Not very good and the old regular updates, it seems. The things you learn about yourself, eh? Although I’ve been back for ages now (it is, after all, fourth week of term already) there haven’t really been any grand breakthroughs. I’m not quite sure why I expect there ought, but it would certainly be nice. Perhaps I’ll first give a plan for the future, in no particular order, especially not chronological:

• Stable homotopy theory: classical calculations and modern structures, Strasbourg, May 7th - 11th. Self and one or perhaps more office mates will probably be attending. If anyone else is do drop me a line, it would be nice to know at least a few of those going! Here the link.
• Junior Geometry & Topology Seminar, Thursday of 8th week. I’ll be giving a talk, tentatively titled “The Classification of Smooth Manifolds”, cunningly not mentioning up to what sort of equivalence to trick some geometers into attending. Unable to resist he continues: it’s up to unoriented cobordism. Thom showed (in the 50s?) that the cobordism ring of unoriented manifolds is polynomials with mod 2 coefficients on a system of generators, on in each dimension not one less that a power of two. Try here.
• Soonish (the precision of mathematicians!) I ought to be getting started on something concrete, though I don’t know what yet. It’s very hard to have any idea of what directions are worth looking in, especially when you know you could spend ages getting nowhere.

And now, what I have done lately:

• Got very excited thinking I had shown something that had evaded the experts, to wit, a sentence in an arxiv paper of my supervisor, Ulrike Tillmann, saying something was unknown. Today I find that that is an old draft and it is now known, proved in the same sort of way. For the curious, all free loop spaces on the circle have nonunique loop structures, and you can easily build an inequivalent delooping.
• Some work in cobordism in preparation for the second item above.

And of course other stuff, to minute to mention. Ta ta.

Excitingly, they were filming for “His Dark Materials: Northern Lights” on my street today, and I would have taken some photos, but the camera was batteryless. As far as I could overhear, it was just background shots, and so no shlabrities. I sat for a while and watched, to notice that it is most unglamorous and repetitive, this filmmaking lark. It also contains extended periods of standing around, and misunderstood requests:

“Dave, could you move that to the left, out of the way?”

“Here?”

“No, to the left, Dave.”

The magic of cinema.

Academically, I’ve been all over the place since last I wrote. However all these papers I’m supposed to be consuming don’t seem to dwindle. They are really quite technical, you see, and interesting, but require a fairish amount of working things out for yourself, which in turn requires all the Time there is.

So instead I flit hither and thither learning about things like principal G-bundles and characteristic classes, that you can actually show things with. I did some work on spectral sequences as well, but don’t have anything to apply them to yet, so I think I’ll wait until I do to learn more.

Unsettlingly, I don’t really know what I should be working on. This is a problem.

Lately I’ve been working on the papers below, but not perhaps as much as I should. I’ve been more interested in techniques to study the cohomology of Fiber bundles: the Leray-Hirsch theorem and the Gysin sequence.

It’s instructive to use Leray-Hirsch (or the Gysin sequence, for that matter) to compute the mod 2 and integral cohomology, respectively, of the real and complex Grassmannians, then define the Stiefel-Whitney and Chern classes to be the pullbacks of the obvious classes via a map that classifies the vector bundle. This is all well-defined, and some properties of the classes are immediate, but the Whitney sum formula for total classes is a bit tricky. I spent a goodish bit of Friday trying (and succeeding, thankfully) to prove it, with the vital guidance of George, a fellow topologist.

I don’t suppose it gives much more insight into the construction of these characteristic classes, but it avoids doing extra work.

There was also a talk on Friday by Eliana, another starting student of Ulrike Tillmann’s, on

C.F. Bödigheimer and U. Tillmann, Stripping and splitting decorated mapping class groups, Progress in Math. (78), Birkhauser (2001), 47-57 (link)

in which we covered approximately the first half. The second looks more complicated, and is mentioning configuration-spaces, so the other paper I am gradually traversing might come in useful. In fact, I may be giving a talk on it the Friday after this.

I’ve also been reading about Brown representability, as I had to give a quick talk on it this morning. Interesting as far as it goes.

I went up to Rhyl to visit my Grandparents this weekend, and found the above print on the wall of what is now an arcade, but used to be the “Left Bank Bistro” many years ago. Progress, what?

It’s not a thing you muse about much, when you’re not marking people’s work, but have you ever wondered what chain of events in someones mind leads them to write down what they do? A rather convoluted one, evidence would suggest.

Not to suggest that I have never been in error. It has happened, I remember it well. But they do write some outlandish things, by any standards. You should see some of the things other people have to mark. Gladly I don’t mark any algebra courses.

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.