retreat.

perhaps it's time to stop researching derιvations.

sure, last week wasn't particularly a good one for research .. but there's something else.

a few new ideas came up in the last 8 months or so,
but their strength is limited; so is their applicability.

in particular, i still can't make any progress towards any of those open problems that i'd like to solve.

i think i've done all that i can with them. i have a new theorem or two, however, and that's good enough.

it's important to know when to stop or to retreat. for much of my first year as a postdoc, i tried the same ideas over and over, to no success.

sometimes i wish someone had just told me to stop, pointed out that my strategy was (ultimately) flawed. that's too ideal of an event to happen, though, and not fair to say.

for one thing, it's too much to expect someone to understand exactly what you're thinking, not even your thesis advisors.

besides, i'd have been too stubborn to listen to that sort of advice. even if there were some way to travel back in time, i'm not completely sure that my past self would listen to the present me.

it's not that i believe in fate. i'm just more aware of how my own mathematical brain works (or fails to work) and what kinds of mistakes i'm prone to making.

that said, i'm not giving up completely. there's still a preprint to finalise and to submit. if i get a good idea during the writing process, then sure:

i'll make a note of it,
put it somewhere that i'll remember to revisit,
and leave it alone to continue writing.

if there's one thing i've learned, in the last few years, it's the importance of "following through" --

for, in the business of mathematics, ideas are NOT good enough. rather, it's important to shape ideas into complete proofs and have them accessible in written form.

it's true that few people will likely read your paper. if you never write up your ideas, though, then it guarantees that nobody will ever listen.

i always think in retrospect that my own work is crude and obvious, probably not worth publishing. (i wonder if everyone feels that way.)

it's better, though, to have something modest but fully-formed, rather than nothing at all.