may: a retrospective.

the summer school in barcelona is fast approaching. soon i will have to be a proper mathematician again.

it's hard to judge the month of may.
simply put, i was unproductive.

it's easier to justify by saying that i was on holiday, visiting my girlfriend and touring parts of the netherlands, germany, and ireland. all of that is true, which made it difficult, at times, to have a regular working schedule.

i did get some math done, though.

most days i was working when the girlfriend was working, but it was speculative stuff. i wrote a little of a research article i planned, but then the outline had to be changed. there's not much to show for it.

so i spent most of the time i spent indulging myself, by chasing conjectures. as you know from previous posts, this didn't amount to very much.

i guess i'll just call may a mathematical loss. maybe the rest of the summer will be more substantive.

more greek, and: "i was right, in the sense that i was wrong."

another passage from ivor thomas's selections illustrating the history of greek mathematics:

"i think you know that those who deal with geometrics and calculations and such matters take for granted the odd and the even, figures, three kinds of angles, and other things cognate to these in each field of inquiry; assuming these things to be known, they make them hypotheses, and henceforth regard it as unnecessary to give any explanation of them either to themselves or to others, treating them as if they were manifest to all; setting out from these hypotheses, they go at once through the remainder of the argument until they arrive with perfect consistency of the goal to which their inquiry was directed.
..
"therefore i think you also know that although they use visible figures and argue about them, they are not thinking about these figures but of those things the figures represent; thus it is the square in itself and the diameter in itself which are the matter of their arguments, not that which they draw; similarly, when they model or draw objects, which may themselves have images in shadows or in water, they use them in turn as images, endeavouring to see those absolute objects which cannot be seen otherwise than by thought."

from plato, republic, vi. 510 c-e.

---

in other news, i was right .. in the sense that i was wrong.

my doubts were well-founded: i finally found the error i suspected. my logical quantifiers were off, and the self-similar set turned out to have positive lebesgue measure.

yes: an embarrassing mistake,
but better now than later.

thinking more carefully about it, nowhere in my most recent arguments do i use information about how a measure distributes its mass, across space.

you'd think that, in attempting to prove a result about singular measures, that i'd account for that. \-:

on the bright side, i've proven a special case of the result. the hypothesis is too technical to be worth anything, but nevertheless: a step forward.

μαθηματικην (or "mathematics," i think.)

one of the workplaces i've found in amsterdam has been the 11th floor of a university building in VU.

my girlfriend showed it to me. it's part of a library, but that floor doesn't have any maths books, so i've mostly ignored what tomes and texts lie on those shelves ..

.. until today.

as it happens, my usual table is between two shelves of greek literature and criticism. right behind my usual chair, at eye level, there is a shelf which houses the following book:

selections illustrating the history of greek mathematics by Ivor Thomas (harvard university press).

this is an excerpt -- in fact, the first entry --

from the works of anatolius,
cited by heron. definitions, ed. Heibur9 160. 8-162.-2

"why is mathemati¢s so named?

"the peripateti¢s say that rhetoric and poetry and the whole of popular music can be understood without any course of instruction, but no on can acquire knowledge of the subjects called by the special name mathemati¢s unless he has first gone through a course of instruction in them; and for this reason the study of these subjects was called mathemati¢s [3]. the pythagoreans are said to have given the special name mathemati¢s only to ge0metry and arithmeti¢, and there was no name common to both."

[3] the world μαθημα, from μαθειν, means in the first place "that which is learnt." in plato it is used in the general sense for any subject of study or instruction, but wit the tendency to restrict it to the studies now called mathemati¢s. by the time of aristotle, this restriction had become established.

huh. gotta love those pythagoreans.

weekend (research) relapse.

[from friday, 22 may 2009]

in other news, i´ve relapsed.

i had really intended to give up working on this one idea, but i´m too close. if it´s just a matter of resolving whether a particular w€ak-star 1imit is nonzero or not, then ..

.. i just can´t let it go,
not that easily.

at any rate, i´ve thought of a convoluted argument which might do it. then again, it´s likely to be gibberish which, in the frenzy of working, i thought was rigorous, and in it might contain an obvious error.

it wouldn´t be the first time, so i´m setting it aside for a few hours, maybe a day. later i´ll write it up and see where the weaknesses are.

(it´s not easy to account for one´s obsession, you know.)

.
.
.
[earlier this morning, sunday 24 may 2009]

the argument is still convoluted and there's still a crucial part that i haven't written up [1]. i have my doubts. in fact, it's the trickiest part and i have a lot of doubts.

"tricky" doesn't mean wrong .. not yet, anyway. the argument is just, well, weird. i'm partitioning sets in a measurable way [2] but because of some self-similarity in the method, i can't see the limiting geometry very clearly.

usually that's bad news,
and likely it means that i've gotten something wrong ..

.. but i haven't spotted an error yet.


[1] as for why these notes are not fully written up yet: today is my girlfriend's birthday. i wouldn't have gotten as far as i have, were it not for the fact that she also wanted to work on her own research, this morning!

[2] in fact, i'm following the standard tactic of splitting space into a "good" part and a "bad" part, relative to these integra1 estimates i'm setting up. let's hope that the bad part isn't actually that bad. \-:

math, math, everywhere.

earlier i found an empty table, fished out some scratch paper, and started to work on a particular research problem. a few minutes later, i overheard someone using mathematical terminology.

it never fails.

mind you, i happen to be in germany today ..
(a university library, to be specific)

.. and the terminology consisted of isolated words amidst animated german conversation. being incapable of forming complete german sentences, i doubt that this was my own imagination at work.

so if "dimensi0n" and "ve¢tor" and "orthogona1" are phonetically the same in german as in english, then i think the next table over consisted of students practicing their 1inear a1gebra.

anyway, work beckons.

who i was, what i thought.

i found this draft from 28 july 2007, but because it was never completed, i never posted it.

re-reading it, i still can't figure out how i wanted to end it. i wonder if i had some idea but thought it wasn't good enough. that's not uncommon: often i overthought what i wanted to write.

anyways, here it is.

---

life gives and it takes.

i've become better friends with this barista i know [1], but now it seems stranger (but not quite strange) to linger in that coffeehouse for hours, doing math in a little world of my own making.

i'm paid to work on mathematics,
with no demands on where or when,
no specifics for how long or to what effort.

it's as long as i get it done, something done .. and "done" is a relative term.

some semesters i teach, but that isn't so bad. it's something to complain a little about, and it gives you more reason to prefer research.

i know this is a good deal.
i know i'm fortunate, that this is a privilege.
..

guilt ..of some sort.. urged me to the office today. i couldn't get very much done; i'd have been more productive working at a coffeehouse or in my apartment.

but it was good to be amongst others who were also doing mathematics, thinking about problems they didn't know how to solve. i wouldn't understand the work they're doing, and they wouldn't understand mine. that's fine, but we share in the struggle,

the futility of a day's worth of thinking,
coming to naught,
with nothing to show for it but pages of paper,
full of scrawl,
destined for the recycle bin.

the perks of academia are fine things, but the creative process ..and problem solving is a creative process.. is a frustrating, almost condemning way of life.

when you're doing something purely because you want to do it, then it becomes painfully more upsetting when it's not working out.

i suppose what i mean is ..

[end of original draft]

---

epilogue: i don't disagree with my past self, which is a little .. disappointing. sure, it's only been 2 years, but i was hoping to have changed more than that, maybe become a bit wiser. probably i am just as naive as i was, then.

[1] some months later, she quit her barista job and was on the road to becoming a spanish teacher. when i last saw her, she was a student teacher, about to graduate college.

we went out a few times. i meant to call her in october (2007), but too much time had passed. this was shortly after the advisor passed away. the longer it was, the less i wanted to explain why it was so long since i called her.

part of me wanted, i think, to know someone that didn't know that juha died, and that i could be sure would speak to me as if i were just me. contrast this with an entire department that walked on eggshells around me, and handled me with kid gloves -- that's how it felt, at least.

ironic, though, that we never spoke since.

letting go, for now.

it's a maddening thing, to let go of an idea.

it hasn't failed yet,
but i haven't resolved it either.

i had such high hopes for it, too;
it would prove a few conjectures, some of which are not even mine!

---

at any rate, it's better that i set it aside, anyway. i've ignored latex'ing for longer than i should have.

papers never write themselves,
no matter how simple the ideas seem.

even in maths: no free lunch.

i must say, i really like the bana¢h-a1aog1u theorem. it's like a free lunch:

let's say that you're building a sequence of functions in some ref1exive Bana¢h space, and odds are that you've built a bounded sequence.

voila: weak-sτar c0nvergent sub-sequen¢e!

of course, there's no such thing as a free lunch [1].

i've spent some of the last few days checking whether a particular weak-* sub1imit (vectorfie1d) is, in fact, nonzero. the frustrating thing is that i still can't tell.

for the record, the space in question is L^∞(Rⁿ), but w.r.t. a Lebeg9ue singu1ar measure.

[1] though a common saying, this is an outright lie. i spent 18 years mooching off my parents, getting free lunches .. and essentially any other meal, for that matter. clothes, too.

thoughts, while on airplanes.

i have this idea;
it won't go away.

it doesn't fully work yet, but each time i check more details and i find it more believable.

---

the idea came to me while i was on a flight to dublin, ireland, on saturday.

as usual i was putting up a struggle, seeing how long i could resist the multimedia screen in front of me and instead, be productive and work out some maths.

i had just woken up -- not that i remember falling asleep -- and the idea came to me: i wonder what goes wrong when you try <this>?

i think i reserved judgment. rather, i scoffed inwardly and decided to test just a special case, if only to avoid excessively technical details. when that didn't fail, i yawned and decided to resolve the general case later.

three days later, it hasn't failed yet.

---

[1] they even have select tv shows available now. i nearly watched the season finale of austra1ia's nexτ t0p mode1.

new place, same (old) goals.

these last few days have been unproductive. i blame the anxieties of traveling and planning. hopefully now i will have a short break from that.

my european visit has begun: amsterdam, not for any conferences or collaborations, but because a pretty german woman invited me to stay for a while ..

.. and well, i couldn't resist.

there will be some traveling in between, but this will be a solitary part of my summer: writing and reading, mostly.

this past year i've retraced my thesis steps but with no avail (or new results, anyway). i wouldn't give it all up and start over completely ..

.. but perhaps it's time to step back and look at the theory again and look at similar or related theories. maybe a different point of view will work better, or suggest a new but related direction in which to work.

(put another way, i'm just saying that the same old ideas of mine never seem to work. best to have some new ideas.)

not the solution i had in mind ..

i know that i can teach 0DE, though not necessarily well;
i know that it's not my favorite area,

but i know that i'm willing to work with them.

---

there's one project that has sat on the sidelines since ..

[thinks]
.. 2004 or so.

there's an 0DE involved, which arises from the minimizer to a geometric extremal problem: a nonlinear, implicit, 2nd order 0DE. it's ugly. i don't know how to solve it analytically.

so yes: it's time to try numerical approximations.

doing so feels .. incomplete. having been trained as an analyst, maybe i've been conditioned to think that way. nonetheless, it's the only thing left to try.

after a few days in tampa with a colleague, maybe we can finally put this project to rest. there's still work to do, but we now see a way out.

de-powered.

a mathematician not having any good mathematical ideas is a little like a superhero without superpowers.

if you believe that you will regain your powers, at some point, then this period of time is an annoying plot device that must be suffered. there are trials and troubles, especially when watching fully-powered colleagues and strangers perform as usual.

if you don't regain your powers .. well, let's just say that the analogy doesn't extend that far.

sure, there are examples of some superheroes being depowered
-- take M-Day, for instance --

but permanent loss of powers? it's like expecting a main character in sci-fi to die and stay dead.

---

i'm at a loss for ideas lately. of half the projects i have in mind, nothing seems to be working.

fortunately, there are also some things that i've been meaning to write .. one of which i promised to too many people for too long a time. (never mind that it won't be a very good paper ..)

at any rate, maybe that will spur on some new directions.