wonderful ideas, none mine.

today was the weekly meeting with the advisor, and short as it was, i think it went as well as can be expected. rather, it went as well as one could expect out of my first week of reading a new paper [1] ..

.. which means that one should have few expectations, if any. i follow that same rule, myself. q:

but enough self-recrimination: it certainly went well, because my efforts are now sharpened towards more specific goals. after a bounty of good ideas from the advisor [2], there is much work to do in these next few weeks, and when this matter of 2-dimensional homeomorphism extensions is resolved, there remains the big lingering problem whose solution will (hopefully) form the body of a dissertation.

there's something soothing about having work to do [3], which is a bit like having a (temporary) purpose in life.

---

earlier today i started wondering (read: worrying) about the nature of research, and i despaired about ever having any good and fruitful research ideas of my own.

sure, i have ideas, but are they any good?
will they lead to anything of passing interest?
will i be able to carry through any of them, even if they are interesting?

as a larger question, how does one obtain a "vision?" how does one think towards a cohesive body of knowledge which explains some abstract phenomenon, rather than merely dabbling and proving what lemmata or theorems which can be done with a few clever tricks?

in mathematics, i listen and less often, i understand. i follow this or that line of reasoning, and with a little twist i can conjure up something of slight novelty but of same spirit ..

.. but i've never created anything. i've left no contributions to the body of knowledge from which i've so consistently borrowed and occasionally earned. in the simplistic words of ayn rand [4], thus far in my life i've been a "second-hander."

somewhat fittingly, i never resolved that line of questioning, but merely forgot it. later i left the office with friends to see a film, stay for a Q&A session with the director, and then off for a pleasant dinner.

[1] it's the one by Douady and Earle, concerning "conformally natural" extensions of circle homeomorphisms.

[2] trust me; the ideas aren't mine, as much as i'd like to them to be.

[3] as opposed to not getting any work done; the difference lies in at which point one obtains the workload. "having work to do" means that you haven't done the work yet, and there is potential in all possibilities; "not getting any work done" means that you've tried and tried and nothing seems to work.

[4] i'm referring to one of the several hundred pages of the book, the fountainhead. for the record, it's not worth reading; if i could trade the time it took to read those pages for a better choice of pages, i would, and i'd probably have chosen Dostoevsky as its replacement.

"the best laid plans .."

yesterday afternoon i told my flatmate alan that i was writing off the whole weekend: no mathematics! since a week's worth of work went to $hit, i thought it was the right time for a short break, and he thought it was a good idea, too.

earlier tonight, alan tells me that my idea was so good that, in fact, he used it himself. after trying and failing to understand a preprint of this recent work, he decided to cut his losses and take sunday off, just like i said that i would.

nodding nervously, i admitted to him that my plan didn't quite work out.

---

yesterday was fine, and i enjoyed myself immensely: i played basketball, drank plenty of coffee, wrote pieces and ideas for short stories in my journal, bought a collection of short stories of Hemingway's from the sidewalk guy,

(ann arbor natives will know whom i mean:
think state street, near the theater)

.. and even read an actual (non-math) book!

then sunday came around, i woke up, couldn't think of what to do, and went to the corner coffeehouse out of habit. i had also brought my bookbag out of habit, and forgot that Hemingway collection at the apartment. the counter had already sold out the day's quota of newspapers, and i had nothing to read; had i left the coffeehouse to fetch the Hemingway, i would have reneged on my free coffee refill privileges ..

.. and i'd be damned if i ever willingly do that! you can take the boy out of the mathematics department, but .. q:

anyways, realising the impasse of the situation, i gave up, fished out my work notes, and sought to understand where precisely my "proof" went wrong. a few basic implications hit me, and i realised, again, how truly stupid my error was.

then, remembering that taxes were due tomorrow and not trusting the wireless networks near the apartment, i marched to the department and e-file on their computers. on the way there, i had a new idea.

diligently i filed my taxes, and then tested out the idea. it doesn't quite work, but it's a new lead, and it might lead to something that may work. only time will tell.

i might think a little about it tomorrow before starting on this new topic. it's the "conformally natural" planar extension of Douady and Earle: a wholly different creature from these harmonic extensions.

at the very least, there's some hope now.

lessons learned, advice given.

on two separate occasions today i gave advice to fellow math grads about thesis advisors and meetings with advisors. it likely means that they don't follow this blog at all, otherwise they'd realise how little i know about these matters, and ask someone wiser. (;

so i did the best i could: i told them what not to do, and why it might be a bad idea, such as

if you have to, read less material in order to understand it better, because

1. inevitably you'll be discussing it with (inserted advisor's name here) and the details will matter. mathematics may be intuitive ideas, but it is also details and rigor.
   
   
2. if (s)he asked you to read it, it's probably instructive and will be useful for your later work. glossing over it now will probably mean that you'll have to learn it properly later;
   
   
3. you might as well read what you can comfortably handle, because with random exceptions, nobody ever reads as much as they expect to read.

as you can imagine, i learned all of these reasons the hard way.

on failures and coping.

i suppose i should explain my last post a little.

over the course of this past week, i proved a lemma and went to work on a proof of the main theorem and tried to generalise it, through various cases as given by the Sobolev Embedding Theorem(s).

as it happens, it wasn't a proof after all. in fact, the lemma isn't true; the counter-example, embarrassingly enough, is the identity mapping from the unit disc to itself.

(well, you might need a neighborhood of the disc,
but let's let bygones be bygones)

at any rate, it means that a week's worth of time and energy has effectively been wasted. being a person who doesn't believe in an afterlife and has a 2-year clock running, this pi$$es me off to no end. worse yet, it was a waste of the advisor's time, which is embarrassing and intolerable.

i was a sourpuss from the time i met with my advisor and learned this cruel truth to sometime around midday today, though pietro, a visiting prof, did make me laugh uncontrollably last night, when we were at the brown jug. [1]

it was around midday when i realised that it didn't really matter that this was the second or third idea in a row which failed, nor does it really matter if i ever prove anything about these 2-dimensional harmonic extensions or not.

it stung, because i could add it to "the list."

it reminded me of how i never resolved the isoperimetric problem on the heisenberg group when i worked with my undergraduate mentor, or how i never resolved my hasty conjecture in operator theory during that summer REU in california. on bad days i usually measure my adult life by how many research problems i wanted to solve but couldn't, for reasons of time, willpower, and cleverness;

it's the price for believing that "anything is possible," given the right inclination of mind. if you believe that you are capable of anything, then any failure is always your own fault; otherwise you wouldn't have been capable of it.

anyways, i now deem it the past: it's one more memory of my stupidity and one more problem unsolved, if only for the moment. the advisor had the tact to suggest another approach for extensions of boundary homeomorphisms in two dimensions, and i'll get to it by sunday or monday .. but right now, i really don't care. i don't care about much of anything except one casual realisation:

i don't have a life anymore. i don't have any hobbies or interests or long-term fun projects. my efforts have been unilateral as of late, and as usual i've put my life "on hold" for so long that it's become frayed at the edges and in some danger of tearing apart.

so i'm spending the weekend away from math: away from the thesis work, away from the joint work with my co-author, away from it all. i've gotten over hating myself for the moment, but that doesn't mean that i like myself very much. i need a holiday and since i won't have any apropos vacation time until late june or so, i might as well take the weekend off and try not to worry about anything academic, professional, or even domestic.

..

chri$t. it was such a fu¢king stupid mistake ..

..

and now, back to forgetting about mathematics. \:

---

[1] the joke is one of the lowest forms of humor: it uses a pun, but i couldn't help but laugh anyway:

A French fry walks into a bar, and says to the barkeep, "Give me a beer."

The barkeep replies, "Sorry. We don't serve fries here."

never mind (re: last post)

my proof didn't work, and the lemma is trivally false; no surprise there. i should have known better, and listened to myself when the result itself sounded too good to be true.

f*ck.   f*ck, f*ck, f*ck.

i guess i never seem to learn my lesson. this always happens, and it probably will happen, over and over and over again. where's the hope in that?

that's the very thing. there isn't.

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!

first floor offices are unproductive.

from a certain point of view [1], i have no love for the daytime. the late evening and nighttime are far superior: it is quiet, people away to their own agendas of play or sleep, and so doing, they leave me in peace ..

.. and i can actually get some work done.

---

east hall is too distracting: there are too many undergrads lollygagging along the corridors and making loud nuisances of themselves. do they deem us old, and hence deaf?

there are also too many grads, postdocs, and profs doing too much interesting mathematics. i overhear someone who is fascinated by this or that problem, and describes it in that same fascinating way. i can't help but listen and then i can't concentrate on anything.

absently i might find myself repeating the same steps in proof, and with good reason: my memory is faulty and my mind too easily confused by this stimuli to be any good for anything. this happens far too often for my taste.

for instance, it happened today. i couldn't accomplish anything i planned to do yesterday .. and not because i tried and failed, because i never had occasion for an earnest try ..

---

paradoxically, afternoons at coffeehouses are amazingly productive, but i have a theory concerning this. you see, noise in such places vary in volume, but it is essentially homogeneous: hence nothing attracts my attention more than anything else, and i don't get distracted.

if i work without my glasses (being nearsighted, it's a matter of lifting books and papers close enough for focus), then i don't have to worry about being distracted by anything, whether it be pretty girls at a distance or a man in a chicken suit running past the window.

it's all a wonderful, forgettable blur. i never thought i'd ever be happy in being unable to see well!

---

[1] yes. i stole this turn of phrase from "star wars: return of the jedi." it's the scene where luke confronts obi-wan with the truth:

"but you said that vader betrayed and destroyed my father!"
"and what i say is true, from a certain point of view."

sneaky old man, that kenobi. q: