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Secondly, for 8th - 14th of July I’m off to sunny Southampton (see right) for a week-long Homological algebra bonanza. Here the meaning of “holiday” becomes hazy, as i) we are doing algebra, and ii) we are not even crossing borders for the pleasure of it. Nontheless, ever optimistic, I will call it a holiday. On this sojourn another office-mate will be my companion, this time George.

Thirdly, an honest holiday: Istanbul with Parminder (of com leite frio fame, the jet-setting bon viveur) for 6th - 13th of August. It seems like a fantastic place, and I fully intend to return laden with outrageous Turkish objets d’art, worthy of a fin de siecle orientalist. Enough with the italics.
Fourthly, Marbella at some point. Which point, exactly, remains unknown. I will try to coincide it with those points belonging to friends of mine. Promise.

And now for something completely different. Strasbourg was as indicated on your left. In fact, that’s not very representative of my time there, as mostly it was spent in rooms with blackboards and writing things down as quickly as you could before the effervescent lecturer lost you entirely.

In other news, the paper that Johannes Ebert and I wrote, about characteristic classes of bundles of non-orientable surfaces, has been sent to the Bulletin of the London Mathematical Society to se if they like it. Unfortunately their rather stringent requirements mean I can’t post a preprint online, nor on the ArXiV. If, however, you are somebody who is interested, just ask me and I’ll send you a copy.

More recently, I’ve been working on understanding the space of d-dimensional submanifolds of a fixed n-dimensional manifold M. This is related to the concept of cobordism in the background space M (which, if you like, is what string theory sort of does: M is some 3-manifold called “the universe”, and cobordisms run in the time direction). I don’t want to mention too much as it might be a fruitful area for me to research…

This of course occupied much of the preceeding few months, but I’ve been doing a few other things too. In particular, I’m getting quite interested in hardcore stable homotopy theory, with formal groups, MU, and all that story. Not things that are obviously or immediately applicable to problems I’ll be working on, but very interesting and clever stuff. It’s best to do this broad learning now I think, because when I’m stuck into a problem I have little mind-space for anything else.

So I’m not going back to Marbella this Easter, as I’m saving up my precious scraps of holiday for the summer, as all the world assures me Oxford is a dull place of a summer. I’ll be going to Marbella for a while, possibly accompanied by this child and this man:

So I’m definitely going to Strasbourg in May now, so do let me know anyone else who is.

• 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.

Anyhow, I have now finished my first term of this game, and couldn’t honestly say that I haven’t learnt much, because the stuff is dribbling out of the old ears. The stuff I enjoy, it seems, is all rather old fashioned (as far as that can be, in a subject that, in it’s modern incarnation is about 60 years old) and so while interesting isn’t going to be very fruitful in terms of research material.

Doesn’t bother me though: I like what I like, and it’s early days, so I’ll stick to it for a while.

I’m off to Marbella for Christmas. See you there, if that’s your thing.

“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.

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?

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.