Martin’s Blog

Algebraic Geometry: This was a pretty scary class and the professor on the last day said “You probably haven’t learnt anything in the last 6 months.” He may have been right in my case. The other great quote was when he said “Hartshorne was a wimp” (he later clarified that he only meant he was a wimp in how he deals with arithmetic).
We talked about schemes, sheaves, Cech cohomology and various topics like line bundles and the Riemann-Roch theorem. The nice thing about the approach taken was that it was very geometric: the horrible commutative algebra that underlies everything was kind of ignored. We didn’t prove a whole lot but I feel that I now have some idea of this vast edifice that is algebraic geometry.
There ended up being no assessment at all in this class but I did spend many hours trying to figure out what the notes I took were going on about.

Integral Lattices: This class was taught by a teaching robot – he was amazing, going for 75 minutes every Tuesday and Thursday without ever looking at his notes or stopping to breathe. He was very good at including all the details but not so good at explaining why we should be interested.
The topic of the course was lattices: basically abelian groups with a quadratic form that give you the length of a vector. Quadratic forms are used all over the place in number theory, geometry, algebra, physics, etc. but I still don’t really know what the point of some of the things we did were. I don’t really care how many even unimodular lattices of rank 24 there are. Some of the applications to sphere packing/coding theory were interesting although they are more useful to mathematicians than people in the real world. I gave a small talk about a coding theory result that implied a fact about lattices that you can look at here.

Channel Coding: This was an engineering course and required a fair bit of work but much of it was programming which is nice. I am now able to write mediocre C code, which will I’m sure be useful to me in my life.
In contrast to the high speed lectures of the two courses above this class went kind of slowly and there were lecture notes provided. It was occasionally frustrating – we covered the Berlekamp-Massey algorithm just by writing it down (no explanation of why it worked) and then going through amazingly complicated examples by hand. I certainly am pretty good at doing finite field calculations by now though.
But overall it was very nice to do something useful in the real world and see a lot of different topics.

As a postscript I realise that I may seem negative in my reviews of these courses but I do think they were taught pretty well. I would say that I’ve never taken an advanced maths course that I was entirely happy with. Still, lectures are the best way to learn things I think so I don’t really know how to improve them.