the problem at hand?

i had meant to write something more substantial on thursday, rather than simply to post photos.

i mean: i still like the photos and all, but i still can't quite think of something to write. in fact, there is nothing to write.

that's the trouble, really:
nothing's quite done yet.

---

i don't know how k. has done it, or how m. is doing it; by this, i mean make progress on a thesis problem.

since i've met him four years ago, k. has radiated the impression that the problem is being solved. there has been ebb and flow, maybe a little drought, but he has solved it and this april he will defend his work. if i believed in such words, it almost seemed his destiny; certainly none of us doubted him.

i haven't asked m. very much about her work, but she seems ever optimistic about it and the advisor says good things about it: good signs, both. in her i see a destiny of success, too, but i can't explain it; i only see it.

friends of mine .. friends i trust, they tell me that the problem isn't really the thing. a problem is just a vessel, a direction in which to collect thoughts from trained minds. a thesis is not always a problem solved, but always theorems proven.

(well, they didn't say it exactly in that way .. but it's close enough and you get the point, i trust.)

so i've listened to them and trusted them. they know, you see. they know from the same experience: having tried without success and leaving problems unsolved, they move on ..

.. to collect victories, where they may.

---

that's all i seem to do, now: wait for the rare opening, and try for a small victory.

when i attack a problem, it's not grandiose warfare, not flags or trumpets or brandishing swords or fine formations. there is no wisdom in plunging in, no certain victory in chances.

instead, it's guerrilla tactics: hit-and-run maneuvers and cautious reconnaissance against something that we don't yet know ..

.. except it is immense and occasionally predictable, but we don't know why that is, either.

i don't have a thesis problem anymore, exactly.

there is a conjecture or two .. none mine, either, but that's not my business. every week or so i have a half-dozen guesses, some borrowed from the past week and occasionally a few new ones.

of these guesses, some i throw out because i haven't the means to verify them. for others i do have the means to realise, soon after, that my guesses were wrong.

amidst all this, i read a little of this and that and keep watch with a scavenger's eyes. maybe i'll stumble upon something useful, something for leverage .. something for a guess or an idea that just might work.

i'm staying productive, at least. i'm working.

when we have few choices or none, we hope;
so, i hope.

works and days: photos!

these days i often work at home and it is a fine thing, because i'm then free to be a weirdo and work as i please.

i'd be free to pace around, maybe mutter to myself, which i cannot do in east hall without acquiring a few odd looks and stares.

best of all, i can bask in the sunlight streaming through my window .. and work like a free man!

but two days ago i was at a loss for a chalkboard, so i fished out my old whiteboards and wrote on them as if they were tabletops. but then i was faced with the problem of having to copy what i wrote on the board .. and then it occurred to me that i could just as easily take a photo.

---

so here are some photos .. not precisely of the work i did, but of the environs. i get the impression sometimes that my non-math friends think that math all looks the same .. formulae and such .. but in my own case, it's often a lot of writing.

it is a little pride, but i like my handwriting ..
.. though some would say that it seems a little effeminate.

as for where i jot things down,

this is one of my notebooks. it used to have full-sized 8.5" x 11" pages, but one day it rained and the bottom pages were soaked, so it gutted it down to size.

this is another notebook of mine. to explain the cover, sometimes i get a little obsessive with neatness, and this notebook is meant to be used to work the gritty things out, and not for pretty exposition!

then there are legal pads, which is de rigueur for any mathematician. you can tell much of the personality of a mathematician from how (s)he treats a legal pad. q:

whenever possible, i like to take notes on loose sheets of paper and collect them in a binder. in terms of actual physical matter, it's the closest version of digital 'cut & paste' that i know of.

it also makes scanning notes far easier, which is something i've done of late for absent friends and colleagues.



lastly, this is a LaTeX'ed version of my notes. i write such drafts when i feel like i need to say something once and for all.

it does look rather nice, doesn't it? q:

lessons for an academic.

it's been a few weeks since i last met with the advisor. i think there has been some benefit, even a little lesson:

if anything, i've had to learn to depend on myself when it comes to research. it hasn't been easy to hold myself accountable for my ideas, and filter out the good ones from the bad by myself.

i've had to tell myself not to pursue certain ideas, but only to write them down (for later) and focus on the goal at hand.

i've had to realise that it's not worth focusing on certain goals, because there is evidence (through examples) that the methods of proof do not work in general.

i've also realised how many mistakes i could possibly make, and how quickly a day can be wasted.

i don't know if this bodes well for me. every so often i remind myself that one day i won't be a student anymore, and i'll be expected to conduct my own independent research and to have good, interesting ideas.

i also remind myself that i have to write a thesis before i'm no longer a student.


it's been about six months and i've no results yet, i'm afraid:
there are only dead ends, and i know why they're dead.

---

on a lighter note, i've made it to 200+ posts on this website.

pondering academia.

(another thing i meant to write, some days ago.)

---

at some point i wanted to write about how mathematics compares and contrasts to science and the rest of academia.

needless to say, this 'magnum opus' was a flawed idea: too ambitious, if it was written the way i wanted it. all i wanted, i suppose, was to explain how i do my mathematics

(or rather, how i regularly fail at it)

.. and perhaps make it easier to understand or to sympathize with my working life, and the lives of my peers and colleagues.

like i said: too ambitious.

i think i'm resigned to be eternally misunderstood, but fair's fair; i don't often understand why other academics do what they do, and why it drives them.

---

take scientists: they try to explain the phenomena of the world, which is fine. but it seems like messy work to me. i don't believe the world to be a simple place, and the complexity looks imposing enough to drive someone mad.

physics looks complicated, because now the mathematics is supposed to have physical meaning. it works at all scales and particles are fearsome, because macrocosmic intuition seems to be of no help whatsoever.

chemistry looks complicated, because it condenses to particles in physics, but it varies in scale and the magnitude of particles is boggling. this is the field which came up with avogadro's number, of all things!

biology looks complicated, because it condenses to chemistry and inherits all its complexities. moreover, the notion of feedback and nonlinear iteration is much more pronounced; the crux of things might be dna, rna, and protein synthesis, but to understand how it all works (or could work) is level with insanity.

ecology is complicated enough, where the scope varies from microscopic cells to ecosystems at a continental scale. social sciences look complicated, because human beings are erratic and hard to predict; an animal's behavior is hard enough to study in how it satisfies its basic needs, but human behavior ..

.. half the time, i don't know why i do what i do.
how is a scientist supposed to know?

---

then there are the humanities. i view them as specialised fields of philosophy, motivated by the complexity of human culture.

they mystify me; i cannot grasp their spirit or center. upon meeting an academic in the humanities and being told what (s)he studies, it will make sense.

but as an aggregate of areas of study, i can only point and shrug. there is merit in it, and often beauty and aesthetic, but i cannot say why.

---

.. and then, there is mathematics.

aftershock.

(before i write anything else, i meant to write this a few days ago.)

---

i finally got around to reading a paper of l. ambrosio and b. kirchheim, titled "currents in metric spaces." that is, i read section 1: the introduction.

it's impressive.

after reading the plan of the paper, i became moody and depressed, and i couldn't work on my own research anymore. it's true that mathematics is not a competition, but this seems such great stuff that anything i do seems to pale in comparison.

the work's not going so well, anyway.

i find myself reacting that way regularly: a seminar talk might end, and i'd wander out of the lecture room, in a bit of a shock and incapable of talking to anyone. it's probably why i feel so withdrawn at large conferences.

where the tracks end .. for a while.

EDIT (AS OF 18 JAN '07, 22:39): call me a glutton for disappointment. i never did get to reading or writing. i thought i'd try the theory of currents again, reached a point where i guessed i would reach, and made no real progress thereafter.

maybe i'll learn my lesson by tomorrow, and then turn to reading.

yesterday and today i resorted to methods in standard measure theory, hoping for some 'silver bullet' to the task at hand.

well, i didn't find any.

in retrospect, i don't think i truly expected to find one, but at the time i couldn't think of anything else that could work.

---

the more i see of mathematics, the more i believe this:

advances in mathematical research are not so much brilliant inventions of the mind, as they are taking the right perspective on the problem.

so it's not so much a grand adventure blazed by the intrepid explorer, as it is a road trip and you finally realised that the map is upside down.

at any rate, it would explain why so many ideas seem 'obvious.'

this is not to say that mathematics is easy .. but then again, we do have a reasonable share of the "making people feel stupid" monopoly.

this is not to say that i've discovered something interesting or useful through today's research, either.

instead, i think i realised that i was obsessing, and for the last few days, i've been an @ss and a right jerk.

---

i haven't been eating well and i've been sleeping fitfully and little.
i haven't held many conversations of any depth with anyone, recently.

i've been pondering different aspects of the same idea for weeks, and so doing, digging a rut.

before that, i was seething while home for the holidays, tempted to work but too distracted by familial roles to concentrate on very much at all.

during those days, there was one idea i had and i really hoped it would work ..

.. but it doesn't.

these past few days i've avoided people and crowds. when amongst others, i've been brusque and treated people coolly; fellow math grads have hello'ed me in the hallway and i'd reciprocate but never break stride.

i just feel angry, at random times and places. maybe i'm tired.
i also feel tired.

---

i think i will do something different tomorrow: maybe not research, but some non-essential reading.

maybe now is a good time to start on the ambrosio-kirchheim paper "currents on metric spaces" ..

.. or parse through the cheeger-kleiner paper about non-embeddability and differentiability theorems ..

.. or sift through the literature of optimal transportation and related topics, say the lott-villani papers ..

or maybe i will start writing. it's been some months since cincinnati and my talk about the schoenflies problem.


the key is to do something that doesn't really matter. if it doesn't matter, and it doesn't go well, then there's no reason to be angry.

bad day.

the advisor is away, again.
life goes on, and so does the thesis work.

in the last few days i've had occasion to look through parts of that tome, h. federer's geometric measure theory. the idea is to recast the problem at hand in the language of currents, and by determining what type of current it is ..

(say .. rectifiable or normal, or a flat/polyhedral chain)

.. we may gain some leverage in identifying necessary conditions for a space arising from n. weaver's construction.

well, that was the idea.

i can't seem to find that leverage .. or any leverage, for that matter. the theory of currents is tricky in its geometry and my formulation doesn't fit very clearly or well, in it.

two days ago, i pondered flat m-chains and how to detect them .. specifically, whether a particular current (induced by the weaver construction) is a flat chain. i think i may know my answer, bit i'm unsure if it helps to know.

if it is correct, then there's little need to bother with the theory of currents; the burden of the problem will remain in standard measure theory.

perhaps i just have to improve my formulation of the problem. i don't know .. and i hate not knowing.

---

anyways, today was a paranoid day. i was mired in examples, non-examples, and pathologies, and it made for a foul mood.

strangely enough, my spirits improved a little, after i went on a short run through the UM arboretum .. in 20-degree weather, of all things ..

.. but only a little. having spent little time at the office this weekend, even spending the evening hours there took some adjustment of habit. i think i was surly and dismissive in my encounters around the department, today.

east hall is boisterous and chaotic. i never seem to think very well when there, with undergrads underfoot and fellow math grads excitedly interruptive and distractingly diligent.

tomorrow is another day, perhaps a better day. i'm not one for optimism, but there seem few viable alternatives. should i expect worse tomorrow and dull my efforts, amidst great sighs of "alas?"

of course not. i have a problem to solve .. or at least, to kill.

when one thesis problem dies, it's hard not to think that the next one could die, too.

a nomadic repetoire.

EDIT (AS OF 13 JAN '07): re-reading this, the list of topics sounds overly fancy and i sound arrogant. this was not my intention.

---

it's strange. over the last few years i could swear that i've wandered through the hinterlands of "the quasi-world."

when i first began my thesis work, i studied Lipschitz manifolds and while doing so, for background i learned

• some differential topology, and even skimmed through that famous paper of j. milnor .. i didn't understand much of it, however, and had to read hirsch's book to understand something;
  
  
• a little harmonic function theory .. i read a little from one of the books of e.m. stein, and learned a little about fractional sobolev spaces and besov spaces;
  
  
• and a modicum about groups of hyperbolic isometries ..
  
  .. say, enough to leave a first thesis problem for dead. \:
  

and now i'm studying n. weaver's construction of measurable 1-forms and exterior differentiation. for that, i've learned about

• j. cheeger's construction of a co-tangent bundle, on doubling p-Poincaré spaces .. but only a little; i've been too lazy to read through the whole thing.
  
  
• Banach bundles and glimpsed .. nay, peeked into the world of C* algebras, which is a puzzling, cavalier mix of algebra and analysis.
  
  
• and now, i may leaf through federer's geometric measure theory, after all, for the theory of currents may be of some use to our work ..
  

i suppose that all these topics came from necessity, and i feel more "mathematically urbane" for it

but only a little; it's a little like travelling abroad for short spells. one witnesses much, acquires broader horizons, but ultimately, realises how little one knows.

.. and as for the ultimate litmus test ..

has my research come to fruition, yet?
have i written a thesis, yet?
have i proven any theorems of any relevance or worth?

the first and last questions depend on opinion, of course, but the middle one is an unambiguous no.

there is still much work to do and little time left .. and perhaps i should stop being lazy and reminiscient, stop blogging for now, and get back to work!

one good thing about having a semester off, from teaching ..

.. is being able to work on mathematics, right now.

(see the timestamp on this post, for why)

mornings used to be productive, because they preceded the rest of the day (i.e. drudgery) and i would focus my most crucial energies towards creative ends.

now time is more flexible .. but i wonder when the novelty will wear off.

not really math, but ..

i just learned that the firefox google search bar can be used as a calculator, and can also compute currency exchanges.

i'm serious. i've just entered in log(13), which gave a correct answer, but the USD conversion for 35 euro was about 50 cents off.