# How not to prove the concurrent normals conjecture

# Video feedback

# Teaching online (April 2020)

To complement these videos/notes, I'm currently running live sessions for MATH105 using M*crosoft Teams (that's what Lancaster uses, and to be fair it is going pretty smoothly). I use my trusty Wacom Bamboo tablet and pen to run through worked examples/solutions and to field questions from the students; I have a team of tutors on hand to help monitor the chat and contribute. I'm finding this a fun experience: many students seem more ready to interact using chat than they do in real life in a lecture theatre. I'm also able to run polls more easily in an online setting (getting them to vote on different options in chat by "liking" them) so multiple choice questions seem to be going down quite well. This week I'm going to try using "breakout rooms", which might be a step too far in complication.

Of course, it's very hard for me to know what the student experience is like. I'm hoping to get some feedback from students at the end of this week, so we'll see if this is as much fun for them as it is for me.

Next year, if all goes to plan and the world doesn't end, I will be teaching Lie groups at Lancaster. I plan to flip this too (this was my plan even before COVID-19 reared its ugly head), so more video lectures are imminent...

# Fukaya categories 1: What is Floer theory?

# Fukaya categories

# Postdoc wanted

See the vacancy page for more information.

# KIAS Workshop

This was helped by the structure of the workshop: over the course of the week, there were only eight speakers, each giving two or three talks. This encouraged speakers to take time explaining illustrative examples and background, rather than rushing over these to try and state their theorem within an hour.

Here are a couple of the things I learned (in this case, specifically from the lectures of Hakho Choi).

# Why open notebook mathematics?

# Scientific communication

Which of the following is the ultimate purpose of scientific communication?

- To impress expert referees and convince them that you are clever.
- To bolster your publication record.
- To get cited lots.
- To publish in top journals.
- None of the above.

# 3-dimensional lightbulb theorem

# Erratum for arXiv:1110.0927

# Open notebook

# Chain relations

*chain relations*. In this post, I will explain the origin of these relations in algebraic geometry.

# manim

[2019-06-21 Fri]

I recently taught a linear algebra class. Before I taught it, Momchil Konstantinov recommended I check out the Youtube channel 3Blue1Brown (produced by Grant Sanderson), which apparently had some nice videos about linear algebra. I was too busy preparing to teach. Once marking the exam was over, I thought I would relax by taking his advice and seeing someone else's take on linear algebra.

My mind was blown.

# First things first

# Mathematical reading

*This used to be a random page on my website, but I thought I would incorporate it into my blog.*

A big part of the practice of mathematics is the struggle to put our own internal mathematical universes into order, trying to fit new understanding in with old.

To do this by reading what other people have written is difficult, because their way of structuring things can be very different from the way you think.

Here are some of the things I keep in mind when trying to read things other people have written.

# Am I allowed?

The question really means: "Will I lose marks for doing this?" which is a perfectly legitimate question. However, it is a clear example of educational "backwash", where the means of teaching or assessing has a negative impact on the way students learn.

# Small resolutions

I was discussing all of this with my collaborator Mirko Mauri and trying to understand when small resolutions exist in more complicated situations. He told me something which blew my mind. "Small resolutions often occur when you blow up a divisor." Don't be silly, I said, blowing up a divisor doesn't do anything. He pointed out that blowing up a Cartier divisor doesn't do anything, but blowing up a Weil (but non-Cartier) divisor can do something. At this point, I was trying to remember what the difference was and whether all Cartier divisors were Weil or vice versa or neither. By the end of our discussion, I had a much better understanding of what "blowing up a divisor" means from the viewpoint of a symplectic geometer. I try to explain this below.

# Associahedra

Mathematicians are big on proofs. We like to work out every tiny detail of an argument before presenting it to the world. Sometimes this can seem like pedantry, and doubtless there are mathematicians who are pedants. But I would like to argue that this process of carefully examining the minutiae of an argument can lead one to new and unexpected discoveries. This is completely analogous to the way that a scientist, confused by discrepancies between theory and experiment, can find a new, deeper theory by careful and imaginative examination of the basic assumptions of the old theory.

I'm going to introduce a mathematical idea which will seem, at first sight, pretty boring. Then I'll show you how it leads you naturally to something completely unexpected.

# Moving

I have had a wonderful seven years at UCL: it is an incredibly friendly place with an abundance of excellent and motivated students and researchers. Thank you to everyone who has made my time here so enjoyable.

Note that this means that my website will be moving too, so if there are people out there who occasionally read this blog or use the resources on my website, be warned!

# Undergraduate pathways update

# Developing map

# Focus-focus singularities

The images illustrate the developing map for the integral affine structure on the base of a Lagrangian torus fibration in the neighbourhood of a focus-focus fibre.

# Elliptic function slide rule

# Fanography

Both such a database and a YAML file from which the database is generated are now available thanks to the efforts of Pieter Belmans.

# Flipping conclusion

# Flipping update

# Mathematical bookbinding

- Maths books are expensive.
- Printing papers out is usually a waste; I lose track of which papers
I have printed and they mount up in enormous, disorganised,
dog-eared piles on my desk. I sometimes find it fun to work my way
downwards through one of these piles, and see the stratified history
of what I have been thinking about for the last year. But it's not a
sensible way to organise one's life.
- Screen-readers are just not the same... I got an iPad a few years
ago so I could minimise printing, and it's very convenient to carry
around all my papers in electronic form. But still, I find it very
difficult to focus in depth on something I'm reading on a screen. I
like to be able to flick through and hold different pages open. I
like books.

# Cyclic quotient singularities, II

# Cyclic quotient singularities, I

# Horikawa surfaces

# New blog functionality

This script is not without its flaws. For example, if a blog post is only on the index page and doesn't have as separate file to link to, I have just linked to the blog index (I should really include an anchor to link to... maybe another day). Probably, I should use ox-rss or some other elisp solution, but I find shell scripts easier to deal with.

One more piece of fun: I recently acquired an Android smartphone (so that I could send photos of my new baby to his grandparents via whatsapp). I became much more enamoured of this phone when I realised I could install a terminal emulator (Termux), use this to install Emacs and Git, and so I can now make blog posts from my phone.

# Farmageddon

I have just finished reading the book Farmageddon by Philip Lymbery and Isabel Oakeshott. It is the most important book I have read in a number of years. It is a dazzling journalistic expose of how industrial-scale agriculture and factory farming has systematically raped the world's farmlands in the last half century.

# A crib sheet for surfaces

**Update:** See also this page of Pieter Belmans and Johan Commelin
for an interactive complex surface explorer!

# Clarification for arXiv:1606.08656

*Geometry and Topology*. Shortly thereafter, Yong-Geun Oh contacted us with some excellent questions where our exposition was less than clear. In case anyone else has the same questions, I thought I would write a blog post clarifying these points.

# Clarifications and errata

# Flipping

In this blog post I want to review:

- why on earth I'm planning to do this,
- what it means in practice,
- what I've already done.

# TikZ and org-mode

# Noether's theorem in field theory

# The Heisenberg picture and causality

# What is a quantum field?

# Pre-QFT 1: the quantum harmonic oscillator

# Quantum field theory reading group

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

# Theorem and proof environments in CSS

# Connecting to wifi from command line

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

# New blog

# Resonances

# Nice paper bump

# Is the speed of light constant?

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

Update (3rd Feb 2018): I have now updated the source code for this to make it easier to maintain. It is now available on GitHub.

# Some simple spectral sequences

Please let me know of any errors in the exercises!

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

# Cone eversion

# Gromoll filtration

# E-learning project report

The purpose of this e-learning project was to test the effectiveness and viability of getting students to film mathematics lectures and the effect on student learning of making these videos available. The project was made possible by an E-Learning Development Grant (ELDG) and by the cooperation of a large number of people who I thank at the end.

Disclaimer. The project analysis is not scientific: there is no attempt made at comparison with a control group, the data sets are not large and the statistical methods used to analyse them are crude. This report is intended to be at best a rough guide to the UCL Mathematics Departmental Teaching Committee as to what action to take on filming of mathematics lectures.

# December: video project update

Read on for some of the results

# Geometry and undecidability

Read on for the rest of this entry.

# Video-lecture project weeks 1 and 2

# E-Learning: Video lectures filmed by students

Read on for more about the project.

# E-Learning: Spring 2013

# Bored now

# The geometric definition of the Johnson homomorphism

The Torelli group of a surface is the subgroup of mapping classes which act trivially on cohomology. Consider the case of an orientable surface with \(g\) handles and one boundary component (diffeomorphisms are required to fix the boundary). There is a famous homomorphism from this group to the free abelian group of rank \({2g\choose 3}\) called the Johnson homomorphism. The usual definition is pretty algebraic-looking (involving the mapping class group action on the fundamental group and its commutator subgroup). This week I read an alternative (extremely beautiful, geometric) definition of this homomorphism in Johnson's survey paper on the Torelli group (D. Johnson, A survey of the Torelli group, Contemp. Math. (1983) vol. 20, 165-179). This definition is probably very well-known, but I didn't formerly know it and I thought it was too nice not to blog about.

# TikZ code for the octahedral axiom

Include the following code in the head of your LaTeX document:

\usepackage{tikz} \usetikzlibrary{arrows}\newcommand{\Octa}[6]{\begin{center} \begin{tikzpicture}[node distance=2cm,thick] \node (1) {\(X\)}; \node (4) [below right of=1] {\(A\)}; \node (6) [below right of=4] {\(C\)}; \node (2) [above right of=4] {\(Y\)}; \node (5) [above right of=6] {\(B\)}; \node (3) [above right of=5] {\(Z\)}; \draw[->] (1) to[out=25,in=155] (3); \draw[->] (1) -- (2); \draw[->] (2) -- (3); \draw[->] (2) -- (4); \draw[->] (3) -- (5); \draw[->] (3) to[out=260,in=0] (6); \draw[->] (4) -- (6); \draw[dashed,->] (4) -- (1); \draw[->] (6) -- (5); \draw[dashed,->] (4) -- (1); \draw[dashed,->] (5) -- (4); \draw[dashed,->] (5) -- (2); \draw[dashed,->] (6) to[out=180,in=270] (1); \end{tikzpicture} \end{center}}

Then the command

\Octa{X}{Y}{Z}{A}{B}{C}

will produce a diagram like this:

# Convex Integration (talk notes)

Also relevant is the preliminary cartoon which contains many of the essential ideas.

**Warning:** Images are large may take some time to load.

# Convex integration (cartoon)

The technical details of the talk will be heavily based on these notes by Vincent Borrelli, which is an excellent place to learn all this stuff from.

**Warning:** The cartoon is big (about 1MB) and may take time to load.

# Kronheimer's argument: Small resolutions and Dehn twists

Read on for an explanation to which I can point people in future.

# Symplectic/Contact Geometry VII at Les Diablerets, Day 1

After three excellent talks today I decided to act as a "maths journalist" and summarise the main ideas from the talks in this blog. I may not be able to keep this up, as there's six talks tomorrow and too much snow to enjoy. Today's talks were:

- Urs Frauenfelder "A \(\Gamma\)-structure on the Lagrangian Grassmannian"
- Yochay Jerby "The symplectic topology of projective manifolds with small dual"
- Alex Ritter "Floer theory for negative line bundles"

# UCL Geometry and Topology Open Day talk: Floer theory

# Why Schrödinger's equation?

I recently overheard someone ask this about Schrödinger's equation. The answer they received was, for me, unsatisfying. "Because it agrees with experiment." Of course, that answers perfectly why the equation was adopted by future generations of physicists and indeed the calculation of the spectrum of atomic hydrogen from the energy eigenvalues of the Schrödinger operator is one of the most convincing and wholesome computations a young physicist can do. But the question that was left unanswered, the question I believe was being asked, was: "Why did Schrödinger write this equation down? Why not something else?" I don't believe for a second that Schrödinger sat down with an array of different equations and worked out what each of them predicted about hydrogen before he found the one that fit...

# HEA course for new maths lecturers

Read on for a couple of ideas I took away (not necessarily maths-specific!).

# Lines through four lines

I was reading Fulton-Pandharipande ("Notes on stable maps and quantum cohomology") the other day and came across the classical result that there are exactly two lines passing through a generic quadruple of lines in \(\mathbf{CP}^3\). I encourage people to whom this fact is unfamiliar to convince themselves of it. It was unfamiliar to me and I found it hard to visualise, so I sat down and drew some pictures until I understood it.

# Quantum computers: Grover's algorithm

I think the easiest way to illustrate how a quantum computer differs (functionally) from a classical computer is by explaining an algorithm which only makes sense for quantum computers and which really outperforms a classical algorithm for the same task (that is the point of quantum computers, after all). The first algorithm they explain in Kitaev et al is called Grover's algorithm and it performs the task of searching a database. Wikipedia has a really nice exposition too, but I was initially confused by what they both call a "quantum oracle" (sounds like something from Star Trek TNG). I tried to explicitly avoid that in what I said below.

**TLDR:** Classically you have to look through all \(N\) elements
of a database until you find the right one (so runtime increases
linearly in N); Grover's algorithm has a surprising runtime of order
\(\sqrt{N}\) to do the same thing, using clever ideas from quantum
mechanics.