# Blog index

## TikZ and org-mode blog org emacs tikz

As an org-mode newbie, it took me a while to figure out how to get TikZ code in my source files to generate images in my blog.

## Noether's theorem in field theory qft

This blog post attempts to explain Noether's theorem in field theory (including Noether currents) in a way that might appeal to a mathematician familiar with symplectic topology and the Hamiltonian formalism.

## The Heisenberg picture and causality qft

This blog post summarises what we learned about propagators and causality in free scalar QFT.

## What is a quantum field? qft

As a mathematician trying to learn QFT, a question that always bothered me was "what is a quantum field?". This blog post explains one point of view (the Schrödinger picture) on the answer to this question which I find satisfying.

## Pre-QFT 1: the quantum harmonic oscillator qft

The archetypal physical system is the simple harmonic oscillator: a ball on a spring following Hooke's law follows periodic motion along a circle in phase space (its displacement and momentum oscillate sinusoidally between two extremes and out of phase like sin and cos). Understanding the corresponding quantum system is fundamental to understanding quantum field theory: indeed, quantising a free (bosonic) field turns out to be equivalent to quantising an infinite collection of simple harmonic oscillators.

## Quantum field theory reading group qft

Like many people, I got into maths because I was interested in quantum field theory and didn't understand what was going on. I have spent a lot of time idly browsing QFT textbooks over the years in an effort to rectify this, but stuff always got in the way.

Ed Segal and I are planning to run a QFT reading group at UCL to improve our understanding. I will post my own notes from the reading group to this blog, as well as some foundational "pre-QFT" material which I always forget and have to re-read whenever I start looking into this stuff after a long break.

If you are interested in attending the reading group, please let Ed or me know.

## Equivalence relations revision mathm205

Equivalence relations are an important concept in mathematics, but sometimes they are not given the emphasis they deserve in an undergraduate course. Having a good grasp of equivalence relations is very important in the course MATHM205 (Topology and Groups) which I'm teaching this term, so I have written this blog post to remind you what you need to know about them. I will kick off with a few examples, then give a more formal definition.

## Theorem and proof environments in CSS blog org

Here is a nice idea from Dr Z.ac, the blog of Zachary Harmany. You can use CSS to create LaTeX-style theorem/proof environments on a website.

## Connecting to wifi from command line linux wifi

I so rarely need to connect to a new wifi network that, when I do, I always forget how I managed to do it the previous time. For future reference, here's how I did it this time (using "NetworkManager Command Line Interface" or "nmcli"):

$ nmcli dev wifi list $ nmcli dev wifi connect NETWORKNAME password NETWORKPASSWORD

## New blog blog org emacs

I've decided that I don't like my old blog and I'm setting up a new blog using org-mode.

## Resonances qft

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?

## Nice paper bump qft

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.

## Is the speed of light constant? relativity light

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.

## Using graphviz to illustrate course structure code elearning teaching

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 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:

For more details, see the readme file.

## Some simple spectral sequences algebra geometry teaching

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. 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!

## A sanity check for the Fukaya category of a cotangent bundle fukayacategory algebra geometry

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.

## Cone eversion topology geometry

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 is one of his images. Reproduced with Tom's kind permission (the copyright is his).