Mathematical Field Notes

Resonances

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How could you “detect” a new subatomic particle, given that it’s so small you can’t see it and (often) so short-lived that you’d miss it even if you didn’t blink?

Let’s suppose you have a process which you think might involve your tiny, short-lived new friends. Maybe you suspect that if you smash protons together, you might produce some particles… which decay into particles…. which decay into the particles you’re looking for… which then themselves decay almost immediately into something else, something that is more stable and easier to detect (something like a pair of photons). How could you check that you were right?

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Written by Jonny Evans

October 30, 2016 at 12:02 pm

Nice paper bump

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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

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Is the speed of light constant?

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I recently came across a beautiful argument due to De Sitter (1913), which gave the (first?) experimental evidence that light moves with a constant speed.

Constancy of the speed of light is one of those things that always bothered me, and I spent a couple of days recently trying to unbother myself. De Sitter’s argument is what finally satisfied me. Below, I’m going to explain the background, then I’ll explain De Sitter’s argument. The De Sitter paper is only a couple of paragraphs long and is available via Wikisource, so if you don’t need the introductory remarks in the blogpost below, just follow the link above and read it.

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Written by Jonny Evans

August 9, 2016 at 5:33 pm

Posted in Physics

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Using graphviz to illustrate course structure

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At some point last year, I got frustrated that I couldn’t see easily the global structure of the UCL undergraduate maths course without trawling through a bunch of PDFs, so I made this webpage:

http://www.homepages.ucl.ac.uk/~ucahjde/pathways.htm

to illustrate it. Hopefully some people have found this useful in deciding which modules to choose or in advising students which modules to take.

To generate the image maps I used a fantastic programme called graphviz. In case anyone wants to adapt what I did to their own ends, I have made my graphviz code for these diagrams (plus some ancillary shells scripts for creating and uploading the webpage) available here:

http://www.homepages.ucl.ac.uk/~ucahjde/choices/pathways.zip

For more details, see the readme file.

Written by Jonny Evans

October 22, 2015 at 11:35 pm

Some simple spectral sequences

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I keep finding myself trying to explain how the very simplest spectral sequences arise (spectral sequence to compute the cohomology of a cone or an iterated cone), so I have taken the time to TeX the explanation into a sequence of guided exercises:

Some simple spectral sequences

This is all very formal and diagram-chasy. One of the off-putting things about spectral sequences is all the indices; in these exercises I have suppressed gradings and concentrated on the very simplest cases to avoid overcomplicating the notation. Once you’ve seen how the proof goes, you should go and look in Bott-Tu or McCleary for some actual examples and computations.

Please let me know of any errors in the exercises!

Written by Jonny Evans

April 28, 2015 at 3:10 pm

A sanity check for the Fukaya category of a cotangent bundle

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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!

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Written by Jonny Evans

February 11, 2015 at 10:10 am

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Cone eversion

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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).

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Written by Jonny Evans

October 12, 2014 at 2:02 pm

Posted in Uncategorized