Posts Tagged 'Analysis'

Graduate Student Lectures: Great Learning Opportunities But You’ll Need Preparation

Atiyah's Introduction to Commutative Algebra

One of the main things that graduate students need to cope with in the sciences is giving lectures. For some, they give lectures to undergrads. Others give lectures in their classes. In my case, since I am not fluent in Mandarin, I can’t serve as TA, which is what most of my classmates have to do. However, in both of my classes, we have to give lectures. This is very different from giving a 30 min to 1h presentation. Students have to assimilate new subject matter and present it to the class, while the prof is watching. In both of my classes⁴, there aren’t that many students³.

This can be quite challenging because you need to prepare fully before you give a 3h lecture. While you give the lecture, the prof will ask you questions about the new topics and proofs, to see if you have understood it. He/She will ask to see if the other students have understood as well. In my case, in my Commutative Algebra class, most, if not all, of the explanations are in Mandarin, but I usually get what’s being explained since I tend to prepare the topics even when I don’t have to give the lecture.

Continue reading ‘Graduate Student Lectures: Great Learning Opportunities But You’ll Need Preparation’

Algebra and Scaffolding

Great Lost Finale Coverage

I’ve been reading some great coverage on the Lost finale. Since I watched it yesterday, I can now read up on it. First and foremost is the great article over at Wired. The article has a bit of everything, but none of the answers that we’re looking for. Today, I was intrigued by this article over at Vanity Fair. Yesterday, I read up on a few different articles at the New York Times. There were about 4 or 5 different ones that I checked out. Mostly it was about the producers answering some fan questions.

Here are some more thoughts on Lost from Vanity Fair.

So, we have no idea (and it doesn’t matter) why there was a polar bear on the island, or why no children could be born there, or who The Others really were, or why a trip through the light cave turns the Man in Black into an undead, nearly omnipotent, smoky antihero but just sort of makes Jack and Desmond tired. There are probably dozens of other plot points that — to me — were presented as important details but tended instead to be diversionary tactics, or, even worse, not worth the trouble to reconcile and account even in a season which added episodes and gave itself a luxurious 2½-hour finale to wrap up whatever it wanted to.

I couldn’t have said it better myself.

I liked how the series ended the way it started. It’s evident that it came full circle. Did Lost become a soap-opera instead of a sci-fi thriller? Maybe.

Francis Urquhart & House of Cards

via Wikipedia

You might think that, but I’m afraid that I couldn’t possibly comment.
Francis Urquhart

When I was younger, I enjoyed two famous dramas on PBS from the UK. They were mini-series and both had Ian Richardson. The first was Porterhouse Blue and the second was the House of Cards trilogy.

In House of Cards, Ian Richardson portrayed the Machiavellian Francis Urquhart and his quest to for power. It slipped my mind that Richardson actually died in 2007. Anyway, I just spent some time yesterday watching House of Cards again. Needless to say that it’s pretty good. I’m unsure if I have seen all three mini-series, but I’ve just started watching To Play King.

Continue reading ‘Francis Urquhart & House of Cards’

Differentiable Manifolds

Morin surface as a sphere, via wikipedia

Morin surface as a sphere, via Wikipedia


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!

Continue reading ‘Differentiable Manifolds’

Measure Theory and Disposable Teachers

Wheeden and Zygmunds book

Wheeden and Zygmund's book


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.

Continue reading ‘Measure Theory and Disposable Teachers’

Rainy Mathematical Days

Lamy Safari fountain pens

Lamy Safari fountain pens²


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.

Continue reading ‘Rainy Mathematical Days’

GTM Hungerford’s Algebra vol 73†

Hungerfords Algebra

Hungerford's Algebra

For some reason, our Algebra prof gave us 40 problems to solve. They are all in the first chapter and review comprehensively what I’ve learned in past algebra and algebraic structures classes. Still, 40 exercises, that’s a lot. Seeing as Hungerford’s book is pretty much a reference for graduate students in algebra, I was looking around for solutions to all of the problems since I am unsure if our prof will give us any or if he will give us hints.

I found Dr. James Wilson’s book on Hungerford’s problem sets. It’s available for free here. If it’ no longer there, you can launch a google search and you should find it easily enough.

Part of me almost wants to print it out. It’s 167 pages long. At any copy shop, it will cost about $3 to print that out. That’s including binding. That’s really cool. It will help me out quite a bit. Luckily, I’ve seen most of the subject matter before, so it shouldn’t be a problem. The prof mentioned that he’d take our midterm exam out of this problem set. Sounds good to me. Midterms are in week 9 and we’ve just finished week 3.

Continue reading ‘GTM Hungerford’s Algebra vol 73†’

More Cheap Graduate Texts

Principles of Real Analysis by Aliprantis

Principles of Real Analysis by Aliprantis

So I went back to the book store³ and went through all of their books⁴. I couldn’t order Measure and Integral by Wheeden because it was just too expensive for my taste. Instead, I bought Principles of Real Analysis by Charalambos Aliprantis. I paid $30 for the hardcover 3rd edition, which is out of print because it’s listed on Amazon for $140. I wouldn’t mind getting the companion problem book as well. The Principles book was a third of the price of Measure. I’ve decided to get a photocopied version of that book. Since I was there, I ordered Real and Complex Analysis by Walter Rudin, a steal at only $15.

While browsing through the aisles, I saw Probability Theory I and II by M. Loève. Both were original hardcover editions that cost me $6.18 each. In the US, you’d bay between $70-90 for each volume. They are vol 45 and 46 in Springer-Verlag’s Graduate Texts in Mathematics. I found a few cheap undergraduate texts as well. Serge Lang’s Calculus of Several Variables, hardcover 3rd edition, David Bressoud’s Second Year Calculus, and Murray Protter’s Basic Elements of Real Analysis. They cost me respectively $9, $6.18, and $6.18.

Continue reading ‘More Cheap Graduate Texts’

Compactification and Metrisation

Some great notes from Dr. Terence Tao’s lecture on compacity and metric spaces.

Metric spaces are sets with a measure of distance between each of its elements. Compact spaces are spaces in which each sequence (xn) admits a converging sub-sequence. This is called the Bolzano-Weierstrass property and such compact spaces are called sequencially compact. There is a more general way of defining a compact space, by saying that each of its open covers has a finite subcover.

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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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