Posts Tagged 'Algebra'

Giving Lectures & Presentations in Graduate Classes (Commutative Algebra)†

Introduction to Commutative Algebra

Last Thursday, I gave my first lecture in a graduate  class of mathematics. There were three other students in that class, and the professor. All of the students were graduate students in Algebra. I was the sole person in Analysis. At first, this intimate setting was pretty daunting. I hadn’t taken the class last semester and the prof obviously didn’t like me being part of it¹.

Categorically Yours & Typographical Musings†

Okami's Lamy fountain pens

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One of the marks of being a good prof is when they see that students didn’t understand something and go over it again. My algebra prof has made a habit of this, especially when he goes over stuff in class too quickly. This happened last week and left me quite furious. Someone must have mentioned something to the prof², and he went over what we saw in the last hour again. This took an hour out of the three-hour class. I appreciated. I realized that there were some undergrads in our class, who weren’t familiar with some of the more abstract concepts of category theory³. It was an issue. Class was great today. I was drinking my strong milk tea and noting stuff down. We saw direct sums and (co)universal objects. Having proofs done with commutative diagrams is so elegant and simple.

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!

Measure Theory and Disposable Teachers

Wheeden and Zygmund's book

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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.

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.

Rainy Mathematical Days

Lamy Safari fountain pens²

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The rain has finally abated. I love the rain in Canada, but I hate it here. Why? You just get wet all the time. You get wet when you get on the scooter, when you drive around, and when you get off. Rain gear does wonders, but it’s annoying to have to carry it around and wait for it to dry. Also, driving in the rain is a lot more dangerous. I tend to be really careful.

Temperatures have cooled down significantly. It’s no longer 30C, but only 24C³. It’s getting a bit chilly when riding on the scooter. I’ll need to take a scarf.

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