It’s good to be back at school. I missed most of last week because I was teaching 30 hours. I didn’t miss much³, but I felt terrible. In the future, I won’t want to miss any school at all. My classmates were actually worried about me, which was kind of nice.
It’s strange that I was actually researching paracompact topological spaces on Sunday and that we are seeing those types of spaces in my complex analysis class. We just started the Berenstein & Gay Complex Variables¹ book and things are pretty interesting. I actually deduced that we were heading there because of some of the concepts that we are seeing.
I spent most of the day reading up on differentiable manifolds, Riemann surfaces, germs, and sheaves. Some of the concepts are extremely interesting since they tie into category theory. This led me to differential geometry. I supposed that differential geometry had more to do with Euclidean geometry, an undergrad class that I didn’t enjoy all that much³, but it’s got a lot more to do with the geometry and structure of differentiable manifolds, which interest me¹.
Since the late nineteenth century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds.
Things got abstract very quickly in complex analysis. We are constructing differentiable manifolds in the complex plane, to see the topology of holomorphic domains. It blends together quite a few algebraic notions, as well as some beautiful topology, and it’s extremely interesting. The prof told us that this would fit neatly into a Riemann manifold or Riemann surfaces class.
Why is this so interesting? It explains exactly why derivatives and integrals actually work in the complex plane. Well, that’s not really true. It’s more than that. Applying calculus to complex functions is certainly richer than for real functions. We delve into the differential k-forms and their construction⁷. It’s quite elegant, I have to say. Some of my classmates were a bit dismayed by the abstract nature of this week’s lectures, but it had my full attention⁴.
I also noticed that we started using Berenstein & Gay’s book, Complex Variables¹. We’re about 5 weeks into the semester and we are on page 10 or so⁵. The level of difficulty in this class just went up a notch. Also, the level of complexity went up. That’s why they call it complex analysis!
It seems to me quite logical on how math classes should proceed. The professor presents some theory, with theorems and their demonstration, as well as definitions and propositions, before venturing into a slew of examples. For some reason, this is completely absent from my classes in Taiwan. The examples. I don’t know what these profs are thinking, but examples are paramount for students to understand some of the theory. OK, my complex analysis prof is good. he gives examples and answers questions well¹.
I’ve been working hard this week at learning more about measure theory. It’s a really interesting research subject and there are quite a few things that I didn’t know about it. In class, we are currently seeing the Lebesgue measure and topics. I’ve read up on the Borel, Haar, Radon, and Daniell measures.
I’ve got quite a few books in this area, including Paul Halmos’ Measure Theory¹ that I got for $6. The Measure and Integral² book that is used in my real analysis class is finally available. I have it photocopied, but I’d rather buy it. It’s a bit more expensive, but not that much. It’s $46. Einstein has it for $69.
The real analysis professor spends 3hrs a week copying that book onto the blackboard. It’s really strange. He doesn’t give any further examples and quite a few of my classmates abandoned the class after the first week.
As I mentioned before, the classes are what you make of them. At my level, having a great professor doesn’t really matter, unless he’s my thesis adviser. I’m actually lucky that 2 out of my 3 profs are good. Since I am going to specialize in analysis, probably abstract analysis and topology, the real analysis class is fundamental to my mathematical development, as it introduces all sorts of concepts that were probably not seen at an undergraduate level. We’ve started the Lebesgue integral and I hadn’t seen it before.
I met a very helpful second or third-year grad student¹ that really helped me out with finding interesting research assistant positions. There are two for which I will be applying for. One will be for next semester, and the other one will be for next year, running for a whole year from July 2010 to July 2011². There was no way that I would have been able to apply for these this year. Still, it’s going to be interesting. That only leaves me to check out all of the profs with grants at my university to see if they need a RA for this semester. I think that it will most likely in PDEs since NTNU received some major funding on two large PDE projects.
One of my colleagues is an SAT tutor to some senior high school kids. This is paid quite well, something around 3 times what a usual foreign ESL teacher makes in Taiwan³. I’m going to see if I can teach something similar. It would be different from teaching English. The good thing is that I have taught math before, as well as science. What’s really interesting is that I wouldn’t have to work that many hours. I doubt that many hours are available anyway. Two hours would be the same as 6 hours of ESL teaching.
I’ve spent about 4-5 hours on my algebra homework. I still have another 27 problems to finish¹. Naturally, they get harder as you go along. Kind of annoying. I like writing easier ones first and then moving to harder ones a bit later. I like this to happen in each problem set. For some reason, I had trouble with cyclic groups and had to review the subject matter before completing two problems.
With these types of abstract math, it’s best to stop when you feel it slipping away or when you hit a problem that looks impossible to let it stew and come back to it. This has been my technique for the last few years and it works well. I have to be really careful with the solutions. I have all of the solutions of the problems that I’m doing in Hungerford’s Algebra².