I'm currently on holiday: this week is Spring Break @ U of M, which still strikes me strangely. Why call it Spring Break when it occurs during Winter Term?
I suppose it would be too disheartening to call it Winter Break, or perhaps too ambiguous. Or maybe it is honesty, because perhaps the administration knows that all the undergrads will travel to warmer climes, and have a proper spring there.
As usual I brought my work with me on the plane and here in California, which suits the occasion: my friend still works during the day, and I try in vain or pretend to work on my maths while he's away. Then on evenings, we set the work aside in favor of some diversion.
It works very well, until I ask him how his day went, and then he asks me mine. Sometimes I wonder whether my friends out in the "real world" think that I live some sort of "double life." One acquaintance whom I met in Manchester, NH, put it this way:
we'll never get to know that side of you, will we? that side of you which is pronounced for a large fraction of every working day of the week. you can understand me because what i do is easy to say, but we mightn't be able to reciprocate to you.
I didn't have a good response to this, but I suppose it's part of the territory of academics and whackos. q: There's a reason why we call it "theory," becuase what we study is not immediately natural to daily life. We suppose and ponder the possible or deem what is impossible. Asking someone to practice this rigorous and technical mode of imagination is a hard thing to ask.
As for the actual work, I've recently been browsing some of the literature on traces of Sobolev functions (cf. Adams's book). There's a fair bit of abstract machinery at work, which makes sense, I suppose.
For those who don't know, a trace operator on the class of Sobolev functions (on a bounded, regular domain) is some method of examining function data on the boundary of the domain of definition. This is inherently troublesome, because the boundary of a regular domain is (n-1)-dimensional and hence a null set w.r.t. Lebesgue n-measure. Since Sobolev functions are defined up to null sets, it's amazing enough that you can say anything on the boundary.
The price to pay is this abstract machinery. In Adams there are norms and Banach space constructions all over the place, and if I recognise it rightly, the context is given as some Bochner integration. It's not as abstract as other fields, say modern algebraic geometry, but I can't envision very easily the right pictures for Banace space theory.
Even trained as I've been, some modes of abstraction still don't sit well with me. Vector bundles are cool, but I never said I know how to use them. Banach spaces in functional analysis should not be taken lightly; in that very realm one encounters the Hahn-Banach theorem, which is a mystical hammer in its own right.
Maybe I've just been thinking too geometrically, as of late. It's easy for me to forget that thinking mathematically needn't always mean thinking geometrically; at the very least, my number theory friends have convinced me of that.
Perhaps later I will write about geometry and logic, and with these, address these recent questions of mathematical proof and verification (cf. talks by Devlin and Hales).
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