(See my previous post, here.)
It was a fine talk, and despite being a colloquium for the "non-expert," there was something in it for everyone: the topologists and the geometers, even the analysts, and of course, the algebraists and algebraic geometers ..
(There is a difference, so I've been told.)
My only regret is that I can't remember the exact definitions, but K-theory has a similar flavor to homology and cohomology theories. In fact, it seems to me a little simpler to digest, even though somewhere one requires definitions from algebraic topology and straight algebra
(At some point one requires the notion of the direct limit topology, which never sat well with me from when I learned about rings and modules.)
However, the analyst in me felt vindicated at the mention of Banach and C* algebras. q:
At any rate, this colloquium was for me a success: I actually learned about some of the words in the title and abstract .. though I might have stacked the deck, this time around. After all, the title was: "What is K-Theory and What is it Good for?"
The rest of the day went reasonably well. In particular, my prelim reading has gone wonderfully over the last two days: the lack of interruptions and commitments have helped enormously. I wish I didn't have that many positive things to say from my advisor being away on business, but fortunately or unfortunately I'm making the most of the time that would otherwise be spent on research and attending classes.
I've browsed through all of Chapter 16 of Mattila, thought about the proofs of the theorems, and I must say: rectifiability via approximate tangent planes and Lipschitz images seems more sensible to me than tangent measures and Marstrand's Theorem.
Today I barely prepared for teaching, but in some sense I wanted it that way. Ours was a computational class today, and if I made it too polished then it would look too easy, which is no good when I'm trying to convince my kids to be careful and to practice more problems.
My Calc II kids have now learned the "Separation of Variables" technique for ODE, and after a hitch or two concerning constants of integration (one or two?) and where to apply the initial condition, I gave my derisive opinion on an abuse of notation.
You can probably predict which notational issue I mean. Consider the computation
dy/dx = -x / y →→ y dy = -x dx →→ ∫ y dy = - ∫ x dx
Now what the hell does 'dx' mean, without a notion of integration? Nothing: it means nothing. The symbol Δx would mean a small but finite length, but dx? Maybe infinitesmals exist in some non-standard logical system, but they are not for common use!
I can accept the fact that it's a shorthand for applying the Chain Rule, but that doesn't mean I have to like it. Maybe it's my inner analyst or my unconscious desperately demanding some sense of rigor and decorum in an otherwise nonrigorous and watered-down course.
At any rate, the kids seemed to take to this ODE technique. I suppose it's the best I can hope for.
6 comments:
Now what the hell does 'dx' mean, without a notion of integration?
Hmm... a differential form, maybe?
There is certainly some difference between algebra and algebraic geometry. There is even a difference between geometric algebra and algebraic geometry. Although it may not be as apparent as the difference between analytic geometry and geometric analysis.
Right! A differential form; I'll give you that one, but I refuse to believe that a textbook for Calc II should use the notation for differential forms when explaining separation of variables.
If anything, they should have said so. q:
In all honesty, it's hard enough to tell what is geometric analysis and what isn't.
My friend Kevin once asked his advisor what he studied, and he declined to answer. Many would say that he studies geometric analysis, but we'll never know his own opinion on the matter, I guess ..
Research Area: Geometric Analysis (from his webpage)
Huh. I guess we do get to know. q:
Actually, we don't. I looked up your advisor instead of Kevin's. Too much late-night grading.
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