foreboding: the enormity of it all.

i remember one scene in the film "wonder boys," where katie holmes reads michael douglas' novel: this voluminous tome. she remarks that a good writer has to make choices.

i don't know if i'll give the talk that i want, tomorrow.
i haven't made any choices.

there are so many interesting detours to talk about, and i can't prepare it all. between thesis writing and trying to work out this one last thing, my mind feels scattered.

i've barely prepared anything;
tomorrow's going to be a bumpy ride.

---

on a lighter note, i heard from another school this afternoon. i'm out of the running for a tenure track job. oh well.

i don't think i'm ready for a tenure track job, anyway.

small victory: slow but steady.

so far, no luck in patching the argument into a proof (see last post).

on the other hand, i wrote another page in my thesis.
i guess that's something, at least.

by the way: does anyone know how to add a horizontal dash to an integral sign? i might have occasion to use many average values of integrable functions.

curses! wrong, again.

about what i wrote in the last post,

today i thought i had proved something great.

well, i found a gap in the argument: last night. it bothered me so much that i couldn't sleep well at all, and i worked on it all of today ..

"all of today" meaning from noon to nine, with some omitted few hours in between. i ended up oversleeping and missing basketball, because an interrupted cup of coffee is more important than sport.

noon to five pm was one long worksession; i thought i nailed it at five, though something seemed .. off. anyways, i went running and it was around mile 3 or 4 that i realised something.

wait.
the logical quantifiers are in the wrong order.
it won't work. it's not a proof.

i kept running, of course, cursing my stupidity and the poor footing from the falling snow.

i still believe that the "theorem" is true, but i'm starting to doubt that the technique will work. maybe it's too much to hope for: it's too close to something else which shouldn't be true ..

rather, it might be true, but it would require a LOT of leverage and in the process, it would probably prove another conjecture.

i haven't thought it through, of course.

i'm tired of this: wanting to prove what i think is true. nobody should want to prove a theorem.

wow. everyone has preprints now.

today i thought i had proved something great.

feeling good, i decided to take a little break and procrastinate, so i searched the arXiv and then a few preprint servers ..

.. and found that an acquaintance of mine has a preprint up. he's the third author.

Rectifiability of sets of finite perimeter in Carnot groups: existence of a tangent hyperplane [link]

Authors: Luigi Ambrosio - Bruce Kleiner - Enrico Le Donne

Abstract: We consider sets of locally finite perimeter in Carnot groups. We show that if E is a set of locally finite perimeter in a Carnot group G then, for almost every x in G with respect to the perimeter measure of E, some tangent of E at x is a vertical halfspace. This is a partial extension of a theorem of Franchi-Serapioni-Serra Cassano in step 2 Carnot groups: they proved that, for almost every x, E has a unique tangent at x, and this tangent is a halfspace.

wow. awesome stuff.

it leads to all sorts of compatibility questions, in my mind.

there is a substantial theory of rectifiability on Carnot groups, these days, using explicit techniques from the sub-Riemannian geometric structure of such spaces. unlike some examples in the analysis on metric spaces, in sub-Riemannian geometry you are permitted vector fields and contact forms and explicit formulas for geodesics, in some cases.

put simply, you can work with pretty smooth things.

on the other hand, a friend of mine (a mathematical brother, actually [1]) has shown that there cannot be currents on Carnot groups in the sense of metric spaces (after Ambrosio and Kirchheim).

in contrast, on Euclidean spaces the theory of currents and rectifiability go hand in hand. it makes, at least for me, a confusing geometric measure theory.

at any rate, people i know seem to do such fine things. take these guys:

john's preprints on the arXiv, of which there are two.
kevin's preprints, also two in number.

.. and all my officemates are writing or finishing preprints of their own. the world is a competitive place, of course.

it's just that sometimes,

you don't realise how good the competition is.

[1] that is, we have the same advisor. speaking of which, he is soon to write up his own work, as well as marie, another mathematical sib.

continental divide.

this is mildly frustrating. there have been a few recent job announcements at several european institutions, and their deadlines are in late january to mid-february. i imagine that hiring decisions are made in early march.

as established by the AMS, the uniform response deadline is 11 february.

never mind the fact that the best and brightest in europe are going after these jobs, and a silly american like me will probably have no chance -- these are very tempting places.

but say i'm lucky and am offered a job in the states, around the time of the uniform deadline. if the european schools aren't sorted out yet, how do you play that waiting game?

bah. i should be so lucky, to worry about what to do with a job offer!

painting in broad strokes.

on a more light-hearted note, today i attended a colloquium where i heard a talk from the mother wavelet herself, ingrid daubechies.

it was really cool.

her research team is doing image processing and signal analysis on van gogh's paintings! there wasn't too much mathematics in it, but she gives an excellent talk.

how fickle: fate.

that one particular conjecture remains standing. i cannot prove it in full generality and it bothers me. let me not say more about it, otherwise i run the risk of sounding like the type of mathematician who calls everything "obvious" or "trivial."

gee, buddy. i guess it's only deep, earth-shattering thoughts for you.
it must be really annoying, spending all day working on math and getting "nowhere."

yes: better that i shut up now.

---

i spoke with the supervisor [1] yesterday, and nothing came to mind.

then, last night, i had an idea. it pushes the argument further, but not quite a proof.

this morning was a confusion. i was stuck and couldn't "prove my way out" of one particular obstacle. i still don't know how to do it.

this afternoon, while recalling what i knew about the problem, i worked out a counter-example to a lemma i wanted. it isn't true.

dejected, i went to the gym and feeling miserable anyway, i decided to run on the oval track above the basketball courts [2].

around lap 10, i realised that the counter-example was valid only for the would-be lemma and NOT the theorem i have in mind. i then cursed my luck, because there were 14-15 laps [3] to go until i could willingly let myself leave.

around lap 15, i remembered a theorem which states that my counter-example is "not generic." yes, i'm not being very precise, but if you really want to know, it's Theorem 10.11 in Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability by P. Mattila.


so: i still can't prove the conjecture, my counter-example didn't work out, but now i know of one more thing NOT to try.

maybe i should stick to writing my thesis!



[1] to honor someone's wishes, i'm reserving the keyword "advisor" for my late advisor. paperwork keeps things running, but morally, i remain his student.

[2} i HATE that track. the conversion is 8 laps = 1 mi, so it takes a seeming eternity to gain any reasonable mileage. there are 3 lanes, and almost surely there will be 2-3 people running shoulder-to-shoulder, shooting the breeze and/or listening to their iPod ..

.. which means that to pass them, you need to switch a few lanes. it's extremely irksome.

[3] i can never keep count of laps very well, especially when swimming. sometimes i run an extra lap or two, just in case i miscounted.

the thesis, so far ..

my (incomplete) draft of a thesis is 25-26 pages long, under the standard amsart package in LaTeX. to be fair, i haven't tried reformatting to minimize the number of pages and lines, and i think that my explanations will test the patience of forthcoming readers ..

things truly incomplete:

completing the second half of a section on metric tangent bundles,

things to add:

an introduction,
more background and basic lemmas,

things optional:

a section on currents on metric spaces (and a few idle theorems),
a section on how this conjecture implies that conjecture, and so on;

an appendix on how our formulation compares with that in the literature,
    (our definition is rigorous, but not quite the same)

well, it's going.

mathematical self-sabotage.

today i decided on an alternative strategy: i would try to construct a counterexample.

but i couldn't.

instead, i encountered object after object which were immune to my machinations; that is, examples which satisfy my hypotheses and none of my sufficient (simplifying) conditions ..

    .. yet not be a counter-example to my claim.
in other words, no progress and just as much confusion as before.

of course, this doesn't mean i'm right;
it might just mean that i'm no good at imagining pathologies.

how perspective changes: "what have you done for me, lately?"

four months ago, i was absolutely delighted that one of my ideas might be applicable to a special case of a conjecture,

that is, someone else's conjecture.
i didn't formulate it myself;
i wasn't even a mathematician at the time!

lately i've been slightly depressed, and this is why:
i can't seem to prove the conjecture in full generality.

emails, writing, kindness of strangers.

this morning i did nothing but write emails, albeit to mathematicians and fellow math grad students. i felt a bit like a bureaucrat (even though i've never been or witnessed one) and it was a little productive.

then again, there is much to write. what i have in mind is correct, but when it comes to mind, it stretches past the corners of my eyes and covers what i can see of the horizon. as geometers we know this is no test for infinity or immensity, but the human part of me falls for this, every time.

i know it's not an impossible task,
but that doesn't mean that it won't kill me;
i also know that it can't possibly kill me,
though i don't know why, exactly.

it just won't.


i know that once i begin, once i write something, then my intent will crystalise. i'll read that first attempt, then say:

"zounds! that looks terrible!"

and i'll know what will be better, in words or in structure. then my mind will turn its gears, form its machinations, and then progress will follow.

but first, i must begin.

---

sometimes i feel condemned to be a kind and generous person, happy to help. it's because too many people have done me much good.

i cannot count how many have been kind and generous to me, in the world of mathematics. not to follow their example would take away from their efforts.


for example, it's why i bother teaching well (or trying to, anyway) when all else tells me to give up.

if i don't, then my old teachers will have tried in vain, because how many of my fellows -- those students when i was a student, too -- how many of them will teach, and train the next generation?

this is no secret: it's a mostly thankless job. in most cases it doesn't pay well. often students will never appreciate their teachers at the time they are taught, just as children don't appreciate their parents when they are being raised. curiosity is is natural and learning is natural, but it is hard to captivate the student with lessons when there are so many other distractions and alternatives that a young person can reach.

but i cannot teach badly, in good faith.


among others, the u.s. marines have a saying: the price of success is continued success.

perhaps what i mean is: the price of kindness is continued kindness.

stuck in a superstore ..

on saturday i boarded a bus to the meijer superstore to buy groceries and a trash can. to get home, i'd have to board the return bus, which arrives every hour on the weekends. the next bus would stop at the store-front at 5:23pm.

by 5:17pm i stood behind two mothers with children and shopping carts loaded with items of all shapes and sizes. the other lines were similar quagmires. i had switched lines twice by then.

i estimated the odds and sighed. looking past the lines and cashiers, i saw a mini empty starbucks cafe next to a haircutters and a bank branch. turning around, i found no one behind me.

then i had an idea.

at 5:21 i made a mad dash and back to the office supply section: ha! found one. in the interim no one had moved my cart away.

by 5:24, i started unloading my groceries onto the conveyor belt and the bus was gone.

by 5:30 i was drinking a small coffee,
shopping bags next to my cafe table,
doing math on a 1-subject notebook i bought for a dollar,
checking details while waiting for the 6:23 bus.

it was surprisingly productive:
i found out that one of my new ideas doesn't work.

[blinks]

i found this post on the arXiv today, under "differential geometry." the abstract is curiously amusing, especially the last sentence:

The material is presented as a sequence of problems, which is peculiar not only to Zen monasteries but also to elite mathematical education (at least in Russia).

knee deep in functional analysis.

you know, i never thought that i'd ever use the weak operator topology, much less in the case of non-Hilbert Banach spaces. i guess i never imagined myself as that sort of analyst.

also, i never expected weak-* compactness to be this useful.

---

on an unrelated note, the first hit for "weak operator topology" on Google is a link to the wikipedia entry.

worries.

i've been getting this feeling that my thesis is too short, that i don't have enough results.

then again, if i believe that, then the inevitable will happen: it will be painfully long and nobody will read it other than those who must.

at any rate, i'm writing,
trying not to worry, yet worrying anyway,
but still trying ..

quotidien anxieties.

i spent wednesday night worrying about mathematics and about career stuff. while doing so, i watched the CBC [1] late night movie, which was "Charade" and starred Errol Flynn and Audrey Hepburn.

they don't make them like they used to, i tell you.

---

also, i still can't patch one argument of proof; maybe the (new) adviser knows a way. i'll ask him next week.

in the meanwhile, i have another theorem in mind. someone els stated it first, some years ago, and hence lay claim to the credit. it would still be nice to see a proof, though.

[1] i think it means 'canadian broadcasting channel.' they seem to discuss many canadian things, during the commercial breaks.