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December 26, 2022
Guillotine Partitions and the Hipparchus Operad
Posted by John Baez
If you dissect a square into similar rectangles, what proportions can these rectangles have? Folks on Mathstodon figured this out for , and I blogged about it here recently. But I was left feeling that some deeper structure governed this problem.
Various people on Mathstodon, including Steven Stanicki, David Eppstein and Rahul Narain, convinced me of the importance of a certain class of dissections called ‘guillotine partitions’. I started suspecting that these were connected to an operad I once blogged about here: the ‘Hipparchus operad’. And last night I put some of the pieces together… though there is still more to do.
December 22, 2022
Dividing a Square into Similar Rectangles
Posted by John Baez
If you divide a square into some fixed number of similar rectangles, what proportions can these rectangles have? We’ve been having fun thinking about this on Mathstodon, and here is a progress report.
December 21, 2022
Free Idempotent Rigs and Monoids
Posted by John Baez
I’ve been having a lot of fun on Mathstodon lately, and here’s an example.
A rig has a commutative associative addition, an associative multiplication that distributes over addition, an element with and for all , and an element with for all .
A rig is idempotent if for all .
Is the free idempotent rig on generators finite? If so, how many elements does it have?
Morgan Rogers raised this issue on the Category Theory Community server, and after a bit of progress I posed this as a puzzle on Mathstodon. By now three people there have independently figured out the answer.
December 18, 2022
Adjoint School 2023
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.
January 9, 2023. Application Due.
February - July, 2023. Learning Seminar.
July 24 - 28, 2023. In-person Research Week at University of Maryland, College Park, USA.
December 3, 2022
Neutrino Dark Matter
Posted by John Baez
I talked to Neil Turok at a café today. He used to be the head of the Perimeter Institute, but now he’s at the University of Edinburgh.
He coauthored a paper arguing that dark matter is very heavy right-handed neutrinos:
- Latham Boyle, Kieran Finn and Neil Turok, The Big Bang, CPT, and neutrino dark matter.
It’s very natural to add right-handed neutrinos to the Standard Model, and if they’re heavy they can make the observed left-handed neutrinos light via the ‘see-saw mechanism’. The problem is to keep them from decaying too fast!