paranoia and uncertain relief ..

except for friday, this weekend has been rather uncharacteristic for me. i actually went out and did social things, just as how a regular person would do. on top of that, i still managed a little math, though in retrospect, it was more a matter of the writeup and a bit of polish.

about that exception: i became so paranoid about a lemma which my advisor emphasized, that i worked on the proof for much of friday night.

eventually i gave up out of frustration, but being more patient on saturday, the pieces fit more snugly and by today i've written up what i think as a clean, careful proof.

to clarify, i had formulated the lemma and provided it's true, it implies a theorem concerning poisson extensions of certain homeomorphisms of the unit circle.

there's some fuss about whether this is the "best" or "most natural" result, but hey .. it's something, right? being a poor and lowly grad student, i'll take what i can get now and try for the sharp result later.

the problem is that the statement of the lemma itself is .. well, on the borderline of intuition.

it isn't so outrageous that it claims something generically impossible, but it seems to relate two bits of data about a harmonic mapping which you wouldn't expect to have any a priori relation.

in all honesty, the advisor's doubt is contagious and now i am doubtful. i can't find anything wrong with my proof, but ..

1. .. yes, i've made plenty of blatant errors in reasoning before;
   
   
2. .. yes, these errors often occur when i compute without careful thought;
   
   
3. .. and yes, my current proof has its share of computation, though i've reduced it as much as possible..

i trust my computations, especially after:

having woken up in the middle the night (between wednesday and thursday) to run a computation,

panicking sleeplessly because i had been making an error all along,

then checking it again the next day, and realising i made an error while discovering one, and that my old work was fine.

if anything were wrong now, it would be some subtle flaw in the logic and would require a clever counter-example. what bothers me at the moment is what the proof says about basic examples,

none of which i've done, yet. it's often the case that i finish a proof just in time for my weekly meeting with the advisor, and it's then when he suggests to me the intuitive reasons for why what i've proven is correct, and hence how the proof can be made more transparent.

i suppose those are next on the agenda. i guess you could say that i've finished up early, this week; the critical piece is apparently done, and now i can worry about the icing on the cake!