Had the induction to the Mathematical Institute yesterday, although all the talks on How Oxford Undergraduate Mathematics Works weren’t terribly interesting, probably on account of already knowing full well how thanks to the long hours I put in as an undergrad here. Still, crosses must be borne, and so on.
For those who like to exercise their clicky, I will be part of the Oxford Topology Group, with Professor Ulrike Tillmann as my supervisor. I don’t know what I’ll be up to yet, but I’ll be meeting her tomorrow and hopefully get something started.
There will be much more to say as soon as I start working, I’m sure.
UPDATE: Transcript of a conversation in New College bar tonight:
Physicist 1 (who I have just been talking to), to his friend: So this guy works in the sexy part of maths.
Physicist 2: Oh Topology?
It’s official folks: my field is sexier than yours.
What, you say, are you talking about? Robots? Have you completely lost it? Not at all, all shall soon become clear. I was looking over in the arXiv today at recent math.AT goings-on when I saw a pile of papers by Michael Farber, who I met at the very cold but quite enjoyable Prospects in Mathematics conference in Durham last winter. He found out that I went in for algebraic topology and gave me some survey papers of his on the field of the application of topology to robot motion planning.
How do you see the relevance of topology to motion-planning? Suppose you have a robotic arm with a few elbows on it. Then the position can be set by specifying some parameters (say, the angle and rotation) at each elbow, and the set of all possible positions is the configuration space, 
. Let 
be the space of all continuous paths in , and 
send a path to the ordered pair of points given by the initial and final points of the path. This will now be a (Serre) fibration, so suddenly we’re doing algebraic topology. Now a motion planning algorithm would take an initial and final point and give you a path between them. This is just a section of this fibration! Thus a “continuous” motion planning algorithm is just a continous section. It is fairly easy to see that such an algorithm exists iff is contractible (you can build a retraction to a point), so generally doesn’t exist. Thus the interesting thing is to see how discontinuous they need to be.
Farber introduces a numerical invariant called the topological complexity which has a few equivalent (for nice enough spaces) definitions, one of which is “the smallest number of disjoint sets that cover the space and on which the section is continuous” along with some technicalities (the sets should be Euclidean Neighbourhood Retracts). There are then bounds for this number in terms of the topological properties of X, but in particular a lower bound in terms of the cohomology, in the form of the zero-divisors-cup-length of the cohomology of the space. Pretty cool stuff.
Now of course you’d like to know the (co)homology of some configuration spaces. That’s exactly what you’ll find in Farber/Schütz “Homology of planar polygon spaces”.
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):
John Milnor has an article on the Ricci flow approach.
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