Mathematical Field Notes

Archive for the ‘Uncategorized’ Category

Nice paper bump

leave a comment »

It’s 7 years old, but I only just came across the following beautiful expository paper of Baez and Huerta on the representation theory underlying the standard model and grand unified theories and I thought I would give it a bump:

https://arxiv.org/abs/0904.1556

It overlaps with some of the material I touch on in the Lie Groups course I teach (using representations to classify particles) but goes into much more gorgeous detail and focuses on fundamental particles rather than baryons/mesons. It is unusually easy to follow (if you know a bit of representation theory) and I learned a lot from reading it.

Written by Jonny Evans

August 31, 2016 at 10:52 pm

Posted in Uncategorized

A sanity check for the Fukaya category of a cotangent bundle

leave a comment »

Yesterday I gave a seminar about Fukaya categories and I didn’t have chance to do quite as much explicit computation as I’d hoped. I thought I’d write a blog post with a basic calculation to show you the kind of things that are involved in doing computations in Fukaya categories. I will show (using Abouzaid’s description of the zero section in terms of the cotangent fibre) that the zero section and the cotangent fibre have rank HF = 1, in the special case of T^*S^1. This is such a trivial result in the end (you could do the computation just by looking at the intersection and seeing it’s a single point) that you should think of this post as more of a sanity check.

The first 3 hours of what I said was essentially covering Auroux’s introduction to Fukaya categories (http://arxiv.org/abs/1301.7056), so if you look at that first, you should be able to figure out what I’m talking about. I hope that all my grading conventions agree with those of Abouzaid – if you notice a discrepancy, please let me know!

Read the rest of this entry »

Written by Jonny Evans

February 11, 2015 at 10:10 am

Posted in Uncategorized

Cone eversion

leave a comment »

Last year, around the time Chris Wendl was running the h-principle learning seminar at UCL, I set my second years an exercise from Eliashberg-Mishachev as a difficult challenge problem: to find an explicit cone eversion. In other words, find a path in the space of functions on \{(r,\theta)\in\mathbf{R}^2\ :\ r\in[1,2]\} connecting r to 2-r such that none of the intermediate functions has a critical point. One of these students, Tom Steeples, got hooked on the problem, almost solved it, and afterwards used Mathematica to produce some beautiful computer animations of a solution given by Tabachnikov in American Mathematical Monthly (1995) Vol 102, Issue 1, pp 52–56. Here, below the fold because it’s quite a large file, is one of his images. Reproduced with Tom’s kind permission (the copyright is his).

Read the rest of this entry »

Written by Jonny Evans

October 12, 2014 at 2:02 pm

Posted in Uncategorized