Sunday, February 08, 2009

research ideas live, then die, then are reborn, then ..?

sometimes i hate being right about being wrong.

so far this latest idea (see earlier post) isn't too much different from its precedessors, and i haven't found any additional leverage from using the language of ¢urrents.

on the other hand, a related observation would give a new approach towards disproving another conjecture i've been thinking about. granted, it would only serve as one of several steps and the most crucial work would still be equally difficult. [1]

research can be soul-crushing. i think i've told this storyline before, but it remains true again and again.

one day you wake up, have an idea, try it out;
later that evening, you realise that you have a counterexample;
despondently you go to bed, sleeping fitfully.

the next morning, as you're about to throw away the scratch paper,
the ambiguity returns:

you learn that you don't need that exact property
something weaker suffices,
one that the counterexample doesn't disprove.

but you don't have time to pursue this;
you have to teach in 2 hours and you haven't written your lesson plan yet.
while hastily writing lecture notes,
you never realised how much you hated calculus,
and how they get in the way of "real math."

[1] essentially, disproving the conjecture would involve constructing a completely new object which has precisely certain abstract properties (c0ntinuity, multi1inearity, 1ocality, etc) yet avoid almost all of the standard constructions (e.g. p0lyhedral appr0ximation).

to be honest, it's not clear to me which side to take. at this point, either answer would be surprising to me, actually.

No comments: