Thursday, March 30, 2006

another teaching stint.

in two weeks i've subbed for two of my g.s.i. friends and i think i've become more boring than i used to be. it's either that or more likely, students really don't find single-variable differential calculus very interesting, at all.

i can't blame them. a basic calculus course is boring, in the same way that most courses in basic methods are boring. there are better words to say this, so here are some of paul graham's:

"I'm not saying we should let little kids do whatever they want. They may have to be made to work on certain things. But if we make kids work on dull stuff, it might be wise to tell them that tediousness is not the defining quality of work, and indeed that the reason they have to work on dull stuff now is so they can work on more interesting stuff later. [1]"

it's assuming, of course, that there is a "later."

i can't quite perceive the reason or cause, but somehow the calculus of Newton and Leibniz seems to take a hallowed role in being "educated," if only to use the right keyword.

for quite a few disciplines, i believe that calculus is quite useless. in a greater sense, mathematics as it exists is often little in worth, and so is the nature of philosophy. As a curiosity, it disciplines the mind without actually being inherently useful for anything.

so it stands. for many calculus is another hoop to jump, through the obstacle course that is college. add the psychological effects of math anxiety and the societal norm of innumeracy, and it's a wonder i can teach calculus with any conviction at all.

[1] short for 'graduate student instructor.' in the mathematics department, almost all the calculus classes (save the honors courses) are taught by g.s.i.'s in exchange for funding and tuition privileges.

earlier this morning; fatigue and balance.

i woke up at 7:30 today and i couldn't fall back asleep. half-conscious, i sat on my bed and decided there were too many things to do in a regular day's time.
so i got up, made myself semi-presentable to the world, and foraged for caffeine. i think the morning manager at the local caribou coffee thinks i'm a nuisance,

a regular who thrifts and spends no more than a few dollars for a muffin and a coffee,

who lingers about for the free coffee refills,

who talks very little to anyone and stares through the windows at nothing in particular,

who scrawls strange symbols and diagrams all over paper, and somehow in obscenely neat lines.

as it happens, i'm not the only mathematical regular there but i think they don't like me very much. i'm hardly the model customer.

on weekends i see there afsanah, a grad student in algebraic geometry, who i think is writing up her thesis.

two or three students have also made that same coffeehouse their watering hole, and i think they are undergrads but not from um. they seem more .. down to earth and unconcerned in their ways. they remind me of eager kids who'd i find in the pittsburgh mathematics library in thackeray hall, just doing their thing and working out a theory or two.

every time i see them they're discussing point-set stuff from the munkres text and seem to take it very seriously .. seriously enough so that i can hear their arguments about limit points and closures from a few tables down.

they're enthusiastic: much more so than i ever remember feeling. i leave them alone and let them have their topology; i have plenty of work and they are happy by themselves.

maybe i'm getting tired and cynical. i was about to say old, too, but that's not quite right; i know plenty of people past me in age but they show none of it. so i'll reserve the word old for when i look like it.



i am tired, though. it's been a long week and work urged me, more so than usual. near noontime i felt like i've reached a peak of my mental powers [1], that i was acutely aware of the nature of the problem, its constraints, and its freedoms.

my thoughts flowed, and today's research meeting with the advisor was wonderfully productive. it's silly to "keep score," but after last week's debacle [2] it was nice to be able to say something of some accuracy and substance ..

.. even though i needed an overly strong, simplifying hypothesis. it was all i could do, being a tyro in univalent harmonic theory and knowing only basic techniques. but there is a little promise left: a little example i computed without much thought is now offering motivation for a reasonable compromise of a corollary [3].

what can i say? my advisor's optimism is infectious.

that went all well and good, but the issue with "peaks" is that they don't arise naturally for me. as in the case of social gatherings and when taking on roles of leadership, it's a matter of harnessing and directing enough energy and willpower toward some effective end.

but that energy must come from somewhere, and as in the example from descartes, where there is a peak ..

.. there is always a valley, and maybe a rut below that. anyways, it's been a long day, and i've burnt too bright for that many hours at at time.

until the next post.


[1] this is not to say that i've reached the epitome of my career, but that i hit a local maximum. there will be more, and also there will be local minima as well. last week was such a latter case, for example.

[2] i meant to post about it, but didn't allow myself the time. the short of it: i gave a false (i.e. non-) proof of a lemma, and the fault rested on a basic inequality - which i thought was true by concavity - but is woefully untrue in general.

[3] as a general rule, i seem incapable of proving theorems, but every so often, i manage a lemma and at best, i prove corollaries to the great theorems of others. it is one of my axioms, along with "i do not dance" and "i am morally opposed to my own birthday."

Wednesday, March 22, 2006

a mathematical battle of gettysburg.

the analysis gods have smiled upon me, once again.

the theorem's not yet proven, but that's not important. what is important is that i have a good idea and after thinking about it carefully, i think it will work. i think i can prove the theorem by tomorrow morning.

i'm starting to believe that the analysis gods accepted sacrifices in the form of discarded pages of scrawls. i think i've used about 10-15 blank pages over the course of two or three days -- front and back -- and i've thrown out two pens ..

.. well, maybe the pens aren't so relevant. i could also credit hard work, but as a friend of mine predicted, the solution is reasonably straightforward .. as always. in fact, an educated guess would have led straight to the methods of proof.

so let me not credit any hard work; instead i should credit and discredit my stubbornness of habit.



first, i should swallow my pride.

in a previous post i dismissed the use of computation to arrive at results about harmonic functions, in favor of more elegant and geometric means to tackle the problem. so i thought of many ideas and not one of them worked, and quickly i wondered if there was enough data in the problem to warrant any reasonable answer.

in the end, the optimistic outlook gave me the current idea. i stopped worrying about formality and rigor, took limits and guessed where they should go. if, for some reason, the quantities converged to what i hoped they would, then i'd at least obtain a foothold and could climb through a proof.

to understand the convergence, i had to compute .. and compute .. and compute. now i think i have some notion of why it all works, without appealing to computations, but the notion would not have appeared had i not computed. it reminds me of something (similar) that a certain prof @ syracuse told me (and the rest of the audience at his talk):

one must be very careful and very patient with computations, but if you are careful then they will teach you morals.

i suppose that prof was right .. at least about my last few days.

anyways, the proof is incomplete. there's work to be done, and teaching preparations to meet. tomorrow shall be a busy, busy day.

Tuesday, March 21, 2006

work frustrations.

it's been a frustrating day .. at least on the level of research.

none of my ideas are working and i can't seem to think of any new ones. the old, unusable ones recur and recur; each time it happens, i wonder if i've overlooked an important detail or insight that could solve the problem at hand.

each time, nothing useful comes out. i only realise that the idea has some flaw, that there is a little gap in my logic, or the number of lemmas and claims i need grow like heads on a hydra: i try to prove one, and two subcases arise from its place.

it even got to the point when i was backtracking through my previous work tonight, i thought a lemma my advisor and i proved was wrong, and it took me half- to a full hour to work out why it was true again. on the bright side, the proof is now about a dozen lines long, and barely a half-page.

i'm almost convinced that i have the wrong perspective, or that my tools are too primitive. for instance, i don't think i can "compute my way out of the problem."

it's also occurred to me that i haven't thought of many examples, and come to think of it, i don't have a very good picture of what is happening. the theorem we're after still seems quite plausible, but it is like a slippery eel: there's no place for a good grip.

it's late. i should quit for the day, wait until tomorrow, and try again. i should sit in my apartment and find some purpose for my non-work hours, which is frustratingly hard on its own.

Monday, March 20, 2006

disparate bits: weekend's end.

it's been a rough week.
  1. i gave two talks during the work-week: a very short talk on tuesday and a very long talk on thursday, so they average out and sum to two.

  2. over friday and saturday i spoke with several prospective students (didn't get to meet them all, this year). to my surprise, some of them are potential analysts [1] and we had a few pleasant chats. i think the odds are good that i'll see them again in september.

now the weekend ebbs to nothing.

research ebbs and i have no results, though an idea came to mind when i was running (read: gasping for breath between sprints) and thus far it hasn't failed .. yet. something tells me that i don't understand harmonic mappings well enough, and there is a deeper, more elegant way to solve the problem. in the meanwhile, i'm doing more explicit computations that i'd care to admit.

it wouldn't be necessarily wrong to say that all my results are corollaries to the notable theorems of better minds, and of these corollaries, they depend distastefully on computations. i suppose i should thank my former math teachers for instilling in me a careful and trustworthy ability in computing and simple problem-solving, but brute force only eeks out a little moral and nothing substantial.

i think i've been out of the preprint loop, as well. apparently there are plenty of interesting results from the finns: here are few notables from the preprint server @ jyväskylä.
  • "Sobolev extensions and restrictions." Piotr Hajlasz, Pekka Koskela and Heli Tuominen - [pdf]

  • "Regularity of the inverse of a Sobolev Homeomorphism in Space." $tanis1av Henc1, Pekk@ K0ske1a, and J@n Ma1y - [pdf]

  • "Continuity of the Maximal Operator in Sobolev Spaces." H@nnes Luir0 - [pdf]

  • "Regularity of the Inverse of a Planar Sobolev Homeomorphism."
    $tanis1av Henc1 and Pekk@ K0ske1a - [pdf]

[1] that is, potential students in analysis, and possibly @ um; i don't mean to say that they study potential theory .. though, in time, they might.

Wednesday, March 08, 2006

work frustrations, and a foot in the sand.

despite careful reading and the topic was part of my prelim, the trace operator from Sobolev functions on a (Lipschitz) domain Ω to Lp-functions on the boundary of Ω still seems a mysterious notion to me. perhaps i simply don't understand the use of capacity well enough ..

.. so much work ahead, tonight and tomorrow morn, and so little yet accomplished. it's frustrating.



speaking of frustrating, on the maths community @ livejournal i've read a few recent posts of users (and probably students) who speak of mathematics with good cheer and unbridled optimism, that they "can't imagine anything better than learning something new every day."

that's fine and dandy, but some of us have theorems to prove and theses to write.

from recent experience, it is very hard to accomplish something which is mathematically worthwhile, every day, and though it is dangerous to compare, i dare say that the difficulty can be compared to making good art, composing a charming song, or writing fine literature. let me set a foot on the sand without tracing a line: mathematics is a branch of philosophy.

it may be a science to the rest of you and that's fine; i'll not refute that. but to me, it is first and foremost a kind of philosophy, and possibly the best kind. in mathematics, up to the fallibility of human mind [1], there is conditional certainty, and that is a kind of progress, isn't it?

[1] at some point i should discuss something about the recent-turning-old events of formal proof, brought up by Devlin and by Hales. it will involve this viewpoint of philosophy.

today and yesterday.

(written today) it's one of those weeks where after meeting with the advisor, i'm either (1) not quite sure what i'm supposed to be doing, since we discussed so much last time ..

(it was break, so there was no anss and we spoke for about two hours, despite my not being very prepared to discuss anything.)

or (2) not quite sure if everything had worked out from last time. in one argument (which settles the 2-dim'l case) we asserted two lemmas. both are the sorts of facts about harmonic functions which are either reasonably easy or in a book somewhere.

but as of today i can't seem to work out the details, though i believe i should be capable of them. as a result, they are bugging the hell out of me. if all else fails, then i could ask the advisor during this week's meeting, but that just seems .. wrong.

call it stubbornness, i guess. if i can't work out details at this stage in my career, how am i going to write a thesis?

(written 5 march, two days ago) this joint work with my co-author may actually reach an end, at some point. if you're wondering what caused all the hope and optimism, let's just say that it was an .. arduous but good mathematics day.
one result concerning geometric measure theory bothered me all yesterday evening but happens, sensibly enough, to be a theorem in federer's geometric measure theory tome. the sole reason why i have no qualms citing the proof in a future paper is only because i read and managed to understand the proof .. well, sort of.

as with much of the maths i see, i understand how the machinery works, that is, the gory details if they are not too many. it's the intuition that is troublesome.

if i think of it like compiling a program, then i can parse low-level assembly language and mid-level language formats, but the high-level stuff - where the algorithm or strategy is shaped - remains fuzzy and ethereal. i know that the program will run and that the output will be good, but i can only guess how the programmer chose that given implementation.

the other result which bothered me isn't so bad after all. i can write a proof on a third of a page, but the trouble is that it requires five or six lemmas and basic facts, as well as more (standard) definitions than one can count conveniently.

still, though .. a third of a page. that's not bad, and all in all it wasn't a bad day. in the late evening i toiled and struggled to no avail against a nonlinear ode, but i couldn't make sense of it. but that is not so bad a defeat.

so lemma 15 of 63 is done (i made the #'s up, for those concerned) and perhaps the final (to-be-submitted) draft will be ready by summertime. a boy can dream, right?

a year ago i'd have kept reservations about how much we "did" for this paper. when extending the results of others, it can be hard to tell if there is enough original thought. but at this point i'll settle for the fact that we're bringing in an additional perspective and a toolbox of techniques to the table. it won't be a a mere reiteration of the variational principle, but as i've hinted, some geometric measure theory will play a key role.

what we'll do is new and worthwhile; i'm convinced of it now, so it's a matter of labor and determination .. but then again, when has it not been so? it is also a matter of tim, and the thesis comes first, after all.