state of the union

Prelim studying is going all right, I guess.

In a ways it feels like downloading a large (legal) media file: at every instant you can read off the streaming speed from the Download Manager, but you cannot predict for certain how long it will take before the download is finished. If you have a deadline, then you watch the screen and hope, then watch the screen and hope some more.

Similarly, I feel like I'm learning at a good rate, but it remains a mystery whether I can be fully prepared for all the topics on my syllabus. We do what we can and I understand that, but I'm having trouble accepting that it's all I can do.

Earlier this weekend I thought I understood tangent measures (arising from blowups of space [1]) but now I'm not so sure.

It can be troublesome to learn from books, because if one is not very clever (i.e. me) then one adopts the perspective of the author by default. There's nothing inherently wrong with this, but one runs into trouble if

1. .. the book isn't "very good," as measured by difficulties such as omitting details, unclear language, possible errata, and the like.
   
   
2. .. despite the fortune of clarity, the book has an unorthodox perspective, and what one had thought was standard terminology is actually "author-speak."
   
   This becomes a deeper issue if the methods of proof differ from one source to another and if one seeks to generalise arguments from a common starting point (for example, Radon measures in Euclidean space).

So I worry. There might be no "right way" to understand a particular idea, but fortunately or unfortunately, there are some "wrong ways" to gain understanding.

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Research is .. well, research. It's shifty, ever-changing, and hence unpredictable.

Three hours ago, I thought I had exhausted every possible idea out there and that the problem is insoluble, despite what M. Morse claimed in the 1950s .. [2]

.. two hours ago, I began to draw the same diagrams, only to realize that I drew them wrongly! Now things look more promising, but they'll require a bit of trickery before I can prove what I want to prove ..

.. and for the record, an hour ago I was eating dinner: vegetarian Indian food which made everyone else in the room, who were Indian food-free, salivate. I felt guilty and sinister at the same time. (;

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Teaching is what it was before: neither good nor bad until you think about it and make judgments.

I tried to switch things a little and make my lectures more example-driven and start matters off with what the students know well (or should know well from their last exam). Time will tell if they are just as confused by the material or if it's actually working.

I must admit, the examples were fun: I introduced differential equations today by giving examples of unhindered and constrained population growth, and perhaps as a moment so that they could catch up with theit notetaking, I idly mentioned the famous viewpoint of Thomas Malthus, if only to mess with their heads and get them thinking.

However, one bit worries me, and I call it the "Heisenberg Uncertainty Principle of Calculus Education." It goes something like this:

The more interesting you think the class topics are, the less likely your students will understand what your talking about. Conversely, the more your students understand what you're lecturing about, the more likely you find it boring ..

That's mean of me to say. Any opinions? q:

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[1] and by that, I don't mean the blowups from algebraic geometry! Perhaps the term "rescaling" is more accurate here.

[2] In one of M. Morse's papers on the Schoenflies Problem (before he studied the problem in terms of conical points with Huebsch), there is a claim which asserts that a smooth analogue of the topological Schoenflies Theorem fails.

Morse cites a paper of Milnor (the one about exotic 7-spheres) and since then, everyone else says the same in a rote manner. I've might have mentioned it before, but my current task is to make sure that Morse's claim is correct.