## DAY 98 Bordered Teichmüller spaces

August 17th, 2014 § 0 comments

Being, like the rest of us, overjoyed that Iranian woman geometer Maryam Mirzakhani has landed the Fields Medal for her work on Weil-Petersson volumes of bordered Teichmüller spaces (not to mention earthquake flows); I thought it would be beneath the high intellectual standards that readers of this blog normally expect to restrict my coverage to the standard demure photograph of the Stanford prof and a few words on women’s mathematical education in the Islamic Republic

(how it’s changed since the days of Omar Khayyam, etc.) Such trivia we leave to the popular sheets like our confrère the GuardianAnd we who spend our evenings watching Press TV and the films of Kiarostiami and Farhadi expect Iranian women to be sharp and feisty anyway. So here’s a quick exposition of the main lines of Mirzakhani’s groundbreaking paper from the 2007 Inventiones Mathematicae, insofar as I’ve grasped it from the usual lightning read-through. As you will realize if you’ve ever given it a try, finding Weil-Petersson volumes of moduli spaces is not exactly a breeze. Mirzakhani’s amazing find is this. Look at the moduli space of bordered hyperbolic Riemann surfaces of genus with geodesic boundary components of lengths $L_1,\dots,L_n$ Calling this ${\cal M}_{g,n}$ then its Weil-Petersson volume (look it up, am I supposed to do all the work already?) is a polynomial in the L‘s. Pretty cool, eh? More precisely, $V_{g,n}(L)=\sum_\alpha C_\alpha.L^{2\alpha}$ where the coefficients lie in $\pi^{6g-6+2n-\vert 2\alpha\vert}.\textbf{Q}$ Yes, good old pi as in the area of a circle. I guess none of us were surprised by the exponent (those g‘s and n‘s had a familiar look); but pi was a surprise. I haven’t been keeping up with the literature, obviously.

Well, I won’t let you into the secret of how this powerful formula is worked out; I recommend starting with the stunning §3 on the genus 0 case, ‘Geometry of pairs of pants’, or salwar khameez as we call them in Iran. It’s lucid, and has

some pretty fancy pictures, such as we toilers in the fields of Riemann surfaces always appreciate.
The more nerdy among you may also be impressed by my new-found ability to incorporate mathematical formulae in these communications (the magic word ‘plug-in’ is appropriate here). Even if I can’t get them to sit smoothly in the text rather than being ‘displayed’ – I’d appreciate some help on this.
Which leads us naturally to the question of Palestine’s membership of the International Criminal Court. This is a bit confused. It seems that in order for the Israeli Government to be prosecuted for war crimes at the ICC, as is obviously sensible, the state of Palestine, or the Palestinian Authority, or whatever, needs to have acceded to the Rome Statute. This in itself is bizarre, since it allows freedom to commit war crimes on anyone’s territory if they have not acceded to the Statute. Well, I thought, there are likely plenty of war crimes in Somalia, and the chances that Somalia has ratified the statute are slim. Right on both counts. But as Somali Rights pointed out a while back, the Prosecutor has the power under Article 15 to instigate an investigation proprio motu into crimes within the Court’s jurisdiction, whatever that is – I refer you to Article 13, there’s plenty of time and money for lawyers to sort that out. In the Somali case, the complaint was about Kenya, who have ratified the statute and are independently embroiled with the ICC over the case of the ‘Ocampo six’. But I digress.
It should be pointed out a) that the PA is in the process of making the application, see Hanan Ashrawi’s interview. and b) that in the larger picture, this all seems beside the point. As I learned a while back, war crimes are very illegal, being outlawed by the Hague and Geneva Conventions, which everyone has signed and which are guaranteed by that wonderful institution the Red Cross in Geneva. Not that the Red Cross will/can punish you for anything; but ratification of the conventions leads to the famous ‘universal jurisdiction’, i.e. the UK government has a duty to arrest a war criminal who lands at Heathrow – as nearly happened with General Almog.
And then we have the UN conventions against genocide et al, which it’s hard to prosecute people for. I think I’d rather go back to worrying about bordered Teichmüller spaces. (Of course, Teichmüller was a Nazi, but that’s by the way.)

To avoid worry and depression, which seem increasingly prevalent these days, I strongly recommend the translation of the Tao Te Ching on www.marxists.org. I think the marxists have taken some liberties with the text, but that’s fine with me. Here’s verse 19 to be going on with:

Throw away holiness and wisdom,
and people will be a hundred times happier.
Throw away morality and justice,
and people will do the right thing.
Throw away industry and profit,
and there won’t be any thieves.

If these three aren’t enough,
just stay at the center of the circle
and let all things take their course.

Many of my younger readers (yes, I have them), have given me stick for not giving space to any of Amy Winehouse’s oeuvre – favouring these foreign artists over the home-grown North London product. Quite right – and so here is the immortal ‘Rehab‘.

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