in which i learn lessons, through struggle.

"Through competition,
we can discover ourselves."

~ 霍元甲 (Huo Yuanjia)

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as frustrating and painful as this ordeal is, i would still recommend applying for a grant, especially to younger researchers like me.

as a base lesson, one learns the difficulty of writing for a general mathematical audience. it's also humbling, in the sense that one realises exactly how narrow and small one's research specialization is.

there's more, though: last year and this year, i learned new things and obtained new ideas when forced to think of new, grant-worthy problems and programs to solve these problems.

in some sense, it's like being ethnically chinese when it comes to food:

you won't put up with spoilt or substandard food,
then again, you're not willing to pay more than you should, either.

in a similar way, if you're going to pose a research problem on a grant,

(1) you should have some idea of how to attack it, or at least an interesting idea to try that would be of independent interest. otherwise, why bring it up? everyone has one research problem that (s)he has no idea how to solve ..

(2) you should be able to explain why the problem is relevant, interesting, and worthwhile. excessively easy problems are frowned upon.

as a very direct example, i didn't realise until this calendar year that certain ellιptic PDE problems with signed measμre data may not necessarily have unique solutions!

then there are lessons one learns, which lead to long-term plans. i never thought i'd consider working on parabοlic PDE, but my colleagues are very convincing. i'm learning about them now and have discovered unnerving things:

for certain nonlinear parabοlic PDE, the corresponding Harηack inequality may actually depend on information from the solution itself!

very strange ..at least to me and my mathematical upbringing.

this is something to which i am unaccustomed; if you work on the analysιs on metric spaces or only with elliptιc PDE, then the constants are always quantitative.

honestly, i don't really understand parabοlic PDE. with the little i've seen, though, i'm intrigued. i think i will spend the next calendar year learning about this stuff, whether it can be formulated in terms of adapted variatιonal problems.

in writing up this grant proposal, i found an open door. i don't know what's on the other side, but i'm curious enough to find out.