## Posts Tagged 'clifford algebra'

### Differentiable Manifolds

Morin surface as a sphere, via Wikipedia

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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!

### Researching Trends

Clifford algebra representation

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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.

### Mathematical Algebraic Structures

Algebraic structures, via Wikipedia

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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.

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