today and yesterday.

(written today) it's one of those weeks where after meeting with the advisor, i'm either (1) not quite sure what i'm supposed to be doing, since we discussed so much last time ..

(it was break, so there was no anss and we spoke for about two hours, despite my not being very prepared to discuss anything.)

or (2) not quite sure if everything had worked out from last time. in one argument (which settles the 2-dim'l case) we asserted two lemmas. both are the sorts of facts about harmonic functions which are either reasonably easy or in a book somewhere.

but as of today i can't seem to work out the details, though i believe i should be capable of them. as a result, they are bugging the hell out of me. if all else fails, then i could ask the advisor during this week's meeting, but that just seems .. wrong.

call it stubbornness, i guess. if i can't work out details at this stage in my career, how am i going to write a thesis?

(written 5 march, two days ago) this joint work with my co-author may actually reach an end, at some point. if you're wondering what caused all the hope and optimism, let's just say that it was an .. arduous but good mathematics day.

one result concerning geometric measure theory bothered me all yesterday evening but happens, sensibly enough, to be a theorem in federer's geometric measure theory tome. the sole reason why i have no qualms citing the proof in a future paper is only because i read and managed to understand the proof .. well, sort of.

as with much of the maths i see, i understand how the machinery works, that is, the gory details if they are not too many. it's the intuition that is troublesome.

if i think of it like compiling a program, then i can parse low-level assembly language and mid-level language formats, but the high-level stuff - where the algorithm or strategy is shaped - remains fuzzy and ethereal. i know that the program will run and that the output will be good, but i can only guess how the programmer chose that given implementation.

the other result which bothered me isn't so bad after all. i can write a proof on a third of a page, but the trouble is that it requires five or six lemmas and basic facts, as well as more (standard) definitions than one can count conveniently.

still, though .. a third of a page. that's not bad, and all in all it wasn't a bad day. in the late evening i toiled and struggled to no avail against a nonlinear ode, but i couldn't make sense of it. but that is not so bad a defeat.

so lemma 15 of 63 is done (i made the #'s up, for those concerned) and perhaps the final (to-be-submitted) draft will be ready by summertime. a boy can dream, right?

a year ago i'd have kept reservations about how much we "did" for this paper. when extending the results of others, it can be hard to tell if there is enough original thought. but at this point i'll settle for the fact that we're bringing in an additional perspective and a toolbox of techniques to the table. it won't be a a mere reiteration of the variational principle, but as i've hinted, some geometric measure theory will play a key role.

what we'll do is new and worthwhile; i'm convinced of it now, so it's a matter of labor and determination .. but then again, when has it not been so? it is also a matter of tim, and the thesis comes first, after all.