Martin’s Blog

So I got my marks back today and got 3 As and a K (for kontinuing).

What have I learnt?

MATH511A Algebra – We did linear algebra and group theory this semester. Linear algebra was as boring as always but I feel a little more confident about stuff like spectral theory. Group theory started off as review but quickly moved into stuff that I didn’t know or wasn’t all that sharp at. Group actions, semidirect products and Sylow’s theorems were very nice and led to stuff like the classification of groups of order seen in the exam below.

The lecturer was good with a couple of caveats. He loves abstraction, for example introducing free groups by their universal property and barely even mentioning that you can think of them as groups generated by things with no relations. That’s nice for people who already have some intuitive notion of what a free group is but if you haven’t though about them before it would be confusing. He also had some timing issues. The linear algebra was all at a slow-medium kind of pace and by the end he went crazy covering what he wanted to – there was a crash course on nilpotent and solvable groups in 50 minutes for the final lecture.

Next semester I think we will do Jordan Canonical form, commutative modules and Galois theory.

MATH523A Real Analysis – This was set theory and measure theory. I knew all of the set theory except for the proof of Banach-Tarski which was very interesting. Measure theory so far is definitely at a higher level than Harold Bevan Thompson’s course. The exercises out of the book are really hard and we’re doing everything at slightly higher levels of abstraction. I feel like I understand how the theory fits together but am going to have to do some work to be able to answer questions under time pressure.

The lecturer is Russian and quite good. However it is just an endless march of epsilons across the board – very technical analysis.

Next semester differentiation (Raydon-Nikodym), spaces, some functional analysis.

MATH534A Topology/Geometry – I had never done any of this stuff before. We studied smooth manifolds and differential geometry on them. It’s kind of an awful subject – there’s about six weeks of definitions before you get anywhere and just showing something is a smooth manifold can take about three pages of work. We’ve been expected to learn most of the material just by reading the text, class time is just for examples and time wasting. So I have some vague idea about smooth manifolds, tangent bundles, cotangent bundles, tensors, differential forms, orientations, integral curves and flows, Lie derivatives, Lie groups,… but I’m not particularly confident about any of it.

The lecturer for this is a nice old bloke who is something of a paradox. He is a bumbling fool at the blackboard, sometimes spending 10 minutes just gesturing and mumbling. However he can write beautiful explanations in handouts and very interesting homework problems.

Next semester we need to clean up a few things so we can state and prove Stokes theorem in full generality. Then DeRham cohomology and onto algebraic topology.

MATH597T Professional Development Workshop in Teaching Mathematics (This is what the K is in. It’s a 1 credit course spread across 2 semesters) – Pretty much a waste of time.