Well, pleasant reader, what have I been doing lately? Latest in the learning-fest is

G. Segal, Classifying spaces and spectral sequences, Inst. Hautes Etudes Sci. Publ. Math. No. 34, (1968), 105-112 (link)

but just the classifying spaces bit for now. This paper isn’t where they were invented, that honour goes to Grothendieck, but it is perhaps where they were first explained in a simple manner. I had a brief talk about it with my supervisor today, seeing how natural things in a category give rise to natural things in its classifying space (which they should, seeing as the space is almost just a picture of the category), like a monoidal category has a monoid structure on it’s classifying space, and so on. Not entirely sure, so far, what the point of it all is, nor how it links in with the usual definition of the classifying space of a group (surely the classifying space of the category associated to a group must be the usual classifying space for the group?)

There is also a small project among the postgrad Algebraic Topologists in the department to read through and try to understand the paper

U. Tillmann, On the homotopy of the stable mapping class group, Inventiones Mathematicae Vol. 130, Issue 2 (1997), 257-275 (link)

(which may not be accessible to those on non-academic networks), but there are several others we think we’ll need to cover first.

I’ve also been doing the usual Hatcher-work, on homotopy theory this week, so loopspaces, homotopy construction of cohomology, fibrations, Postnikov towers and obstruction theory (which is something I’ve wanted to know about for ages, because it sounds so cool. However, now that I know about it I can’t think of anything nontrivial to apply it to. Ho hum).

Plan for this week:

• Mark my first set of scripts, for Professor Nigel Hitchin’s Geometry of Surfaces course, and attend the class bright-eyed, willing to help, and able to.
• Find a good exposition of (semi-)simplicial sets, read, digest, ruminate. I think I have found on, by May.
• With this knowledge, apply myself to the first paper mentioned above, while maintaining a beady eye on how it applies to the paper in my last post.
• Pick out choice morsels from the Extras Disc in Hatcher. Brown Representability, H-spaces, Limits and Ext should do.
• Read about things I’m already supposed to know about in this fantastic new book I’ve found, who’s name currently escapes me. It may be by May though, who for this week is my idol.
• Mark my second set of scripts, this time with the steely eye and grim resolve of an old hand, for Professor Marc Lackenby’s Topology and Groups course, and attend the class with optimism, as this should be something I’m quite happy with.

I had a brief talk with my supervisor yesterday to see what I’ll be up to. It looks like the short-term goal will be to understand

G. Segal, Configuration spaces and iterated loop spaces, Invent. Math. 21 (1973), 213-221 (link)

but putting in the hours with Hatcher’s Algebraic Topology will carry on as usual. Last night I discovered Whitehead products on homotopy groups, and it was nice to know I haven’t forgotten everything. In fact, it looks like I’m meant to give a short talk on either Obstruction Theory or the Cohomology of Fibre Bundles. No doubt it will appear here if I do.

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.

After the Malaga-Manchester mob-lynch debacle, we have another instance of not-especially-covert racism, in the form of

A 32-year-old man speaking Tamil and some English about a sporting rivalry was questioned at Sea-Tac Airport and missed his flight Saturday because at least one person thought he was suspicious.

and then

The man was speaking Tamil, a language largely used in India, Sri Lanka and Singapore, on his cell phone at the departure gate and on the aircraft. An off-duty airline employee heard the conversation and informed the flight crew.

The man also apparently said something in English about a sporting rivalry at his alma mater.

but don’t worry because

“It’s a big misunderstanding,” said Parker [airport spokesman]. “He had a perfectly innocent explanation that all added up.”

Excuse me? An explanation for speaking Tamil? Is such a thing needed? What I’m really looking for is your excuse for being such an overt buttmonkey.

[The victim] told officials that he would not speak in a foreign language on his cell phone at an airport in the future.

I hope he was smirking when he said this, sporting a nonchalant and world-weary gaze, and didn’t have the actual fear put up him.

So I’ve moved back in at Oxford, just a skip down the road to where I lived in my innocent and carefree first year. This is my room:

Room aside, things seem to be going well. The induction to the Mathematical Institute isn’t until Tuesday, so I won’t be able to sort out the supervisor situation until then, but I think I have my mind made up. I won’t say anything yet though, just in case something changes.