an annoying splinter of an idea.

i had this research idea from last week, but sometimes i wish i hadn't thought of it.

you see, it doesn't work.

on the way to the airport last thursday, i worked out a simple example of a "measure-induced metric" on the unit interval, and that didn't work.

then again, i didn't expect it to work because it shouldn't; there are underlying reasons, and that later idea was meant to exploit those reasons.

so during the random moments i could muster over these holidays, i tried a more complicated example and hence: the idea that i mentioned earlier.

it not only fails to work, but it makes no sense; if i can finish one particular argument, then it will lead to a contradiction, and that's no good at all. it would mean that there is some undetected error in my work -- an error i can't quite understand, and that irritates me to no end.

i don't mind making errors; they're inevitable, and i've made my peace with that. but i can't stand making an error whose consequences are evident but whose source is secret.

it's like having a friend who knows a good riddle, knows that you don't know the answer, and won't tell you no matter how many times you ask ..

.. but perhaps i should focus on the positive. at least, everyone seems to tell me that.

---

at any rate, my best guess is that by modifying the example, i've turned a purely metric condition into something .. sobolev, or more aptly, function-theoretic.

the end result looks like a variational problem and i don't quite know how to formulate it. it's almost as if the functional is "in the wrong place," but i can't make that precise ..

moreover, is it even worth formulating? the example was meant to generalise into a setting which hasn't enough structure for variational problems!

on the other hand, i'm curious to see if other perspectives will illuminate the situation.

if only by analogy, the modified example reminds me of .. something like elasticity in light of finite energy, and i'm tempted to ask the folks at syracuse if they've seen anything like it.

then again, it could be something simple or crackpot. i'd hate to waste their time .. that, and look like an idiot.

---

usually such conundrums are worth even a little something, if only for the trouble they cause. however, i remain of the opinion that i might be better off without this idea.

at least, i would be less disturbed if the idea came in early january, when i'd be back in ann arbor and more inclined to work.

unhappily, it is still very late december, i am visiting family, and the holiday season hasn't yet passed.

my family are already of the opinion that i am a little crazy and "not quite right," and i'd rather not convince them any more of that.

inspiration .. or rather, motivated confusion .. strikes at such inopportune times!

it's a conspiracy.

it's been said that the last mathematical universalist was J. Henri Poincaré, and he died sometime early last century. since then, mathematicians have demonstrated the tendency to specialise in their own fields and then subfields and then (sub)²fields ..

but occasionally, mathematics appears as a conspiracy. there are too many coincidences ..

.. and i don't mean this upcoming film called "the number 23" with jim carrey (see the trailer @ apple.com), which looks really atrocious.

more precisely, i mean how concepts are related.

for example, earlier this summer i was browsing through c. villani's book, topics on optimal transportation because i had heard in a lecture, years ago @ Carnegie Mellon University, of a way to perceive the Wasser$tein space of measures on a metric (length?) space as having some sort of Riem@nnian structure.

the analogy is initially formal, though in the case of euclidean space there is a way of discussing this rigorously, through a complicated construction of gradient flows.

so i decided that this is really a formal thing, and not much can be done. of course, i was wrong.

lott and villani have recently worked on optimal transport and Ricci curvature (in the aleksandrov sense), and lott has even made sense of a Riemannian connection and curvature of the Wasserstein space, in the case of a compact smooth manifold.

as i may have said once, i'm perfectly willing to be both happy and wrong. the days when i am right are the pessimistic days, when the worst-case scenario actually happens.

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the other case at hand involves the notion of 'concentration of measure' which i learned from hearing lectures a year or two ago in a class @ um. the ideas were wholly nonintuitive yet intriguing, and it was really quite something!

but i thought that it would simply be an idle pursuit, that i'd never actually see it used in my field of interest. again, i was wrong.

more, lott-villani have just recently related notions from the 'concentration of measure' phenomenon to local/global Poincaré inequalities on length spaces -- the latter being the 'bread and butter' to metric analysts like myself.

amazing. if i didn't have to research and write a thesis, i'd love to explore these notions and see what can be said. but alas .. if i want to finish in five years .. \:

grading day.

6:30am

the second alarm goes off. i groan and stumble out of bed and due to sleepiness, it takes me twice as long to brush my teeth.

7:45am

i make it to the exam room, ready to proctor. we begin late, as always.

10:15am

deliberations begin on grading rubrics, and then actual grading. by happy fortune, someone has brought a communal supply of coffee, as well as a batch of donuts and pastries.

4:30pm

the grading is finished, and we enter exam scores on the computer. we'll worry about the curve another day.

5:30pm

alcohol and darts ensue @ ABC. my dart throwing is idiot savant; in one turn i hit the bullseye, another turn i miss the dartboard.

between a long day and alcohol-diminished inhibition, i make wisecracks to the surprise of my company. apparently i'm not often humorous.

9:00pm

i tabulate tentative course grades; they will be finalised after tomorrow's office hours.

12:00 midnight

my body crashes from sheer exhaustion. (looking forward to it, actually.)

wow.

EDIT (AS OF 14 DEC '06): below, the first link may only work for subscribers. the second link may be more accessible.

holy cr@p. i'm listed on mathsci.net ..

.. or rather, 'we.' it's a co-authored article, after all.

i mean, i realise it was accepted .. but i still don't believe it. it hasn't sunk in. after looking at a host of research problems over several years and never solving any of them, this is the one which makes it.

this, a fun one that my co-author/friend and i worked out over our common travels, and played email tag with latest drafts for months.

maybe the universe has a sense of humor, after all.

at any rate, for those who care: here's the title/abstract and a PDF.

'conjectures' and writing.

i feel like i spent today reading raven's bones, or shaking a bag of runes.

apart from a little writing, most of today's thinking time went to making guesses pertaining my research, and on a whim, i called some of them 'conjectures.'

i don't really know what makes a conjecture, in the same way that i don't know when a fact is a proposition or a theorem. i suppose i've never asked anyone before.

a conjecture sounds to me something deep, a really good guess based on good evidence, or one of a few possibilities which exhaust the known methods and tricks but which still eludes proof.

random guesses should not be conjectures -- not without a great deal of contemplation first.

so today, i made guesses: guesses which 'feel right,' but i cannot determine how prove them, or even how to begin to prove them.

it makes me feel like a crackpot, or less unseemly, i feel like i'm playing 'grown-up' and making 'conjectures,' pretending to do what research faculty actually do.

---

this writing feels like 'pretend' as well. it began some days ago, when i felt the need to codeify the progress of my research in some final, clear form.

in the past few months i've been following several threads which are loosely related; they wind in some common directions and fray in other directions. my mind isn't very good and i cannot keep all of it straight in my head.

week after week, i make claims and form arguments of proof between mental doldrums, and they change depending on new insights or discovered errors.

the 'big picture' has never been easy for me, and complexity is not my strength. so i write and i clarify in hopes that some things become obvious,

.. much like how i keep a personal journal in order to avoid seeking a therapist .. well, among other reasons.

if i were clever enough, or if my memory were better, i wouldn't write so much and my notes wouldn't be so neat. maybe it's a crutch and does me both harm and good, or maybe it's like a prosthetic leg and lets me keep up with the herd.

i don't know. but it makes my research easier.

i have four pages of "progress" and part of a page of these guesses. i haven't done any problem-solving today, proven anything new, or exhausted a line of argument. my guesses aren't even conjectures yet.

if it weren't a weekend, it would be a waste of a workday. then again, progress is rare and most research days are days wasted.

on book request spam.

november turns into december: it's time to unclutter the in-box again.

so today i deleted 318 emails that i've collected over (less than) one month's time, and most of them are spam: the majority are dead giveaways, by the nonsensical title and unknown sender.

then's there's unintentional spam from the math department. today being 2 December 2006, in the last week four emails were forwarded to the entire department in the form of book requests. three were from grad students, and one from a full professor.

these all take a common shape: someone wasn't able to find a book from the library, then emails the entire department. the idea is that the book in question is so specialized that someone else in the department must have checked it out.

the request is very reasonable: the requester wants to borrow the book for an hour, in order to check a fact or two. at worst, they want to make a copy of one chapter and return the book to the borrower.

of course, this situation is absurd. as grad students and faculty, our library privileges permit us to keep renewing our borrowed items indefinitely. when you are the borrower, then this is very convenient .. provided that you don't abuse your privileges.

for instance, i'm guilty of this vice. i once had a book checked out from the library for 1 1/2 years, possibly longer.

this is exactly the purpose behind loan recalls for university library resources. no one member of the university should be able to hoard library resources which are for the good of the many.

so to those who make book requests: if you need the book, then recall it. you might need it only for an hour or so, but chances are, the borrower also needed it for a little while. if they truly needed the book, then they would buy his/her own copy.