DAY 218 – The magic wand

July 22nd, 2017 § 0 comments

So today, considering the complete disasters surrounding us all, which range from Grenfell Towers to Calais to Chios, and farther afield to the starvation in Gaza and Yemen; and to the slaughter in Mosulmosul and Raqqa; it seemed like a good time to turn this blog’s attention to more long-term, nay eternal questions. There is no more poignant illustration of what I’m talking about (how’s that for a flowery phrase then?) than the juxtaposition of Maryam Mirazakhani’smaryam monumental and timeless ‘Magic Wand’ theorem with her recent death from cancer at the age of 40. It seemed the least I could do to abandon. for today, the daily trivia of massacre and brutality, and try to explain the theorem to those of my readers who aren’t already familiar with the MWT. The theorem is clearly credited as joint work with Alex Eskin, but I hoped that Alex  - and indeed Amir Mohammed, also credited – would forgive me for concentrating on their beautiful and prematurely lost collaborator, as I tried to dash through the main points of their 204-page paper.

A long time ago when I was in analysis (did it work? does it ever? what does that mean?) my analyst asked me to explain my work, i.e. what was it to ‘do’ algebraic topology, which was the racket I was in at the time. Can you give me a session? I said; and, in what I still think was an amazing feat, I produced a passable explanation of what algebraic topologists do for the mathematically illiterate inside a 50-minute analytic hour. Who paid? Who do you think? Analysis, the ‘talking cure’, works on the basis that the analysand pays to talk, about n’importe quoi. Why would the analyst pay, even if he/she learned something?

Anyway, back to the 204-page paper. The more I tried to explain, indeed understand it, the more plain it became to me that I was hopelessly out of touch with today’s mathematics; so that even the neat 15 page summary by Anton Zurich in Gazette des mathematicians 142 left me struggling amidst measured laminations and other heavy stuff unfamiliar to me. So while I refer you without hesitation to Zurich who  adds a biography of Mirzakhani, and a character study, and much more, I decided to see where I got with her earlier work – as one might prefer to present Beethoven’s Razumovsky quartets rather than, say, the C minor op. 131 to folk who don’t really get on with chamber music in the first place; his account of the theorem (when he gets to it) is still a tad opaque to those of us who are quite happy dealing with the fine points of LGBT+ oppression or intersectionality or appealing expulsion decisions on the grounds of Article 8 of the ECHR but who go all weak at the knees at the mention of non-ergodic flows and such.

So, what’s the picture? You have a surface. You cut it up into bits. You’re cutting it up along ‘straight lines’, or geodesics as they’re called in the surface trade. Screen Shot 2017-07-20 at 09.08.55How many are there? Well, the longer they are, the more there are (is this obvious?) How long are they? And do they divide it it up – i.e. if you cut the surface along one, does it fall apart? Many of these questions, with their ramifications, are some of the things which preoccupied Maryam; you can see and hear her talking about them rather excitably here.

In her early work, such as http://annals.math.princeton.edu/wp-content/uploads/annals-v168-n1-p03.pdf, or ‘Growth of the number of simple closed geodesics on hyperbolic surfaces’ (and you can get it free, compared to all the evil social science journals which force you either to pay Sage loads of money or to be an academic if you want to read their stuff), she still wants to know how many of these curves there are, how many of them split the surface, and so on. And, rather than tire my poor brain and overstretch yours, I thought we could zoom in on one – to me – astounding statement near the end of that paper: ‘Roughly speaking, on a surface of genus 2, a long, random connected, simple, closed geodesic is separating with probability  17‘ . Note how precise we mathematicians are, with our ‘roughly speaking’ and ‘long’

Screen Shot 2017-07-22 at 12.00.48A surface of genus 2, and some rather short simple closed geodesics on it. The one  in the middle is separating, the one on the right isn’t – OK?

and think of a long random connected simple closed geodesic on a surface of genus 2. Is it separating? My picture is supposed to help with the concept, if not with the reasoning; and you’ll have to picture a long nonseparating closed geodesic for yourself.

But what you’ll find in the paper which leads to that weird figure 17, is an actual calculation (pp. 122-123) of the number of (non)separating closed geodesics – or the volume of the space of such, which looks equally magical. If I could have mastered the typography, I’d have given you the formula. Pretty cool stuff, eh? Go off and think about it; and stop trying to create wi-fi systems for refugee camps  or unburn-out burnt-out aid workers, if only for a moment. That limiting ratio of long closed simple geodesics will still be there, like Kepler’s laws or Darwish’s poems or the chromosome number of the fruit-fly, when you and I and the Home Secretary and the refugee camps and the volunteers and the NGOs and the UNHCR, and the PMLA for that matter,  are the shadow of a shadow, the memory of a memory.

I have been thinking about it, and I can’t say it really helps me much; but it’s worth thinking about such things, and I recommend it to you. And then we’ll return to the old rants.

And here – with no particular relevance – is the troubadour Guillaume of Aquitaine’s ‘Song of Pure Nothingness’; or at least the beginning.

Screen Shot 2017-07-22 at 13.05.42  

 
Which, ideally, would be a cue for Tansy Davies’tansy setting of the song (which I heard at the Victoria in Dalston which I’m ceaselessly recommending as a music venue); except that Tansy hasn’t put it on Youtube or anywhere else that I can find. We’ll have to be content with about two minutes of female troubadour music from her ‘Troubairitz‘.

 

 

 

Leave a Reply