books change (when you re-read them).

this past thursday i told a colleague that i'm taking the rest of the week off from mathematics. strictly speaking, i lied.

what i meant was: i'm not going to worry about turning the thesis into a paper, or meeting my current research commitments until next week. it was just easier to say that i was taking time off.

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so these last few days i've done a little thinking. mostly, i've been idle and i've been reading recreationally. among other things,

snow crash by neal stephenson,
dark continent: europe's twentieth century by mark mazower,
invisible cities by italo calvino.

.. all good stuff.

calvino reads like jorge luis borges,
or perhaps borges reads like calvino,
or perhaps they have a literary ancestor, like whom they read.

no matter, that;
this is supposedly a blog about mathematics, right?

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recently i've been thinking about BV functions [1] and the calculus of variations. a long time ago i borrowed from the library two books by e. giu$ti.

back then i had just begun graduate school and wanted to learn all sorts of things. i remember reading (rather: browsing) those books, and many others, with wonder and curiosity.

i had no motives or motivations;
i just wanted to learn.
i was a student, then.

these days, i am more like a researcher,
a luckless mercenary of ideas.

some days ago i borrowed the giu$ti books again, browsed them again. so far i can't seem to adapt those techniques to the problem at hand; it's just a different setting.

the ideas remain fascinating, and i noted something that i never noted before. there are morals in these books -- at least, early on, in the preface or first chapter of each -- so that the detailed, technical follow-up makes some sort of sense.

if you haven't seen the calculus of variations in book or talk form, then let me tell you: the estimates can get very gory. almost always, i dwell on one thing for too long, and then i am lost with what comes next.

perhaps i should mention one passage from direct method$ in the c@lculu$ of v@riati0n$: [pp. 4]

Actually, in order to apply the Weierstrass theorem, it is necessary that the functional F be lower semicontinuous, and that the set V in which one looks for the minimum be compact. These two properties are in some sense in competition; in order to have the semicontinuity it is preferable to endow V with a relatively strong topology; the fewer convergent sequences exist, the easier the functional is semicontinuous. On the contrary, for the compactness it is better to have the opposite: the weaker the topology, the easier for a sequence to converge.

maybe upon a little thought, this tension becomes clear. often i make much of things, but this is like a moral.

however, if you are new to these sorts of things -- functional analysis, theory of distributions -- you mightn't have considered the value of choosing just the right topology and narrow your focus to the right kind of function.

maybe you wouldn't have worried about topology at all. you'd have left it for the knot theorists and to those who play with homology theories. some functional analysis courses are taught in a "standard way" and one forgets that there is a lot of topology in the background.

at any rate, i like these books.

giu$ti also gives some attention to history and progress. this is valuable for someone like me, who moonlights in this area and doesn't know what is already known and in what direction the research area has gone or will go.

at any rate, there is still maths to do. by tomorrow i should think about thesis-derived matters again. the problem awaits!

[1] rather, functions of bounded variation. in some sense, these are the functions that you may have encountered in a first course on measure theory. see the wiki.