Well, what I shall do, actually. This requires a fairly liberal interpretation of “holidays”, encompassing anything that is done away from ones home city. The first thing is the Postnikov Memorial Conference, 17th - 24th of June, in Bedlewo, Poland. The first thing to notice about Bedlewo is how it’s never even heard of the beaten track, nevermind been on it. Consequently planes, trains and automobiles all need to be involved. Luckily I’m going with my office chum Liz, so I won’t be entirely alone in the enterprise. Compare and contrast with the Strasbourg conference.

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…

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?

I just returned last night from my “summer holiday”, if you will call it that, as it already lies within a holiday, in Portugal with Parminder. As is to be expected, the flights were all a bit or a lot late (can there be an excuse for a 2 hour delay on a 40 minute flight?) and it didn’t help that I still had a bit of a bastard behind the eyes from my mystery illness at the beginning.

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