My exams...

...are over!

Topoggaly was horrible. I am not in any danger of failing, but I don't think I did well.
The main problem, in hindsight was the fact that I insisted on going for the Part 2 question on edge equations. In all the practice I did, this was dead easy, and before I even went into the exam hall, I knew I would do it. And as I was struggling my way through Part 1, I was always happy to know that some easy points, that also wouldn't take long, were on their way. It was also going to be a gift in that, generally, it entailed doing the same thing twice, for two different edge equations. This was true again.

Only it wasn't dead easy. First of all, I should have relabelled everything, rather than stick with the fiddly x₁, x₂, x₃ labels they used. That would have made things much easier. But there was the added fiddliness of the edges being separate and having to be combined before working to canonical form. I kept on doing this wrong. I got myself in quite a tizz, and eventually, having eaten into half of the time for my other Part 2-er, opted to move on without finishing the second half. For this I chose one on topological spaces, connectedness and so on. This started off straightforward, but proving condition T2 on the intersection of open sets involved about a hundred different cases to address. That's what it seemed like at least. The following questions about connectedness were easy, and just as we got to the last minute, there was a two point question on path-connectedness which I managed to scribble through just as the time ran out. If only I had chosen a different first Part 2-er, or given up on the edge equations sooner, perhaps I would've got through the rest in time. I think I should've gone for the one on metric spaces.

I also skipped one Part 1 question, because I took one look at it, thought "I don't remember having to do this for a shape with two types of face" and for once, used my brain and just moved right on. I did have every intention of coming back to it, but never had the time. Both of the people I spoke to after knew what to do, so apparently we had covered it, which made me feel rather stupid.

Oh yes, and I wasted far too long trying to prove that a sequence of functions was uniform convergent, without once realising that perhaps the reason I couldn't get the conditions satisfied was because it wasn't. This question in particular has burdened me with a week's worth of flashbacks! I think it was worth three points, and I probably got 1 for saying what I had to do, so it likely only cost me 2. Despite the fact that skipping the question above cost me 8, and the unfinished Part 2 ones cost me about the same, THIS is what kept bothering me.

Groups and geometry, on the other hand, was much kinder. I only didn't finish one question in Part 1, which are only worth 5 each. Although, I suspect that I didn't give enough information on another one. The one I left part of involved proving that a set H-intersect-N was a normal subgroup of H (or something). I kept getting muddled on proving inverses, and I don't know whether I got closure right. Rest assured, I got the identity. I wonder if I should have gone for the old if its non-empty, and if x and y in it implies x^-1y is in it, then its a subgroup routine. I did afterall, in my collection of bullet points of things to remember, note three times (unintentionally by the way; I must've thought each time I came across it, "Oh yes, I should make a note of that"), not to forget that you can do this.

For Part 2, you choose two from geometry and one from groups or the other way round. I would have been overjoyed if one of the questions on the specimen paper was in it. Question 13 I think it was, just asked you to classify three wallpaper patterns. This was really easy and took less than five minutes. For 15 points! I knew it was too much to hope that'd be in it though, and it wasn't. Nevertheless, I did choose a question on wallpapers. That was the second Part 2-er that I did; the first was on colourings of a brooch. This was fairly straightforward, but coming up with the pattern index was very very fiddly. I was overjoyed then, when I divided by the order of the group, 12, that I got integer results for the various equivalence classes of colourings. It took a long time though. Polya's Theorem, I think is the result we used for this.

My groups question was on Abelian groups. That old chestnut of find the Abelian groups of a certain order. This wasn't very hard, and I had had much practice at it. The final part of it, which involved finding how many elements of order 45 each of these groups had, took the longest. And there was plenty of scope for basic arithmetic mistakes, and I must have made some along the line somewhere. Then again, this wasn't worth too much. The whole final part was worth 5 points, and this bit of it came after "stating" (which is always nice since no justification is required) how many elements of order 9 there are in Z₂₇ and of order 5 there are in Z₂₅, something which even if I didn't know already could be worked out quite quickly. They were probably worth 1 or 2 of the 5 points already.

Well that's that then. All over. I am quite happy with how that one went, and it cheered me up a lot after the debacle that was my topogally exam. Too bad it had to be that way around, because it doesn't really reflect how much I liked each course. Or perhaps it does reflect it, in that it's the opposite :)
That said, I enjoy M336 much more this week than I did last. Not because of the exam, but because in the course of revising it, I think I finally "got" a lot of it. And now feel that groups in particular has potential. Too bad a lot of the year was wasted on wallpaper patterns, when we could have gone onto rings and fields...

And so, onward, to my final level 3 course, MT365. I am not happy about the fact that a lot of people choose this because it is supposedly quite easy, as I don't want that to be thought about me. My reason, in fact, is a conversation I had with my topology tutor a few months ago, about how I don't really enjoy applied that much, but am all out of level 3 pure courses. He said that although it is not a pure maths course per se, it is a lot purer than any of the other level 3s. As well, I am interested in the connection between graphs and topology.

Good luck with your results, those of you who also took exams recently. And I hope you enjoy the few days break before the next course(s).
Neil H