Wednesday, August 20, 2008

old ideas don't die; they don't go away, either.

thinking about it now, it's been months since i've discussed research with a fellow mathematician. so i suppose this means that i've been working in isolation. maybe that explains how i feel ..

.. because i feel slow and stupid.

my ideas aren't quite right -- i'm almost sure of that -- but i cannot think of better ones which will do. so i alter and adjust them and try those variations.

i feel that, mathematically, i spent the last few months making small permutations of the same basic, probably flawed notions and seeing the slight variations in their common failure. [1] this borders on obsession, of course, and i hope that when the postdoc begins, someone in the department can knock some sense into me and suggest more fruitful areas of math for me to pursue.

today i couldn't remember the definition of a curr3nt, in the language of f3derer and f1emin9. [1.5]

the space of k-curr3nts is meant to be dual to the space of differential k-f0rms, and a 0-curr3nt is equivalently a distri6uti0n; i remember the motivations. so there's probably a LCTVS structure [2] on the space of k-f0rms, just as there is such a structure on the space of test functi0ns, by way of partia1 deriv@tives and sup-n0rms.

i may be wrong; but i can't remember exactly and my copy of f3derer's gmt is some hundred miles away, so let's go with that. at the very least, there should be some continuity property for an arbitrary curr3nt, otherwise a 0-curr3nt couldn't possibly match up with a distributi0n, right?

sometimes i really hate myself, especially my poor memory.

i can deal with ambiguity. however, i can't deal with confusion. like anyone else, i looked up "curr3nt" on wikipedia: here's the entry, but i can't reckon what the continuity axiom should be .. if i'm even remembering it correctly.

argggggggggggh.

anyways, i'm bothering with the "classical" definition because i don't want to throw away one of my ideas.

to patch up one of my currently flawed approaches of proof, i need to use some property of metric curr3nts that an arbitrary (FF) curr3nt should not have. otherwise, i would "prove" something for all (FF) curr3nts (of finite m@ss that is conjecturally untrue.

often i wish i were smarter, but i'd settle for better ideas. \:

[1] non-mathematically, i spent the summer unemployed, recovering from thesis writing and thesis corrections, wall-climbing, and as they say in good will hunting, "seeing about a girl." q;

[1.5] this means, of course, that ethically i should renounce myself as someone who studies ge0metric mea$ure the0ry. on the other hand, i was trained to be a metric ana1yst, and until recently, all curr3nts to me were curr3nts in the sense of ambr0si0-kirchh3im. so: put one way, it's not my fault. (; [2] short for "l0cally c0nvex t0po1ogica1 vect0r sp@ce," as you might learn in functi0na1 ana1y$i\$.