The n-Category Café

Because I’ve been battling exhaustion, quasi-depression and the UK higher education system’s conflation of customer service with demands to filter according to a perceived external standard?

(Also trying to finish collaborations where I owe my colleagues, and look after my graduate students.)

These are things I can identify with, Yemon. I didn’t intend this as a guilt trip! I’ve added a paragraph which is intended to address this somewhat.

I’ve been giving talks online and some have been videotaped. Does that count?

• Mathematics in the 21st century

• Theoretical physics in the 21st century

• Symmetric monoidal categories: a Rosetta stone

• Can we understand the Standard Model?

• Can we understand the Standard Model using octonions?

I had a go at explaining linear algebra, and quantum physics, via “path counting”.

This is the approach used in graphical linear algebra. If you go back and look at the Feynman lectures, volume III, the rules he presents for combining amplitudes are the same as the rules for counting paths. So this just looks like a great way for people to appreciate something about linear algebra and how it applies to quantum physics directly.

One big idea here is to “categorify” amplitudes. These are inner products in a Hilbert space, so it makes sense that these would categorify as Hom’s. The path counting idea is a tiny bit of success along these lines. But amplitudes can be negative, and it seems no-one knows what negative numbers are counting, especially in quantum physics. It’s a mystery.

In the video above, I get to the point of explaining the Born rule, which has a nice interpretation in terms of paths. You just rewind everything back to the start! So amplitudes count paths from point a to point b, and the corresponding probability counts loops from point a through point b. I wonder if there is some interesting homotopy theory interpretation of this.

I have been meaning to create more maths videos. I used to love giving seminar talks at grad school and I feel like there should be more high level maths videos on the internet.

I have been playing around with the manim software, and while the results are gorgeous, the time it takes me to animate, narrate, and edit a talk are enormous. Grant’s talks are great fun, but I doubt you could make an entire lecture series this way (at least not on your own). The workload would be prohibitive. For example, (admittedly as a beginner) I coded and recorded a two hour lecture using manim and audacity and cinelerra… and I was still unhappy with the results after 6 months. This does not scale.

Furthermore, I don’t think that “video” is necessarily the best format for all content. At the higher level, so much mathematical communication is text based and having a video (that is purely graphical) representation of text is unsatisfying. There ought to be something better. Furthermore, when we imported lecture series to the internet by taping the lectures and putting the videos on youtube, we get a sub-optimal result there too. A lecture that might be enthralling in real life can become tiring and hard to follow on youtube. The choices you make for an in-person talk (content, pace, level of detail) are not optimised for digital consumption.

I could go on… but luckily I have already dumped my thinking on the matter at spoken.page but have not pursued it further. I’d welcome any comments :)