
Re: Lawvere’s Work on Arms Control| golem.ph.utexas.edu
You can define a rooted planar binary tree recursively by this equation:| golem.ph.utexas.edu
First lecture in a 4.5-hour minicourse on combinatorics with species.| The n-Category Café
Second lecture in a 4.5-hour minicourse on combinatorics with species.| The n-Category Café
Species and their generating functions| golem.ph.utexas.edu
The monoid of n×nn \times n matrices has an obvious nn-dimensional representation, and you can get all its representations from this one by operations that you can apply to any representation. So its category of representations is generated by this one obvious representation, in some sense. And it’s almost freely generated: there’s just one special relation. What’s that, you ask? It’s a relation saying the obvious representation is nn-dimensional! | golem.ph.utexas.edu
Stellenbosch University wants to hire a mathematician. Apply before April 30th!| The n-Category Café
Categorical Linguistics in Quanta| golem.ph.utexas.edu
Nuclear physics forces you to imagine blobs of protons and neutrons wiggling around in a very quantum-mechanical way. Nuclei are too complicated to fully understand. We can simulate them on a computer, but simulation is not understanding, and it’s also very hard: one book I’m reading points out that one computation you might want to do requires diagonalizing a 10 14×10 1410^{14} \times 10^{14} matrix. So I’d rather learn about the many simplified models of nuclei people have create...| golem.ph.utexas.edu
Before I can describe this categorical viewpoint on potential functions, I have to explain what potential| golem.ph.utexas.edu
The inclusion-exclusion formula tells us the cardinality of a union, which| golem.ph.utexas.edu
This different viewpoint comes from the observation that in the category of| golem.ph.utexas.edu
The story for specifying subsets goes like this. Given a property| golem.ph.utexas.edu
Maybe these facts are ‘coincidences’. There are lots of obstacles to understanding them. But we can at least try to get some idea of where the numbers 24 and 240 are coming from, in the homotopy groups of spheres. They’re connected to Bernoulli numbers.| golem.ph.utexas.edu
Here is his talk:| golem.ph.utexas.edu
Introduction| golem.ph.utexas.edu