Differentiable Manifolds

Morin surface as a sphere, via wikipedia

Morin surface as a sphere, via Wikipedia

I

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!

A torus, which is a Riemann surface, via Wikipedia

A torus, which is a Riemann surface, via Wikipedia

II

I asked a colloquium coordinator how long our report would have to be⁶. He said about one page to be handed in some time in December. I think that I’ll write a 10-page review on Clifford and Azumaya algebras. Until now, this was the most interesting presentation and it made me want to discover more about these algebras.

I mentioned the presenter a few weeks back. It’ s definitely interesting and my guess is that it won’t take me more than a few hours to get something out. I’ll use it as a information gathering tool.

III

I’m getting  a sponsorship deal through Velocite bicycles and to top it off, I’ll be able to try the bikes out before I try. I’ve got my eyes on a Helios carbon fiber ISP frame as well as the Millennium titanium frame. The Millennium is perfect for long hauls and training. The Helios will be my entry into the CF bicycle world. Why Velocite? They adhere to stringent EU QA norms. Prices are good and I can actually try them out before I buy.

They’ve got great deals on Shimano, Stronglight and TRP components. Velocite also makes their own carbon clinchers and other components. I’ll be getting a premium bike for not that much, which makes me kind of happy. All in all, the Helios and Millennium frames will make a great combo and aren’t that expensive. I’ll probably get a Helios first and then a Millennium. I was pretty much at a standstill before, since I had serious doubts about the QA of the other small brands that I was looking at.

IV

It’s definitely hard to explain to people who have never done any calculus what I’m exactly studying. One of the reasons why I enjoy abstract algebra is that differentiable manifolds and intrinsic differentiability is scaffolded onto these algebraic structures. Most people don’t really think about them, but they are prevalent.

I’ve tried explaining it to my wife, but she didn’t get most of it. Neither did Chad. It’s strange to realize that what you are studying goes over most people’s heads. I find that it puts me into a whole new frame of mind. For example, today I was working on an interesting algebra problem. I couldn’t stop thinking about it, even during this week’s colloquium, which was on surreal numbers³.

VI

I spent about 6 hours on abstract algebra. I managed to finish almost completely. I’ve got 5 problems left out of 39 and they involve permutations, which I find boring. I still have to review some of the more funky problems that I solved. Cyclic groups and quotient groups lead to interesting problems.

Now that the algebra problem set is pretty well handled, I’ll move onto the real analysis problems. There are a total of 56. The prof didn’t give any hw, but I’ll try them all. I don’t know why this class is called real analysis since this is a measure and integral class, an advanced abstract analysis class. The title is misleading. Anyway, this topic is most interesting, but the prof is boring. He spends 3hrs a week copying the Measure and Integral² book onto the blackboard. Stupid, but I haven’t missed a class. I noticed that some students have started missing class.

Out of all of my classes, I expected to like modern algebra the least, but I’m enjoying it. Complex analysis is the most interesting class with some extremely abstract notions. But I think that I’m making the most of my classes and studying by myself. I haven’t been diligent enough over the lat week, but if I push myself for the next few days, I’ll get a lot of work done.

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* * * * *

[¹]: Berenstein & Gay, Complex Variables, 1991, Springer Verlag, GTM Vol 125
[²]: Measure and Integral by Wheeden and Zygmund
[³]: It wasn’t really interesting as I’m not interested in number theory. I’ve actually read up a bit on them and it’s an interesting concept. I think I was distracted by algebra today.
[⁴]: So much that I didn’t really pay any attention to the farter/sleeper in my class. So much that I actually had follow-up questions for my prof.
[⁵]: The first few weeks were spent reviewing and presenting some important results in complex analysis, that I had already seen in an undergraduate class. However some of the math education people and some of students with different backgrounds hadn’t seen these results.
[⁶]: Colloquium takes place once a week and lasts an hour. Different professors and researchers present. All you have to do is to be present and write a report on 1 or 2 presentations at the end of the semester.
[⁷]: In college and in undergrad school, linear algebra has always been my strongest subject. That’s why I did so well in numerical analysis. It’s interesting how linear algebra is used to tie these notions together. It’s quite elegant.

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ranjitwithkinginbehand.jpgI'm Range, your host. On the menu, photos, art, stories, entertainment and reviews. Links, maths, education and social issues. I'm in Quebec (Canada) or Taiwan (R.O.C.). Follow me on Twitter.

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