Differentiable Manifolds

Morin surface as a sphere, via wikipedia
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!

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Mathematical Algebraic Structures

Algebraic structures, via Wikipedia
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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