All I want for Christmas is a Chumby. Too bad they won’t be out by then. Oh well.

Well, a bit of excitement at home last night. I think we had an attempted breakin. I was just going to go to bed, and I was in the kitchen getting a drink when I heard a sharp knock at the back door. We never use it, so this was very odd. I turned all the lights off (thinking that I know the layout of the house and whoever is there probably doesn’t) and waited around the corner to jump whoever came in, but I just heard a few more noises and then nothing. Then I thought I was just being silly and carried away, so I carried on getting ready for bed, and just went to check the back door before I went to sleep. It was unlocked. Hmmmm, I thought, now I’m kinda scared because maybe there’s someone already in the house. I got one of my brother’s golf clubs, turned all the lights off and called him to ask about the back door, and he said he was sure he’d closed it the last time he’d used it. He was only down the road so he came up and got himself a golf club too, and we went around the back of the house. We were quietly walking down the back when my brother said he heard someone run around to the other side of the house, so we ran around that way too and my brother suddenly thought he was in a Guy Ritchie film and started doing a bit of shouting in his east end accent. We could hear a bit of movement in the bushes behind the house, but it was probably nothing.

Anyway, we come back in saying it was probably nothing and we’re just getting spooked, and my brother goes to bed. I’m just getting ready to as well, and I hear something out the back through my window, so I turn my light off and stand there looking out the back (because I could see a bit with the moon, but you wouldn’t have been able to see me in my room). I hear a few noises and thought I saw something moving off to one side, then the noise at the back door happens again. I go to wake my brother up and probably scare him half to death because it’s pitch black. I didn’t want to turn the lights on, it feels spooky knowing people can see in the windows but you can’t see out. He tries to call one of his bruiser friends but there’s no luck, so I get a golf club (yet again) and quietly move around the house to see if I can hear anything. The noise at the back door happened again and just as I’m going back into my brother’s room, there’s another noise on the roof, which as we’re now completely wound up sounds exactly like a tile breaking as someone steps on it. Anyway, my brother calls the police and we sit there in the pitch black both holding golf clubs and waiting. A while passes and we see a light come from the kitchen, where the back door leads to. We rush it with our golf clubs but there’s nothing there, and we see another light at the front so deduce that it’s the police (you’d think they’d let you know it’s them, but maybe they don’t want to let the crims know).

My brother opens the hole in the door and sees it’s the police, and they’re saying “Are you the proprietor?” and he walks out with his hands up in the air, in just his boxers, saying “I’m the son, I’m the son!” This made me laugh, as now six policemen waving guns (welcome to the continent, where the police are armed and keen for practice) were there I could see the ridiculous side. They had a look around and couldn’t see anything, and I went out to look at the lock on the back door but couldn’t see anything amiss. They got our details (and unloaded their guns, some sort of protocol I suppose. One of them said “If we found someone here tonight we’d leave him like a colander. It’s a good thing you’re doing some building, we could put him in the concrete.” Policia Local: brutality with a smile) and went on their way. Now me and my brother were quite happy again, and joking about him walking out with his hands up in his boxers, about how funny it would be if that’s all a neighbour saw of the night’s proceedings! A normal night then.

This just in from the War on The War on Terror: get a “I Am Not A Terrorist” T-shirt and wear in on your next flight to really put the shits up The Man. The Man, it is known, fears what he doesn’t understand.

This is going to be slightly uncomfortable for people who just want to know what I’ve been doing this summer, but this post is going to be about Quillen’s Plus construction. Why? Because I was trying to remember the details when I was having dinner in Bar Uno the other night, and couldn’t quite get it all out. Also I want to test out my new renderer, which incidentally looks a bit bad as it is giving images to put on a white background, and has to be recompiled to change that, and I don’t have shell access yet. Incidentally, after looking up the bible, as mine was too heavy to bring over from Oxford, I notice the Allen Hatcher/Terri Hatcher Google war is being dominated by the topologist. Good stuff. (more…)

I was going to talk about Grigory Perelman turning down his Fields medal, and that if he wants to avoid publicity this is probably not an ideal way of doing it, but that has been covered well enough already. In particular, Gooseania has all you probably want to know about it. Show off time: people involved in the media storm around this that I’ve met: Nigel Hitchin (my supervisor next year), John Ball (president of the IMU, and took a manifolds course with me last year, which was odd) and Marcus Du Sautoy (media spokesperson for mathematics, it seems).

Instead I thought I’d talk a bit about what he’s meant to have done, as this gives me a reason to find out myself. As anyone on the street can probably now tell you, it’s the Poincaré Conjecture, which asks if a 3-manifold with trivial fundamental group must be topologically a 3-sphere, or in plain talking that if you have a 3-dimensional space where every loop can be contracted to a point, then must your space be a deformed 3-dimensional sphere. So far so good. Many have tried, though, and many have failed, but they found out lots of cool stuff about 3-manifolds, and so probably started the field of Low Dimensional Topology, so it’s not all tears. Perelman got around the problem by instead solving a much harder one, Thurston’s Geometrization Conjecture which concerns, in broad strokes, the decomposition of 3-manifolds into pieces which have certain, well- or better-understood geometries (the Thurston model geometries).

How then did he do it? He used an idea of Richard Hamilton (so-called Ricci flow), whereby you distort a given 3-manifold by letting it’s Ricci curvature follow a differential-equation analogous to the heat equation. The idea is that this should smooth the Ricci curvature out until it is uniform over the whole manifold, and what you then have is some sort of canonical form for the manifold, which should be one of the Thurston model geometries. The interesting thing about dynamical systems is their singularities, but what needed to be shown was that these would be well-behaved ones that can be resolved. This is what Perelman did: he formulated a variant of the above (Ricci flow with surgery) in which singularities develop but can be removed and so controlled.

For the keen, his original papers on the arXiv (may be down due to demand):

• Perelman, Grisha (2002). “The entropy formula for the Ricci flow and its geometric applications.” November 11
• Perelman, Grisha (2003). “Ricci flow with surgery on three-manifolds.” March 10
• Perelman, Grisha (2003). “Finite extinction time for the solutions to the Ricci flow on certain three-manifolds.” July 17

John Milnor has an article on the Ricci flow approach.