epilogue (or work-in-progress, part 5: the sequel)

my luck never ceases to fail.

i send off the preprint to a few colleagues, rest on my laurels for a little -- i.e. wait for their comments, if any -- and one day, while checking one particular post on the arXiv ..

.. i find out that i've been scooped!

..

..

*sighs*

more about this later .. but not now.

work-in-progress, part 4 of 4: triumph!.. i guess?

so this is the last of this series. maybe there will be others like it in the future, but i've told my piece for now .. maybe more than that.

honestly, this should be part 5; then again, why restrict ourselves to integers? (-;

on a more serious note-- had this story ended otherwise, then i like to think that i'd still have told it .. but that's what i'd like to think. the truth is that i don't know. the most i can say is that i'd have written noticeably less, if only not to dwell too much on one failure and start something new right away.

some of you are students, graduate or otherwise, and perhaps this account just shows you that nothing ever goes smoothly, even years after the ph.d. .. but on occasion, that things move along well enough. (i could say the same about this blog as a whole.)

for those of you who know me, though: i still don't know why you trust me with anything ..

// initially written: mon, 30 july 2012 //

i patched the proofs and my old theorem's back from the grave. i feel pretty good, like not only have i dodged a bullet or two ..

.. but disarmed the bad guys, saved the day,
all the while muttering, "i'm getting too old for this sh-t" [1].. q-:

so if i may indulge in a little shameless gloating .. i feel like i really understand what's going on now, in the sense that my intuitions have become rigorous. along the way, i dare say that i've developed a few new techniques ..

.. and strangely enough, for Euclidean spaces!

it's either that, or i don't know the literature well enough so that i'm ignorant of previously similar constructions.

this has happened to me before; i thought that approximating Hölder functions by Lipschitz ones was novel,
but semmes has done it before, more elegantly; my colleague rοger's observations had preceded mine, too.

in particular, i am proud to say that some of these ideas involve geοmetric measure theοry. to me, this is a stroke of good fortune!

you see, this is a field that i have always admired; if things had gone differently, then perhaps i would have specialised in it as a student [2]. these things are hard to say, of course.

// initially written: thurs, 2 august 2012 //

you'd think that finishing a project would make me feel elated, but then you'd be giving me too much credit. i now feel empty, like i lost a worthy opponent.

at this point i'm reduced to typesetting, checking citations, and other minor corrections: hardly anything worth regaling anyone.

sometimes i feel like i've taken the ascetic ways too far: there is this common advice to not worry about the destination, but enjoy the journey there. now that this journey is over and i see the destination, i cringe.

that's it?

these last few mornings i've been scribbling on the same conference legal pad as i've done before, back when i thought i might lose those theorems. most of the time it's picking up loose threads from proofs, seeing where it takes me. i'm slowly collecting a small list of problems to work on next.

every so often, though, i try to construct a counter-example for the theorem again.   you never know ..





[1] suffice it to say that, as a hot-blooded american boy, i grew up on all sorts of action flicks: ah, the 80s .. (-:

[2] i feel like that about a lot of topics. for example, sometimes i think it would have been better if i had really focused on optιmal transpοrt or in stοchastic games, when i was younger and could pick ideas up more easily; maybe even banach space theory. for worse or for better, i'm firmly a metrιc-space guy now.
..
.. "if things had gone differently" .. that's a dangerous direction towards which to ponder.

work-in-progress, part π of 4: despair.

as i said before, this was the day when i learned there was something else in the preprint to patch. i didn't take it too well ..

// originally written: thurs, 26 july 2012 //

at some point i felt like giving up. i feel that way now. every time i try a patch i either find a hole in it later or i go back to another part of the manuscript and something else is wrong.

layers atop layers, patches upon patches,
water's still spilling in this leaky old boat.

i'm tired. it all seemed like such a good idea at the time, but it's gotten so ugly and i can't stand the thought of a simple idea — my simple idea — become this pockmarked thing:

bruised by technical details, carrying scars
from the surgical arguments that filled in earlier gaps.

i'm trying to make it simple: simple and clean. it's hard.

i can show you a highly technical statement that remains true, but it's not worth saying and wouldn't be worth writing down.

i feel like throwing away the result.

it was too much to hope for, anyway, and dangerous besides: it would have turned a perfectly good theorem into a tautology, sullied a very compelling proof that is quickly becoming a kind of template, a meta-argument.

---

it's late in the evening and i want to sleep, give it another go tomorrow.
things always look better in the morning, right?

i'm worried, though, that if i don't go to bed with a good idea, then i'll have nothing to try first thing in the morning. a bad start might ruin the effort, and i'll spend pages and pages, making desperate kinds of estimates and computations, draw many diagrams only to scratch them out after a few moments.

i want to sleep. i can't.

work-in-progress, 3 of 4: a suspicion or two, but generally proud.

this post is a little more cheerful, since by this point i start to see light at the end of the tunnel. in fact, at this point i thought i had fully patched the paper .. only to find, the next day, another gap in another part of the paper.

it never ends, does it?

no matter: the ending (that you'll later read) is near and this post gave signs of things to come. it is also a bit more technical, as you can tell right away. (to explain, all of my colleagues were away on holiday, so there was really nobody to share the details with.)

// initially written: tues, 24 july 2012 //

i'm a little suspicious now.

i saved one lemma with a new proof, and i think i may have patched the full one. for the latter case, i've narrowed down the technical issues to measure theory, so it should be a matter of checking standard facts ..

.. but i'm still suspicious:

i've never seen luzin's theorem used like this before, and i wonder if the statement is as general as i would like it to be.

added later: yes, the version of luzin's theorem
that i need is, in fact, true.
..
added even later: there's a more elegant approach,
so i don't need a luzin-type theorem after all!

---

part of me is seething, upset with myself for having been so stupid again.

how careless could i have been, to miss that error?
maybe i should collaborate more in this metric geometry stuff:
having another pair of eyes around has got to help, right?

something good .. or at least, something useful has come out of this, though. i know my proof much better than i did before, and i'm more aware of the subtleties involved.

moreover, i feel like i've become stronger with these techniques now.

to compare with rock climbing, it's like realising that, sometimes, you don't need cracks or holes in the rock to have a good grip.

if the rock is granite-based, then the surface is already rough with friction, i.e. a slab climb: you just have to be careful and not lose all the layers of skin on your fingertips, while doing so.

so i feel stronger and more able, and now i want to climb harder things ..!

---

in particular, part of me is proud. i honestly think that i invented (or at least, re-invented) a new construction ..

work-in-progress, part 2 of 4: lamentations.

there's more to the story, of course;
i wouldn't advertise four parts otherwise!

for those of you who are curious and want to know, right away .. yes: there's is a not unhappy ending. bear with me, though ..

// initially written: sat, 21 july 2012 //

when i think about it, there would have been an easy way to avoid all of this stress:

all i would have had to do [1] was recycle a talk from before,
instead of having insisted on discussing something completely new.

it would have been the safest bet,
the easiest approach .. but when have i ever been that wise?
then again, "easy" is overrated. q-:

---

stay long enough in this business, and you can hack your own habits.

over the years i've learned that i work well, but not necessarily better, under a deadline or some other constraint. this is especially true for ones that i cannot circumvent easily or without cost.

the reason, i think, is that there is neither room nor time for any debate:

when there is something that must be done,
in the here and now, then you do it [2];

it's that simple.

sometimes thinking is the enemy to progress;
other times, it is our closest, most valuable ally. [3]





[1] for a past conditional statement, i think it correct to use the pluperfect subjunctive mood, though i could be wrong. at any rate, who says that english is easy? q-:

[2] that said, i am terrible at keeping open-ended promises. (ask my collaborators and co-authors.) in fact, it's for close-to-the-same reason: when you stack your schedule and your mindspace with urgent tasks, then anything that can wait .. will wait.

[3] i get the feeling that most people don't have this problem. then again, this blog is called "the (frustrated) overanalyst" for a reason, you know ..

work-in-progress, part 1 of 4: confessions.

this is a 4-part series of posts that i wrote in the last two weeks, which might explain why i've not been updating very much lately.

// initially written: wed, 18 july 2012 //

so i found a gap in the proof of one of my "theorems" [1] and now i'm scrambling to fix it. one of the main lemmata still holds true and is interesting in its own right, so there's still a short note if this all goes to hell.

what bugs me most is that i gave a preliminary report about this result, about a month ago at a conference. though i didn't realise it at the time, this means that i was lying .. which bothers me to no end.

this is exactly why nobody should ever trust me ..!





[1] if it is not a rigorous proof, then the original statement is not a "theorem" but just a claim .. and possibly a false one. that's the whole point of the axiomatic method!