full of good intentions

One of the big themes of my time here in Manchester has been the conflict between my responsibilities as a grad student and my urge to relive my ‘wild’ undergraduate days. The conflict here isn’t about timing, or anything: yes, I do get to be very flexible with when I turn up and work and yes, sometimes motivation is hard and my bed is warm, but as a whole I’ve managed to find a decent working routine and stick to it. It helps that I do actually enjoy what I’m doing. Umh, see below.

The conflict comes from being this person with a good degree and four years of university – with all the lessons learnt and growing up that entails – under my belt, and essentially with a job (have I mentioned that I get paid to do maths?) , and yet still living a university life on a university campus, with all the drinking and irresponsibility and drinking from saucepans that entails. There’s an underlying friction between those two ideas, and I’ve yet to reconcile them. I’m a grad student. But you can’t really be both at once. The question is, am I a grad or a student?

So it’s a balancing act for now, mainly facilitated by spending the weekends reliving the heady days of, um, six months ago. It helps that lots of my friends are Masters students, who are basically undergrads, so I can go out with them to parties and indie bars and so on at the weekends. And it’s a good life! I’m certainly not complaining. I’ve just been feeling a certain amount of unease that I’ve only really started to understand very recently.

This is a very long way to say that last Friday was very nice, and the sort of evening (morning?) I will miss dearly once I’ve left academia for good. And you’re not getting the rest of that story from me.

* * *

My research is getting really cool. I think I like what I’m doing now more than any math I’ve seen yet. I’ve been thinking about fibre bundles, which are like generalised product spaces. To start with, consider a cylinder: that’s just the product of a circle and a line. If you were to map the cylinder onto a circle by squashing it down, and then take any single point on the resulting circle and look at what parts of the cylinder were squashed to that point, we would get a line: everything above and below the point that got squashed.  So have a fibre bundle with total space the cylinder, base space a circle, and fibre a line. Now, what about the Möbius band? It’s clearly not a product space: it has a twist right through the middle! But if you squashed it down as before, and then looked at the parts of the band that were squashed into the same point, we’d once again find a line. There’s a bit of technical stuff I’m just skimming over, but we say that the Möbius band is locally a product: it’s not a product as a whole, but at any point, it looks like one. The total space here is the Möbius band, but the base space and the fibre as the same as in the previous bundle. The cylinder and the Möbius band are made from the same basic ingredients, but the fibre bundle captures the twisting phenomenon here rather vividly.

These examples are very simple (if I could be bothered, I’d draw some pictures and you’d say ohhhh, guaranteed), but they’re just baby examples to give you the basic idea. I’ve been studying a much more complicated fibre bundle called the universal U(n)-bundle, which in a sense classifies all ‘nice’ bundles with fibre U(n) (which is an important object in mathematics and – I’m told – physics). Fibre bundles are really nice objects because they have so many nice homotopy-theoretic properties, which let us really attack them and find out lots about them. It’s fairly straightforward to compute the homotopy groups of a fibre bundle, for instance, because they fit into a powerful exact sequence, and there are lots of powerful methods that let us tackle the (much more mysterious) cohomology of these spaces, too. It’s neat stuff. Honest.

* * *

It’s practically a year since I met Jilly (and by the time I next update, it’ll be long past a year, so I thought I’d mention it here). I do miss her, but in a good way. I spent my last few weeks as an undergraduate finishing up my dissertation, basking in the sun, and spending time with her. They were carefree times. Nostalgia doesn’t have to be bittersweet.

2 Responses to “full of good intentions”

  1. 1 anders! April 9, 2011 at 11:58 pm

    lots of my friends are Masters students, who are basically undergrads

    Hey! :[

    • 2 rootpot April 10, 2011 at 10:22 am

      I felt like an undergrad when I was a Masters student, although the MMath is an “integrated” masters, which means it’s basically an extra year on the end of your bachelors. Which explains why I didn’t feel like a postgraduate, because technically I wasn’t one. :P

      Certainly, I’ve found the difference between being a fully-fledged grad student and a Masters student a lot larger than the difference between being a Masters student and an undergraduate in Sheff, although there are lots of external factors at work here (new city, etc.).

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