- i (or someone else) would spot an error or conjure up a counter-example to one of my main theorems. the writeup would then become a meaningless list of latex symbols, the outcome of wasted months.
- i (or again, someone else) would discover that i've been dense all along, and there are trivial/two-line proof to all of my theorems.
these fears are hard to shake off, but i like to think that this is not exactly paranoia. call it experience.
- 13 or 14 months ago, i thought i had proven something great, and from this one result you could get so many interesting consequences.
11 months ago, we found an error in my argument. we were never able to patch it explicitly, and i thought i'd have to move to a third thesis project and learn a new theory all over again.
since then, you could say that my recent work has been some sort of professional vindication. as it happened, that project worked out, but unexpectedly so [0]. - when i was applying for jobs last fall, i had no papers or preprints about my current work (..i still don't, to this day..) and in lieu of such materials, i sent a research statement to my letter-writers.
one day in that fall, my supervisor and i were talking, and he mentioned that one letter-writer called him and asked him something: "i don't understand. why don't you just do <this>?"
the supervisor then proceeds to tell me what <this> was. i opened my mouth a few times, then promptly shut it and thought a little more. huh. that does work. later on, the other letter-writer pointed out the same <this>.
i still feel like an idiot for not having seen it. then again, if not for that suggestion, the thesis would probably be 9 to 10 chapters long, and not 8. so i'm not complaining.
just earlier i was thinking about one particular claim that should be true, but doesn't quite fit into other techniques of proof. jotting a few things down, suddenly my blood ran cold.
i sped to my bedroom, fished out my bookbag and a copy of my thesis draft from the bookbag, and leafed frantically.
i read, and re-read, and sighed;
thank god. i assumed N+1, not N.
as the saying goes (in translation, anyway):
"after you've been bitten by a snake,
even coils of rope will make you nervous."
it's going to be a nervous few weeks!
[0] 10 months ago, i broke a promise or two [1] and started thinking about that problem again. nothing worked: that is, none of my old ideas or new variants of those ideas.
after a while, i traced my reasons of argument to their core motivations in 1-dimensional examples. i wrote down "assume that _____" quite often and at the time, i thought that these were wholly unreasonable assumptions.
i wrote them down anyway.
it seemed like i would run out of time: 9 months ago, i flew across the atlantic and attended a few conferences. one night in england and after one too many at a local pub, i stumbled to my room and decided something.
all right: i'm going to do the most naive thing possible. this is NOT going to work, but it doesn't matter anymore. i want to know WHY.
an hour later, i couldn't figure out why it didn't "not work." deciding that it was the ale, i slept fitfully. the next conference day, during the first coffee break, i re-read what i wrote and couldn't find the breaking point. then i began to suspect something.
over two weeks and three conferences, among hyperactive social mathematicians in collaboration and good cheer, i became an antisocial ghost. i was constantly suspicious of my work and constantly expecting to be disappointed when i found the flaw.
but i never did.
from that one naive idea that i entertained, out of desperation, there appeared something interesting enough to make a thesis.
[1] that's a story for another day. i might tell it sometime.
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