haunted by wonderful things.

i had meant to post yesterday, but today was a meeting with the advisor and i was hoping to prove a theorem or two for the occasion.


i think i've been convinced of the importance of currents on metric spaces. yesterday i heard a talk by urs lang that was very challenging to follow, but fascinating ..

.. and no, i don't think i can explain it very well. i can say that there was the spirit of hyperbolic geometry in it, but it was a very metric talk. it invoked the metric currents of ambrosio & kirchheim and passed to the world of gromov: metric hyperbolicity, asymptotic cones, even the metric version of the "euclidean" isoperimetric inequality due to gromov and to s. wenger.

it was the sort of talk that, afterwards, i was in a daze. this was terrific stuff, but comparing it to my paltry efforts at research, it left me slightly depressed. i couldn't concentrate for a while, and i left the office to forget about things and to have dinner with friends.

it was hard to work after that, and it was only after a good cup of coffee this morning that i felt like my old self again.

it's a silly thing to feel so overwhelmed, or rather, to let myself get overwhelmed again.

what can i say? sometimes it is a difficult thing to live with -- knowing what is possible -- because by seeing it, we realise our limitations, and our shortcomings that are not due to limitations.

these men have dreamed and dared, and so have discovered something which inspires others. i don't know if i'll ever be capable of the same, but the fact that it can be done ..

.. it makes my inner voice ask me, what's stopping you, to do the same, to dare and to give others a dream? and i can never answer.

did i get the picture?

i knew that someday it would matter, whether or not i went to kindergarten.

today i kept drawing the wrong picture and was confounded for hours as to why i could get one particular proof to work. happily, i can now explain why the fact is true in a sentence or two.

i think i should have spent more time drawing, as a child.
then, perhaps, this mightn't have happened.


on a completely unrelated note, i would say that my favorite mathematician is david hilbert.

suggested reading, for a techie.

i must have mentioned it before, but i really enjoy reading paul graham's essays.

maybe it's because he flatters mathematicians and other techies, or maybe it's because i see in his work what i would have liked to have done, but wasn't brave or deft enough.

for example, his latest essay is "Is It Worth Being Wise?" and weighs wisdom vs. intelligence in a way which i couldn't scoff at. he's willing to approach issues in philosophy, but in a clean way; his reasoning reminds me of a mathematician's.

then, maybe i like him because admittedly, he writes things that i would like to believe. this is different from flattery, but a related matter.

for example, here's an excerpt.

To me it was a relief just to realize it might be ok to be discontented. The idea that a successful person should be happy has thousands of years of momentum behind it. If I was any good, why didn't I have the easy confidence winners are supposed to have? But that, I now believe, is like a runner asking "If I'm such a good athlete, why do I feel so tired?" Good runners still get tired; they just get tired at higher speeds.

People whose work is to invent or discover things are in the same position as the runner. There's no way for them to do the best they can, because there's no limit to what they could do. The closest you can come is to compare yourself to other people. But the better you do, the less this matters. An undergrad who gets something published feels like a star. But for someone at the top of the field, what's the test of doing well? Runners can at least compare themselves to others doing exactly the same thing; if you win an Olympic gold medal, you can be fairly content, even if you think you could have run a bit faster. But what is a novelist to do?

and here is another:

The path to intelligence seems to be through working on hard problems. You develop intelligence as you might develop muscles, through exercise. But there can't be too much compulsion here. No amount of discipline can replace genuine curiosity. So cultivating intelligence seems to be a matter of identifying some bias in one's character—some tendency to be interested in certain types of things—and nurturing it. Instead of obliterating your idiosyncrasies in an effort to make yourself a neutral vessel for the truth, you select one and try to grow it from a seedling into a tree.

graham inspires. he inspires me, at least; few people can do that anymore, and at this point of my life, i need all the inspiration i can muster.

doubts.

at random times during last few months, i've been wondering one thing:

is mathematics really for me?
say i leave it all behind; what would i do, instead?

it seems important enough to say out loud .. or at least make a blog post. the problem with asking such questions is that people think the worst of them ..

.. or perhaps i only think they do.

i've said my piece about doubts and questions before, from a previous post:

questions do not erode good foundations; they only show us which parts are worn and rotten, so that we may build better foundations ..

.. and even if the foundation breaks, then how well-built could it have possibly been?

so the doubts linger. i might be more productive without them, but they having some purpose, i can't bring myself to dismiss them without resolving them utterly.

---

for the record, i don't know what i'd do if i left mathematics. the first thing which came to mind was to become a math and computer science teacher.

despite the stress and cynicism, it would be a good thing to act on the opinions that i've formed, over the years: that kids don't learn good mathematics in school anymore, and that the problem of innumeracy permeates to universities.

on the other hand, i don't think i would be a good teacher. i lack the patience and tolerance needed to be a mentor, and my nurturing skills are woefully shot. over the years i've become too cold-blooded and dispassionate, i think.

it just goes to show you that sometimes the first thought isn't always the wisest. \:

a quasi-addendum, on science, and on dreams (ambitious!)

i should add that i think i'm the sort of older student who unconsciously spooks the younger grad students.

"spook" probably isn't the right word.

is there a better word for saying that i cause others to be more somber after talking to them? 'sobering' doesn't sound right, and i wouldn't call myself a 'killjoy' .. though i've always liked that word.

killjoy: it means what it sounds, like 'guttersnipe' or 'ne'er-do-well.' (:

i think i should steer clear of recruitment weekend this year, and henceforth. it's better to leave that sort of thing to more light-hearted people.

then again, i could become a better liar;
that always comes in handy. (;

---

i think i'm going a little crazy.

every day, up to permutation the same question or two come to mind: how does the weaver construction work? when is it nondegenerate and does it depend on geometry, or measure theory, or is it some hidden factor, a quality or quantity that we haven't noticed?

someone once said that insanity is repeating the same action and expecting different results, but i forgot whom.

so i'd run through a few arguments .. some which are the advisor's ideas and some mine, and they'll progress through their lifespan. but they never fully explain the situation.

i feel a little like a scientist, sometimes. instead of experiments i run examples, which is a sort of experiment; some call them 'thought experiments' after all.

on an unrelated note, i used to think that the scientific method was inherently illogical. one forms a hypothesis H, reasons a conclusion C, and supposes H → C. through experiment one demonstrates C, and somehow concludes H: the fallacy of the consequent.

a friend of mine corrected me, quite diplomatically: i forgot about control experiments. the goal is actually ~H → ~C, and hence H .. by some version of modus tollens.

he also explained to me why the rutherford experiment was really cool, and at the time i was convinced that it was actually really cool. but now i have forgotten ..

.. which means that, later in life, i'll get to rediscover how really cool it actually is .. again. (:

as for mathematics, i'm convinced that i'm not very good at it .. or maybe just at geometric function theory.

i must have made it this far because of my good looks and my charming, winsome personality! how unethical!

well, we all have egos. (:

it's not the lack of cleverness or skill which bothers me, but quite surprisingly, imagination. too many questions come to mind, and i can't answer them all .. or many, or even a few. i suspect too many things and make much of trifles, and i consider too many crackpot ideas.

as descartes writes in his discourse on (the) method ..:

After all, it is possible I may be mistaken; and it is but a little copper and glass, perhaps, that I take for gold and diamonds. I know how very liable we are to delusion in what relates to ourselves, and also how much the judgments of our friends are to be suspected when given in our favor.

it can be a terrible thing to dream, to know what could be and to elude what is. \:

travel: my advice to the young.

lately i found myself giving advice to younger grad students about graduate school. sometimes they ask, other times it arises in conversation, and often it is about the summertime.

in any case, i say much of the same thing:

1. make sure to take a full month off from math, or any serious work whatsoever, especially if you'll be teaching next fall for the first time. it is important to be well-rested, if only for prepare for when we cannot be .. say, during the fall and winter terms.
   
   
2. if you're planning to take a qualifying exam at start of fall, unless you really don't know the subject, don't spend the entire summer studying for it. you'll get the jitters and start second-guessing habits which would have served you perfectly well if you had simply trusted them.
   
   
3. travel. attend a conference or a summer school.
   

i'll explain further the third suggestion, below. it's full of obvious observations, but i just felt like writing them all down.

besides, it's my blog. (:

---

years ago the advisor told me (and the rest of an audience of 50+ people) that young people should attend conferences [1] which is good advice.

i think it a good thing when students first observe current mathematical research in an unfiltered way, if only to see what impressions they form.

correct or appropriate or otherwise, these impressions are honest and are worth something, much like the fleeting fancies and imagination of childhood.

as every adult knows, childhood is a naive time and a foolish time, but it is not without value. it is the most honest time in our lives and where our potential, though not yet realised, is greatest. when we are children, we are at our most possible; by not having the burdens of experience, we can do anything. [2]

but despite such latency, such potential, as children we make silly choices. even if we can choose whom we want to be, we may not choose well, and this is where the analogy breaks. a new graduate student is not a child, and has the capacity to make the most of his/her nascence.

a graduate career is predictable in its steps.

graduate students will choose advisors, and over a span of years they will learn more than the methodology of mathematical research. they will adopt from their advisors viewpoints and philosophies about mathematics and academia in general, even mannerisms and rivalries.

but a graduate student may never develop a healthy skepticism in the sociology of mathematics. (s)he may never learn to question what (s)he has been taught holistically by an advisor and a department; the default is accepting that reality is what you've experienced, and that your experience is true and special.

mathematics may lie in the realm of platonic Forms and embrace canonical choices, but the sociology of the world is not canonical. it does not fit so conveniently into a single worldview, even if that is the worldview of an assorted collection of mathematicians in a single locale.

this is why it is useful to travel. what graduate students learn from their department may be gospel and serve them well for a lifetime of inquiry, but it does no harm to permit a reasonable skepticism.

questions do not erode good foundations; they only show us which parts are worn and rotten, so that we may build better foundations ..

.. and even if the foundation breaks, then how well-built could it have possibly been?

---

writing all of this, i didn't say precisely how traveling and 'worldliness' are related. in all honesty, i can't say that i'm fully sure of the connection.

one outcome of travel, especially attending conferences, is that we can quickly encounter ideas and techniques that we have never seen before. this is an invaluable resource, but as for what ..

we will be confused by something .. not that it would be a new feeling for any graduate student, of course, but the source of the confusion arises as something novel from the daily grind. so travel can be a source for novelty and further possibility.

human minds dislike confusion, because their thoughts will run less smoothly in spite of it. an academic mind will seek to understand and resolve the confusion, because of firm convictions in the rationality and consistency of human thought.

when we seek to resolve confusion, we then begin to ask questions out of individual curiosity. so in travelling, we learn to ask our own questions.

it is important to ask questions. some people, despite attending school and college, never learn this.

many graduate students solve research problems given to them by their advisors, and that is fine. graduate school doesn't last forever, however, and students will become independent researchers.

there will come a time when our inquiry is our own responsibility, and it will matter whether we can ask good questions: questions that are interesting and nontrivial, yet questions that we have some chance of answering.

---

having made a short story long, let me be brief:

graduate school is not just a road to a ph.d. thesis; it is also becoming an independent researcher. an advisor is invaluable in helping you along the road of research, but you will not always have an advisor.

so learn to be independent: form your own ideas and ask your own meaningful questions. traveling is one means of developing such skills out of circumstance, with the convenient bonus of novelty and possibility.

[re-reads]

wow. i can go on and on, can't i?
at this rate, i'm probably become an ideologue.

[1] then again, he said this before i became his student and before i started graduate school, and i never figured out exactly how old or young is "young." but i agree with the idea.

[2] well, not exactly: there are always constraints upon us, but if you never learn there are constraints, then are they really there? i'll leave that question to the philosophers.

in which basic trigonometry helps (link)

every so often i stop by overheard in new york for a laugh. the one below is surprisingly good, if only because it involves math and the beatles.

Jerk in back row: Paul McCartney should have stopped after the Beatles. I mean, what the f*ck else good did he do after that? Nothing. Not a goddamn thing. He couldn't go from point A to point B. What's the shortest distance from A to B, again? Like, the hypotenuse of a triangle? He never found the hypotenuse without Lennon.

Annoyed man in front of him: Dude, the hypotenuse is the longest side. Now shut the f*ck up.

Annoyed man's girlfriend: That was so hot.

--Carnegie Hall

[link]

a book(?) on Banach spaces (w/ link).

there's a link from a recent post off the weblog "ars mathematica" to a 101-page manuscript by w. johnson and j. lindenstrauss.

title: "basic concepts in the geometry of banach spaces" [DVI file]

i've only thumbed through it and cannot say much .. except that the paragraphs are often long. if anyone has or intends to read it, i'm curious as what you think.

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: