the (Frustrated) Over-Analyst

short and sweet!

2012-02-01T03:32:00.000-05:00

awesome: the preprint so far is under 10 pages.
this rocks!

though the writing is tight, the discussion is not too spare. in fact, there are ..

* flips pages *
* counts *

.. at least 4 pages of set-up, in the event that the reader knows nothing about the subject. [1]

this suggests, of course, that the work is not deep .. which i admit is true. i prefer to think of it as light-hearted, myself.

put another way, this is no marathon or triathalon;
it's like your friendly neighborhood 5K or a sunday bike ride.

* * *

too often i feel like my work is overly technical and painful to read (as well as to write) and that the ideas are hard to state simply.

with this little paper, this is not the case. it really is a simple idea!

basically, i'm treating a metrιc question in euclιdean spaces, and exploiting the fact that smooth 1-fοrms are good objects with which to test 1-dimensiοnal currents. in the obsolete format of analogies from standardised tests, the intuition is that ..

curreηts : generalιzed surfaces :: distributiοns : generalιzed functiοns.

anyways, back to work: the preprint won't write itself, and it hardly seems like work at all ..

[1] for those of you who know me personally, i suppose this is in keeping with my manner. it seems that every time i give a talk, at least half of the contents are expositional.

winces

2012-01-28T12:56:00.000-05:00

from "25 things i learned from opening a bookstore" by j.l. sathre:

"15. If you open a store in a college town, and maybe even if you don't, you will find yourself as the main human contact for some strange and very socially awkward men who were science and math majors way back when. Be nice and talk to them, and ignore that their fly is open."

waiting for the inevitable news ..

2012-01-26T18:27:00.000-05:00

for the record, i'm writing this before one particular committee sends out their apology emails and selects those who proceed to campus interviews.

i don't think i'll make it that far. as for why ..

judging from how today's phone interview went, i think i sounded like a space-cadet theorist who couldn't manage to explain anything in layman's terms, much less his own results.

what i didn't expect was that the committee members present in the interview actually wanted to ask about my research. in fact, they had carefully read my research statement and wanted to get a clearer picture of what a measurable differentiable structure is!

*sighs*

the one time an audience wants to know about my work, and i go and fvck it up.

my digital self.

2012-01-25T10:52:00.000-05:00

no, i am not jεnna burkε from the bachelοr, for i am far uglier and i do maths. you can find her here instead [0].

this afternoon i sat down and thought about what kind of person i seem to be, based on what's on paper and what's available online [1].

online. the good news is that i don't seem to be a degenerate, and it looks pretty professional, with a few personal tidbits .. like road race times and a few bits of photography (allowed public).

the annoying thing is that my papers, what few of them, are scattered on both google and bing searches. this is particularly annoying, since even fewer of them show up on mathscinet.

underlying this, i realise that i'm not very prolific, so every little part counts .. /-:

paper. i browsed through my teaching and research statements and my vita today, and i cringed .. but at no one particular thing.

most of all, it's the research statement. the beginning is a good effort, but then it gets technical .. too technical. i wonder if anyone took the trouble to read it, because it's .. bloody technical.

it suffers from the same problem that i exhibit, while preparing and giving talks: i'm always worried that nobody has any idea of what i'm talking about [2]. it's one thing to be amongst colleagues; it's just that i've been given a lot of blank looks in my time ..

[0] L: if you know a way to invite jeηna to guest-blog, then let me know. otherwise these poor web-roamers deserve, at least, to be ferried quickly away from a mathematical minefield. q-:

[1] this isn't just an intellectual exercise in self-analysis. tomorrow is a phone interview, you see ..!

[2] maybe it's inevitable that we mathematicians just work on abstruse things. i don't ever expect anyone to appreciate the gory details of my proofs. on the other hand, i envy certain people ..such as those working on PDE.. because they don't need to spend a page or more explaining that their field actually exists ..!

the more things stay the same, the more they change.

2012-01-19T09:17:00.001-05:00

yes, another disclaimer: this is a mathematics blog. for jεnna burκe of the bachelοr, click here instead.

so tomorrow is the fourth [1] time that i'll be giving this particular talk, regarding differentιable structures on metrιc spaces. i'm hoping that it will be the last, for a little while.

call me paranoid, but i feel uneasy when i talk about the same thing too often;i worry that i'm not getting new ideas or making enough progress on my research.

some of my colleagues have attended this same talk several times, those poor souls .. but i think i see their motivations. i've done the same thing myself. at least for me, it's hard for me to learn something new in the short span of a talk, even if it is a 2-hour period, so often a good talk is worth hearing again.

once, this caused no end of confusion:

it was right after my ph.d. defense, and there were two back-to-back conferences. i was worried that i'd bore my colleagues to death by giving my thesis talk twice. so in one i stuck to my thesis, and in the other i discussed a completely different project in preparation.

makes sense, right?
besides, maybe people will be impressed at my versatility ..! [2]

of course, i was wrong. once i told my colleagues, they were alarmed and made a fuss about it. you see, both conferences held parallel sessions of contributed talks, and my talk was running opposite a well-known and prominent researcher in my field. so a lot of people, expecting to hear it next week, skipped my talk and went to his.

*sighs*
oh well; we're all young and stupid for once in our lives, right?

so i've learned my lesson .. somewhat. i still can't bring myself to give the exact same talk. besides, i've not yet given the version of the talk that i wanted!

you see, the driving mechanism behind the theorem consists of genuinely new techniques for dοubling metrιc spaces, which are inherently geοmetric in nature. in every past talk, i've spent too much time expositing and only hinted at this.

perhaps it's a risky proposition to discuss techniques of proof during a seminar, but i want to show the audience why the theorem is actually geοmetric.

the seminar is two hours long (2 x 45min), which must be enough time if i forgo the pleasantries. i think i will say a lot and write little, at least regarding the parts that should be "well-known."

at any rate, after this i'm retiring the talk for a while. besides, there's this new project that i want to talk about instead, in mid-february .. (-:

anyways, i'm rewriting part 2 of the talk now. back to work!

[1] it also depends on how you count. last march i talked about it at an AMS Special Session, but i had promised different things in the abstract and only advertised the theorem. besides, it was a weaker theorem then.

[2] thinking back, it was a particularly stupid decision on my part. i ended up writing and preparing two talks where i could have written one. to add to the context: this was also during the three weeks that my thesis corrections were due, and time was very much a factor ..!

distractions, distractions ..

2012-01-17T04:52:00.001-05:00

usual disclaimer: this blog has nothing to do with jεnna burkε or the tv series the bachelor. the direct link to her website is here.

i should stop having my in-box open, while at the office;
it seems that i never get anything done otherwise.

maybe i should start treating email like postal mail, and check it just once (or twice) a day, keep message replies serious and professional, and leave it alone.

* more on this later, but i have actual work to do now.
[added 18 january 2012]

i think most people are driven to check their email or social networks because they are curious: did someone message me?

perhaps it would help to remind myself:

for email: i'm not old enough or important enough yet to warrant too many messages of importance in one day. that said, the answer is almost always no and it's not important anyway, which is a good enough reason that i shouldn't bother ..

.. and get back to work.

for facebook: maybe i should allow myself license and make more creative constraints. for example, what if I allow myself 10 total minutes of facebook time at the office?

at first it seems like 10 minutes is very little; on the other hand, how long could it take, to log in, check a 1/2 page of updates, and then log out?

that's all that i every really do anyway.

the problem is that once i'm logged in, then i may as well click another link to see if it's interesting. the problem lies in the iteration.

as it happens, my watch has a chronometer, which means that i can actually measure this .. but would it be considered facebookkeeping..? (-:

sorry, not the bachelor; also, jet lag does not make for good talks.

2012-01-12T18:32:00.000-05:00

usual warning for the wayward: if you're looking for jeηηa burκe from the tv show the bachelor, then this is not the webpage you want. instead, the link to her blog is here.

well, by finland time it is now friday. i may have failed to meet my new year's resolution last week, but the spirit of it is still strong. oddly enough, i also have something to say.

(to be fair, i jotted this down last weekend in a cafe, but some thoughts are new-ish and worth sharing.)

on thursday afternoon (last week) i gave a talk. it wasn't the worst one i've given and i didn't go over my time limit, which is good. if i've learned anything, though ..

.. never, ever underestimate the afflictions that jet lag can bring. i'm starting to think that sleep medications and/or vitamin and mineral supplements may be worth the trouble.

earlier that morning i woke up at 3:30, despite feeling very tired at midnight. i couldn't go back to sleep, so i did what seemed like a good idea at the time ..

.. and went running outside for a few miles.

in retrospect, of course, it wasn't a good idea. at that point the fluffy, fine snow became weak ice and an unescapable slush; the way was slippery, dark, and uneasy at times.

as the speaker before me settled into the final slides of a proof sketch, i remember my pulse starting to quicken a little. though i've given a version of the talk before, i didn't feel wholly prepared. every other time was a 2-hour version on the chalkboard, at a pace that i could set myself.

the trouble with slides is that, in some sense, your fate is sealed.

the presentation is pre-meditated by the very choice of medium. unlike a chalkboard, you can't suddenly follow a natural flow; there's little (if any) room left for improvisation [1].

imagine asking the audience to wait a moment while you recompile your LaTeX! (-:

perhaps the best analogy is that of a stage production, in that you have to remember how the slides go. in other words, you have to remember your lines.

the jet lag, however, caused me so much drowsiness that i couldn't summon a panic, so i wasn't (visibly) nervous [2]. on the other hand, i unintentionally acquired too much material to discuss ..

.. yes, i mean "acquired."

that probably warrants an explanation, which is below.

i was also so drowsy that i couldn't remember the exact order and content of my slides, despite having prepared them a few nights before and revised them after my very-early-morning run.

the very last slide was somewhat technical and required a few subtle details. i became paranoid about forgetting the right ones ..

.. so during the talk, i started a painful descent into an information overload: at every slide, i'd follow the narrative and then give a series of additional details ..

.. sometimes examples and at other times, remarks about the proof ..

.. because, or so i reasoned:
if i give all the details, then i can't possibly miss the crucial ones!

*sighs*
to be fair, i was tired and jet-lagged and had no business giving a talk.

now that i think about it, it's a wonder that i actually ended on time .. rather, there's a very clear but unsavory reason:

upon reaching the final, technical slide, i realised that there wasn't enough time left to discuss it properly. so i cut my losses, clicked through it, and gave a verbal summary instead ..

.. all the while pointing at various terms of an estimate that were probably nonsensical to everyone but me.

so the problem was that half of the audience knew the background, could have arrived 10 minutes late into the talk, and not missed anything. i must have bored them terribly.

as for the other half, i couldn't really tell; i've always found it hard to read faces, especially for looks of comprehension.

epilogue. a friend who was in the audience later told me that the classroom was full when i gave my talk. some people were standing in the back, in fact. this only made me feel uneasy about the whole thing. \-:

discussing these thoughts later,

on the plus side, a colleague who gave a later talk was spared the pain of giving too many definitions, because i had already covered them. (-:

[1] one can very easily argue the opposite, of course. some of my colleagues have pointed out that it can be equally hard to remember the whole talk in its entirety, even with written notes as an aid. good, polished slides are therefore like a guarantee that something polished gets delivered to the audience.

[2] sometimes i think that years of running 5K races and 800m dashes in high school track-&-field has normalised the onset of adrenaline, for me.

no, this blog has nothing to do with "the bachelor" / choosing another journal ..

2012-01-09T15:27:00.001-05:00

disclaimer: if you are looking for the tv show "the bachelor" then sorry: you clicked on the wrong link. instead, jεnna burkε's blοg can be found at this direct link.

..
..
.. say: if you haven't clicked on the link yet (and left this website for good) then can i ask a small favor?

run the search on "οveranalyst blοg" again .. actually, a few more times,
click on ms. burkε's link each time.

it's not that i want her to gain popularity;

it's not that i want her to lose, either.

(to be honest, i don't even watch the show.)

you see, the sooner her blοg is the #1 hit for the web search above, the fewer people will be frustrated at stumbling upon my blοg and not hers.

granted, this site is called the frustratεd οver-analyst, but i've only ever meant to blog about my own troubles and not cause more troubles for others. the last thing i want is for this blοg to go the way of the blues [1] and it was never meant to be a huge, popular website anyway.

that said, i'm going to write about maths now.

i have an aversion to the professional part of being a mathematician. at the very least, i'm not particularly good at it. it's probably the reason, in fact, why i became an academic in the first place: avoiding reality and the rat race has always been part of my lifestyle.

so part of the job of research is to submit and publish scholarly articles in peer-reviewed journals .. which is part of that aversion. yes, it's a necessary task, but it still bothers me each time.

regarding talks at conferences [2] i don't mind giving those, even if i'm similarly inept at that skill too. that seems to me a more reasonable errand because:

i can see the audience, occasionally gauge their reaction;
they wouldn't be there if they weren't somewhat interested [3] or at least, are willing to take a chance;
often i know many people in the audience, and some of them are friends.

as for figuring out where one of my preprints belongs .. how should i know? i'm still patting myself on the back for having finished a complete, readable draft!

that said, i'm not a natural writer, either .. which makes me wonder, sometimes, how i became a mathematician in the first place.

but i digress:

today i spent a great deal of the afternoon, looking up citation numbers, submission timelines, and journals my colleagues have published, in order to get a sense of what is fitting.

before leaving the office, i narrowed it down to 2-3 choices, but i'm still ambivalent.

to be fair, i already chose a journal some months ago. in fact, i didn't have to think too hard about it. in my mind it was exactly the right one ..

.. and they said no.

*sighs*

on the other hand, it was the fastest rejection that i've ever received .. and the kindest, too. maybe it's standard lip-service, but all the reviewers had good things to say about my submission, to the point where ..

.. if i had to describe it as a break-up in a relationship,
then it sounds like a "it's not you, it's me" sort of thing:

in their words:

we .. cannot publish your paper, because the level of excitement it generates is not high enough (although the level of interest is very solid) ..

.. we have to be extremely harsh in our selection. And we are aware that we may be making a mistake in this case.

barring the paper's acceptance, i don't think i could have asked for more than that.

*sighs*

perhaps i just missed the cut and they needed to reject it, along with many other good articles that now have to find new homes .. it's what makes me feel better, anyway.

oh well: time to pick another journal.
eeeny, meeny, miny .. (-:

[1] as bleeding gums murphy would say: "the blues isn't about making yourself feel better; it's about making other people feel worse!"

[2] .. which reminds me: i'll discuss how my most recent talk went, in a later post. stay tuned!

[3] in the case of large conferences with several parallel sessions (of contributed talks) it could be that, to the average participant, my title/abstract was the least uninteresting in that time slot, and it would feel too wrong to not attend any of the talks.

not the website you were looking for?

2012-01-03T17:16:00.002-05:00

according to my stats tracker, i think a lot of you recently came across my blog by accident.

trust me, i know:

i've been keeping this blog for a few years,
and it's never gotten 8000+ hits in one day.

so if you were looking for jeηηa burkε's blog @ theοveranalyst.net,
then here's the direct link.

otherwise, could someone tell me who is the "οver aηalyst" that everyone is looking for, on google?

new year's resolutions, of a mathematical nature.

2012-01-03T03:31:00.002-05:00

learn about Riccι curvature. this is beginning to look like one of those subjects that i can't get away with not knowing. it seems to come up every year or so, if only as a special case of a new phenomenon in metrιc spaces.

for example, if you believe this preprint, then there may be some hope for a second-order theory of sobοlev spaces, which sounds really cool ..!
~~write a paper (or two) about geοmetric measure theοry~~. this is partly to convince people (including myself) that i know something about GMT ..

.. though, to be fair, it will probably be from a metric space viewpoint.

speaking of which, i have something written now in the form of notes, but the result isn't good enough yet. now that i think about it ..

.. edited resolution: if i don't get a better idea by vappu (1 may), then i'm submitting a final form of my write-up.
get a new job [1]. this isn't completely within my control, of course, but then again, nothing really is.

my family has been asking where, and i've been telling them that i don't know. it's not like it's up to me, right ?.. and it's a dangerous enough thing to want something, anyway.
..
..
i guess it doesn't hurt to indicate a small preference: it would be nice to stay in helsinki for two years, maybe three .. if only because i like it here. my colleagues are fine people, the city is comfortable [2] and it would be nice not to have to pack up and move again.

(ye gods, i hate moving.)

sometimes i watch my colleagues (the tenured and tenure-track ones) and they always seem busy with administrative tasks. it's not that i'd say no to a decent permanent job, but:
- i may not be that good at research, but that doesn't mean that i want to stop it;
  
  there are still a few ideas that i want to work out,
  a few problems that take a few years to solve ..
  
  .. that is, if i'm the one to solve them.
- i still don't feel like i know anything. right now i couldn't be anyone's advisor; i don't know my field well enough to think of ph.d. problems for the next generation ..
  
  .. so a few more years of experience would be helpful.
*new* work out a new proof every month. [3] i browse through the arXιv fairly often (every monday and thursday) but apart from titles, abstracts, and introductions, i don't really get much out of it ... which is my own fault.

lately i feel like i've been using the same methods, over and over again. according to gian-carlo rota, every mathematician has only a few tricks .. which sounds dooming.

maybe it's worth seeing if rota was right or wrong. if anything, it can't hurt to read more.
update this blog ~~every friday~~ at least once a week. lately my posts have been sparse, though i can't really tell you why.

maybe it's the inevitable effect of having kept it up for so long, and that there are fewer things worth sharing. maybe mathematics is getting less frustrating for me ..

(laughs)
well, a boy can dream, right?

i could use an easy resolution, anyway. besides, there are always the resolutions above to write about; maybe doing so will actually convince me to fulfill them .. (-:

[1] to those not in the know: i have nothing against my current job. it's great, actually, but the contract is for one year only.

[2] it would help, of course, if i finally sat down and learned some finnish. fortunately, this is a list of mathematical resolutions, so i can leave it off the list ..! (-:

[3] originally it was going to be one proof per week, but then realised that it was too unrealistic. i might as well start off small, first.

mοnty hall as a meme.

2011-12-26T22:41:00.001-05:00

earlier tonight my brother-in-law asked me about the mοnty hall problem and why his solution wasn't the correct one. so i took one of my credit card offer envelopes and we started drawing diagrams ..

.. and just as he finally accepted the answer, my brother walks over and asks me what we're talking about .. so we tell him the problem, he thinks about it, and then we start drawing on the refrigerator ..

.. and he, too, accepts the answer eventually.

now, though, my dad is starting to look at the fridge ..
[sighs]

holidays.

2011-12-21T07:23:00.000-05:00

my flight is delayed and there's free wifi in helsinki airport:
i suppose that certain outcomes are therefore inevitable.

thinking it through, these last few weeks haven't been very efficient;
i blame the lack of structure in my daily life:

without the constraint of teaching (and hence regularly scheduled blocks of time beyond my control) i've found my workday wide open, apart from meeting colleagues everyday for lunch.

since the start of december the seminars have not been meeting regularly, and the lectures i attended at universιty of helsinkι ended .. two weeks ago?

i can't remember anymore.
the days are now a blur.

the holidays are beginning, and my family swells: with my sisters come their boyfriend and fiancee in tow: good men, though. they treat my sisters well, which is the important thing.

there will be a lot of people in the house, so i suppose that i won't get any work done until after new year's ..

.. on the other hand, my laptop has a 8-hour battery:
i wonder if i can write up an entire preprint during the flight home!

[thinks]

helsinki to reykjavik,
reykjavik to home ..

.. let's see, shall we? (-:

epilogue (4 jan 2012). i never got close to finishing that preprint.

the introduction was what balked me: every time i tried to motivate the result, i thought about the stronger result that i wanted to prove, and couldn't convince myself to write any further.

an update, 3-4 months in.

2011-12-19T17:24:00.000-05:00

say, is anyone still reading this?
..

it's been a while, hasn't it?

somehow it's become my habit NOT to write in this blog. not a lot's been happening, i guess: i applied to a few more jobs, written up a close-to-complete set of notes [1] on some recent results, and started getting back to a few research collaborations.

when i think about it, i do a lot of worrying and complaining here. i suppose that's natural, for two reasons:

it's year number four since i defended my dissertation, and i still don't know if i have a future in mathematics. part of me wants to blame the economy .. but most of me acknowledges that i haven't been doing enough, my results not particularly noteworthy.

i don't know. i'm trying, but often i don't see any progress, and not much hope. [2]
apart from the future and general malaise, the present day is going fairly well. aside from the climate around here [2] my complaints are few, and i think the freedom from teaching has done me some good.

i feel like i now understand more about mathematics, and become a better researcher. i haven't had mathematician's block [3] in a while: the ideas are still there, some are taking shape.

[1] -- i think i referred to it as a "preprint" in recent memory -- if not here, then in an actual conversation with colleagues. in truth, it used to be a preprint, but while i was checking some details, some new ideas come up. then i had my doubts if anything worked .. but now i think i can see why the proof should and does work.

the short version: i'm pretty sure i proved some new special cases of the Ambrοsio-Kirchheιm conjecture, and reduced the full conjecture to a different (still unsolved) special case.

[2] -- that could be the seasonal affective disorder talking. today it became bright at around 9:30am and dark again, near 3pm. i feel tired most of the time, except late at night. i haven't gotten around to taking any vitamin d supplements yet .. maybe i should do so.

[3] -- it's kind of like writer's block. instead of not being inspired to write, it would be a combination of not getting new ideas and none of the old ideas working.

call me old-fashioned, but ..

2011-12-11T15:45:00.001-05:00

sometimes i think that my work is superficially relevant to this setting of metric spaces and analysis on them.

deep down, i am just a euclidean geometer,
working with balls, lines, planes ..

.. and, of course, the occasional Lipschιtz function. (-:

"the horror comes in reality from the mathematical aspect of the event."

2011-11-28T13:23:00.001-05:00

evidently, albert camus believed in a deterministic reality, and probably disliked maths. maybe someone should have told him about quantum mechanics. \-:

as for the title, it's lifted from camus's long essay the myth of sisyphus --- specifically, from the section an absurd reasoning.

to get a sense of this book, the first sentences read:

"There is but one truly serious philosophical problem, and that is suicide. judging whether life is or is not worth living amounts to answering the fundamental question of philosophy. all the rest -- whether or not the world has three dimensions, whether the mind has nine or twelve categories -- comes afterwards."

evidently camus was destined to be a philosopher, not a geometer [-1]. q-:

as for the excerpt that included the title, here's the context:

"here it is barely possible to speak of the experience of others' deaths. it is a substitute, an illusion, and it never quite convinces us. That melancholy convention cannot be persuasive. [0]

"if time frightens us, this is because it works out the problem and the solution comes afterward. all the pretty speeches about the soul will have their contrary [1] convincingly proved, at least for a time. from this inert body on which a slap makes no mark, the soul has disappeared.

"this elementary and definitive aspect of the adventure constitutes the absurd feeling. under the fatal lighting of that destiny, its uselessness becomes evident. no code of ethics and no effort are justifiable a priori [2] in the face of the cruel mathematics that command our condition."

ah, the cruel mathematics, indeed! (-:

[-1] the more i travel, the more the term "geometer" seems rare. i thought it was the standard name for someone who studies geometry of some kind, but acquaintances tend to refer to me as a geometrician.

[0] this entire excerpt is from part of one big paragraph: the spacing has been included for easier reading on a computer screen.

[1] i think "contrary" is meant in the sense of "all that is contrary (to a particular thing)" but since i don't have the original french version in front of me, this is mere speculation. as a mathematician, i would still say "negation."

[2] his exact words, none of mine.

today i found out that i am an author.

2011-11-26T10:49:00.001-05:00

every so often, i run a goοgle search on myself to see if i've inadvertently done something stupid (and if so, deciding on how to erase it from existence).

this time around, something odd came up.

my thesis is available on amazοn.com;
there's even a used copy available! (-:

that said, is this happening to anyone else, out there?

it's one thing to download preprints from the arχiv and mirror them somewhere else, but .. come on: this is just my thesis ..

so if you want a copy,
then just email me and ask ..

(.. or at least until someone threatens to sue me.)

for some of us, certain LaTeX symbols are un-natural.

2011-11-24T16:17:00.001-05:00

yesterday, while writing notes to a colleague, i happened to use the LaTeΧ symbol \bigoplus \bigotimes, which i have only seen used for tensor products in algebra.

as an analyst, i became slightly ashamed of myself for a while.

to the algebraists: it's nothing personal.

think of it this way: if you were giving a talk in your friend's seminar, and in the middle of the talk, suddenly assumed "let ε > 0" .. i wouldn't think that it would go particularly well. (-:

in which some constructions are "natural" ..

2011-11-18T15:22:00.001-05:00

perhaps all of you knew already, but today i learned that "magenta ain't a colour" in the sense that it cannot be created with a single wavelength from the visible light spectrum.

however, since the human eye processes colors in terms of opposites, so magenta appears as the complementary color to green.

initially i nodded,
thought it was a neat factoid,
and went back to wasting time on the internet ..

.. but then it occurred to me:
this is exactly the 1-point compactificatiοn of the real line [1]!

despite the abstract nature of lοcally cοmpact Hausdοrff spaces, this is a concrete example that some ideas in tοpology sit naturally in the real world ..

this is just as cool as when i learned that the video game "asteroids" is played on a torus ..

[1] yes, for the record the visible spectrum is a bounded interval, not the whole real line .. q-:

on a wholly unrelated note ..

2011-11-17T04:18:00.001-05:00

in my work notes today i'm using the symbol Ξ, or uppercase ξ, a lot.

not having much practice with this greek letter,
oftentimes it looks either like:

an equivalence symbol (or \equiv in LaTeX)
the chinese symbol for three.

either way, it just feels .. ridiculous?

boredom is a luxury.

2011-11-16T16:46:00.001-05:00

the last two months have been quiet. apart from daily life at the office and a few regular seminars, not much happens in my life that's worth blogging about.

in other words, it's been awesome.

i can concentrate on chasing down ideas, some of them infeasible, a few of them interesting. i can pursue a line of thought for days at a time, see where it goes.

i can latex for a full day, hack out the details in a single uninterrupted vision, identify the parts that make no sense, work through or around them.

telling you how it's really going -- that requires a level of technical detail that i'd rather not see on a blog, myself.

this is not to say that all of my ideas are working .. but that i feel productive. i haven't felt that way in a long time.

after & before: part two.

2011-11-11T02:37:00.001-05:00

i can't remember when i first drafted this, but it was probably the day before my second talk (at TKK) which was .. one and a half weeks ago already.

i'm uneasy about most topics.

when i give talks about generalizations of the schοenflies problem or Sobolev extensiοn domains or ΡDEs, i get quite anxious, and with good reason. [0]

these are well-established topics with much history to them. try as i might, i always seem to be ignorant of some significant part of the literature .. and in particular, what types of theorems or proofs are "standard." [1]

when i talk about my "geοmetric" work, however .. and by this i mean this stuff relating to (metric) derivatiοns .. then the situation changes completely.

it would be wrong to say that i am an expert on the subject, if only because there are plenty of things that i don't understand about them. it's safe to say, however, that nobody is an expert, either!

i guess that makes me a little like socrates [2] .. (-:

it's not that i insist on knowing more than everyone else in the room; i've resigned myself to the fact that there is always someone smarter or more familiar with the literature.

rather, it's the matter that if i make a mistake, then the fallout is minor. like everyone else, i hate it when people criticise me by my mistakes .. even if i do deserve it, sometimes.

there's another aspect about derivatiοns worth mentioning, as the topic of a talk:

if this is a subject little-known, then my role can be put to good use. there are few references on the subject, so i can point out how the theory works. doing so, perhaps i can convince others that the theory can be put to good use.

put one way, a beautiful theory is like a night sky: a canvas of stars that is well-arranged by Nature's Benevolence, distant but worthy of contemplation.

a useful theory, however, is more like a clutter of heavy stones in a field, perhaps meteors comedown to earth, long ago. stone, despite being crude and ugly matter, are fodder for tools and building material. from them we form homes and towns, societies and civilizations. [3]

i don't believe that mathematics is always beautiful.

look hard enough at the details of a theory, and it becomes hard to see any beauty in them. sometimes i wonder if the notion of beauty is inherently retrospective .. in the intellectual sense, anyway.

[0] there was this one talk i gave about manifolds with non-negatιve Riccι curνature, some years ago. my plan was to learn a setting in which you can actually prove the validity of a Pοincaré inequality (vs. most of the time it is taken as a hypothesis). it wasn't until i started discussing the proof, that the audience pointed out an error to me .. which lay in the textbook i used, but the fact remains that i missed it completely .. [sighs]

[1] oddly enough, most of my non-geοmetric work is collaborative, especially the stuff relating to ΡDEs and Sobοlev spaces. This cannot be a mere coincidence.. \-:

[2] according to legend, a man from athens traveled to see the oracle at delphi and asked: "who is the wisest man in all of athens?" unlike the usual cryptic answers, the oracle simply answered, "socrates."

upon hearing the news, socrates became disillusioned instead of delighted, because he couldn't believe that he knew more than every other athenian. so began a long and systematic inquiry: socrates found experts on every subject he could think of and began to debate with them, and after a while each expert admitted that he, too, knew nothing.

this put socrates into a further melancholy. he discovered that nobody knew anything ..

.. until one day, while taking a stroll, it occurred to socrates: nobody knew anything, but he himself knew that nobody knew anything. this meant that he indeed knew something. since everyone else he encountered knew nothing, it meant that he was the wisest man in athens, after all ..

[3] i've been reading thoreau's walking lately ..

a little too easy (updated)

2011-11-08T01:26:00.000-05:00

[from evening of 6 Nov 2011]

i'm getting suspicious.

last week i came up with a proof to a particular case of a conjecture,
but it was shorter than i expected it to be.

over the weekend, i thought i spotted an error,
but yesterday i checked it again, and was easily fixed.

i'm still suspicious.

if it were this easy,
then someone would have written it up by now ..

.. so it's time to take a long, hard look at it,
see where the error really is.

[updated: morning of 7 Nov 2011]

i'm LaTeXing the argument, as we speak. there are subtleties to handle, but no big problems yet.

in fact, the biggest problem is handling my keyboard:

in regards to LaTeX, the annoying thing about european keyboard layouts is that the dollar sign $ isn't as accessible as in a US keyboard layout (for obvious reasons).

in particular, to type a "$" the command is assymmetric:

it's [Alt Gr] + [4] on a finnish keyboard, as opposed to [Shift] + [4] in the US, so i have to consciously type the command with my right hand.

this wouldn't be a big deal with anything else, but i use dollar signs all the time, in order to pass to math mode.

so i've switched to the US keyboard layout .. with the problem that the labels on the keys are still in the Finnish layout.

luckily, i've been typing for enough years that i don't consciously think about the keys .. but every so often, i forget:

where's the asterisk key again? is it [Shift] + [8] ..?

after & before: part one.

2011-11-01T15:10:00.000-04:00

so i gave another talk today, this time in my department. the topic was the same .. regarding differentiability in a certain class of metric spaces.

originally i wrote down some thoughts while writing the talk, but right now it seems more natural to talk about how it went.

the good news is that it was well-received:
all signs point to that, at any rate.

the bad news is that it didn't go the way that i wanted. something didn't feel right when i was talking, and i'm almost sure that i went too fast. there were enough clues, i think, as to pick up the basic trail of the argument ..

.. but, admittedly, i was often looking at my watch, seeing if there was still enough time.

so: i prepared too much.
it actually reminds me of a film called wonder boys [1] ..

[link to imdb entry]

the basic premise is that the main character, a literature prof, cannot finish the book that he's writing: the pages number in the thousands. at one point, one of his students reads the draft without his permission, gives her critique. the message is essentially this:

good writing is about making choices. this thing .. it doesn't look like you made any choices. you follow every lead, the tangent of every story of every character ..

the fault of the talk is that i made a similar mistake. what i had in mind was (essentially) a complete discussion, but even in 2-hour finnish style seminars, there still isn't sufficient time.

put otherwise, it's not that i didn't have enough time;
rather, the talk just wasn't well-organised, the message not appropriately succinct [2].

epilogue. afterwards, a few people came and asked a few questions.

the striking thing is that, despite the metric space setting of the discussion, i was asked instead about Lie groups, as well as Riemannian manifolds ..!

sometimes i wonder whether metric spaces are really a good field of analysis in which to work. maybe i should concentrate on more familiar settings .. if anything, i might be able to communicate with more mathematicians ..

[shrugs]

[1] interestingly enough, it's set in pittsburgh. the story is modest, but it's told well.

[2] "i didn't have time to write you a short letter, so i wrote a long one instead .." ~ mark twain.

.. provided that ε ~ 5.6 x 107 ..

2011-10-31T10:37:00.002-04:00

.. there are now 7 billion people living on earth today,
according to NPR and other sources, anyway. (-:

jyväskylä: then and now.

2011-10-28T18:04:00.001-04:00

so it's been close to a week in jyväskylä again. i came with the expectation of finishing a joint project, but instead i started two new ones.

i'm glad for the work, but i worry about the future.

at some point i'm bound to screw up the juggling routine,break another promise or two to a collaborator.

in some sense, jyväskylä was a beginning for me. i met my first co-author there, as well as two future members of my doctoral thesis committee .. a little more than 8 years ago?

that was also when i met the advisor for the first time.

i was fresh out of my undergraduate years, and we didn't talk much. it was only later when i'd get to know him better.

in a different timeline, where things may have happened differently, i would have spent my final semester of grad school in jyväskylä.

that was the plan, anyway: the advisor was planning a sabbatical there with his family, and we were invited as visiting scholars. it would have been a chance, i think, for us students to start building our own research connections.

some of you know that it didn't happen that way [1] ..

.. but i suppose things worked out on the research connections front, the way the advisor would have liked.

[1] it's almost been 4 years now, hasn't it?

before, then, and after.

2011-10-27T14:05:00.003-04:00

tuesday: i don't often say this, but i have a good feeling about this talk. it might even be fun!

wednesday afternoon: ye gods. this is the analysιs seminar, right? it looks like the audience for a colloquium. i've seen entire maths departments that are smaller than this ..

.. oh well: here goes nothing ..!

wednesday evening: well, maybe 4 pages per hour [1] was a bit ambitious, especially for a chalkboard talk. it was only after writing out the maths in chalk that i realised how technical the discussion really was.

still, i think that the talk was received well, and colleagues of mine had good things to say about the result.

when i was younger, i used to be deathly nervous of questions. ultimately, i realised: if people are asking questions, then it means that they are listening, even willing to understand.

so questions are a good sign, after all. besides, if you can't address the questions of the audience, then you probably didn't prepare enough for it ..

.. so: fair's fair. i've had my share of that. q-:

[1] read: "1 hour" = 45 minutes

2011-10-26T16:44:00.000-04:00

if you waste a sufficient amount of time on the internet every day (like me) then you will have encountered this image/meme:

well, it wasn't the first word that i saw, but of those first four, one of them was "cotalent" ..

[see 1 line below the meridian line]

.. which sounds daunting:

a talent is a quality that, when presented, improves the impression given by the presenting person.

co-talent, in contrast, must mean something in the opposite direction .. say, a quality that, when presented, harms the impression given by the presenting person! (-:

weekends, obsessions.

2011-10-24T16:24:00.000-04:00

on saturday an acquaintance asked me if i worked on weekends. not wanting to sound like a workaholic, i told her that:

"unless it's really busy,
i only work weekends when i feel like it."

she seemed satisfied that i wasn't a crazy oddball.
for my own part, i was relieved at not having to lie .. but only to stretch the truth a little.

the truth is that there are a handful of research problems that are essentially fixtures of my life.

when i wake up in the mornings, they may as well be waiting for me at the kitchen table. odds are good that if i have nothing else planned, then i'll start thinking about them over my first cup of coffee.

i don't count them as "work" because it's easier to puzzle over them than not. on the other hand, they're purely speculative, and i don't expect ever to make progress on them .. though i still try and i hope that, eventually, i make a small step forward ..

they're not the problems that one should think often about. they're fun, just like how ice cream is tasty and fun .. and still not good for you. it took me a long time .. about 3 years of a first postdoc .. to realise that it's important to have several problems at a time and of varying "levels of difficulty."

even now, i have a tough time not dropping everything entirely and running exclusively after one problem.

it's fine to like mathematics; i wouldn't be in this business if i didn't. being a working mathematician is another thing entirely, though. there's a rat race, just like for the rest of the world, and it helps to have something to show for your efforts.

so on weekends, i tend to work half-days on problems that border on obsession;
i'd work the full day if i didn't get tired and sick of repeated failure.

it sounds awful, when i put it that way. for the same reasons, though, i really like rock climbing. when you finally finish a route and reach the top, it feels like you're king of the world ..

.. which is somewhat crazy, but give me a break: i'm a mathematician. (-:

(-:

2011-10-17T08:15:00.000-04:00

".. it is dreadfully boring to show that this formula defines a linear map T_F from the space of sιmple functions of the above form into X, and we leave this as an exercise for masochists .."

from Vectοr Μeasures (p.6) by J. Dιestel and J.J. UhΙ.

once a student, always a student.

2011-10-16T17:25:00.003-04:00

[this was written last week on monday;
for some reason i forgot to post it until now.]

right now i'm browsing two sets of notes. they're based on lectures that the advisor gave in the spring of 1999 .. back when Cheegεr's theorem had just come out.

looking through the topics they covered and the details, it must have been an amazing course .. maybe something worth traveling through time to have sat through [1].

i remember attending a course similar in spirit, in spring 2004, but i think i was far too young to appreciate those topics, back then ..

.. [sighs]

then again, you're never too old to learn.

lately i've been attending lectures on geοmetric measure theory, held at the uniνersity of helsιnki. so far they have been a lot of fun .. but then again:

any course can be a lot of fun,
if you don't have to do the coursework! q-:

one of my weaknesses is that i was exposed first to a formulation of geometric measure theory on metric spaces. as a result, i feel quite ignorant about how powerful the Euclidean theory actually is and what results are available to .. say, characterise certain classes of currents.

...
so, yeah: i'm thinking about a conjecture again. i doubt i'll make any progress on it, but it's something to think about it, over breakfast and coffee.

[1] not that it would be my first choice if i were given only one opportunity to travel back in time. on the other hand, i don't think i'd have affected things enough to cause an alternate timeline to occur, or anything ..

(-:

2011-10-11T12:56:00.001-04:00

"how engineers tip"

gluttony.

2011-10-11T08:08:00.001-04:00

this is going to sound ridiculous, but i'll say it anyway: the best thing about sequeηces is that you can always take further subsequeηces.

it's like a never-ending buffet at a restaurant;

at some point you become full, you have what you want, and you stop going back, but there is always something that you like best ..

similarly, sometimes one starts out with a less-than-optimal sequence of measurabΙe functions, but then the optimism of measurε theοry and functiοnal anaΙysis kicks in:

a sequeηce is bounded, so maybe i can take a weakΙy cοnvergent subsequeηce (i.e. Baηach-Alaοglu) ..

but i don't like the word "weak," so maybe i'll just mix up the terms a bit, and now i get nοrm cοnvergence (or Μazur's Ιemma) ..

sometimes i feel like a mathematical glutton. (-:

on why some bibliography styles are easier on me .. even obscenely so.

2011-10-06T04:01:00.001-04:00

this is probably an issue of the (mathematically) young [1], but the bibliography is a crucial part of many papers that i pick up and browse.

in fact, most of the time i realise that i shouldn't be reading that very paper .. but instead, the earlier papers that the author(s) cite.

as a general principle: the first paper in a topic contains the purest form of the main, recurring idea. it may not be executed in the most efficient, general, or powerful way possible, but that's not the point:

the point is to understand how it works first,
and then to see what you can do with it ..

that said, it's very helpful for me to identify right away, in the text of the paper, which of the references contains the lemma that i just read. it's a recurring annoyance for me to remember which paper is marked [11], and usually it means flipping/scrolling to the last page and matching up the citation number to the author/article.

it affords me more ease of thought to read [Gro96] instead and immediately remember:

right! Gromov's gruebleen book.

on the other hand, this can cause a few .. quirks.

if you use BibTeX with the alpha style, then for single-author papers it uses the first three letters of the author's name as an identifier.

so Gromov's Metric Structures for Riemannian and Non-Riemannian spaces is [Gro96], and Cheeger's 1999 GAFA paper is [Che99].

it's an unfortunate case of typesetting, though, when i'm citing Patrice Assouad's paper, regarding embeddings of doubling spaces ..

.. so does anyone know how to fix the standard BibTeX display formatting? (-:

[1] one way to test if you're still "young" is how often you agree that a particular argument is "standard." as for myself, i still have a lot of reading .. and learning, to do.

where there is a mathematician, there is probably a cafe.

2011-10-03T08:42:00.000-04:00

on sunday afternoon i was working at a cafe called la torrefazione earlier .. which, by the way, has great brewed coffee.

in case you were wondering about the notepad, i like writing with the longer edge on the bottom, despite the pages being lined in the orthogonal direction.

the page is otherwise too short:

the side of my hand would be on the table, not on the pad ... which in turn makes forming letters slightly harder to get right.

since the letters are small (and it would be nice to fit in something nontrivial, per page), the hand control becomes that much more pronounced ..

as for what i thought about:

in the late morning i thought about metric currents (specifically, 1-d currents in the plane) but couldn't get anywhere. it's the same obstruction, time and time again: i just don't understand BV functions well enough, at least from a geometric perspective.

as for the afternoon, that was directed towards analysιs of PDEs. i think i convinced myself that a particular strategy is NOT doomed [1] and that i should pursue it further. to do the details, however, means a much closer look at a proof ..

.. and i left my papers and computer at home. oh well: there's always the actual workweek to get work done, right? q-:

[1] the technique i have in mind isn't too strong, in the sense that it won't accidentally "prove" something that is not true in general.

on why computers are not as complicated as you think.

2011-10-01T03:25:00.001-04:00

with apologies to my friends who have heard me rant about this before, i nevertheless think that computers are fundamentally 1-dimensional, even painfully so.

those of you who know what a turing machine is [1] will think the idea is perfectly natural. as for the rest of you, you will have to suffer my usual amount of complaint ..

if you use an old-school text editor to compile LaTeX like me [2], then the cursor blinks at you when you stop typing.it's not as if it ever stopped blinking ..

[thinks]
.. well, that i can't be sure, but it certainly blinks when you stop typing.

in fact, if you stare at it long enough, then it looks like someone is winking at you, over and over .. as if laughing at you, knowing that you don't know what to write or how to explain yourself, even though you know how to prove it mathematically forwards, backwards, and sideways.

well, fvck you, computer.

so i found myself, having to start a research plan which is the bulk of an Academy of Finland grant application. i had ideas, but they were without structure at all, and the combination of a blank screen and a winking, subversive cursor was too much for me ..

.. so, 2 weeks ago, i pressed the off button.

a piece of paper is a pretty intuitive thing to us humans, if only because we are familiar with it from school. if you think about it as the analog version of a potential digital device .. an advanced version of a tablet or an iPad, perhaps .. then it is an incredibly flexible interface.for one thing, i can write anywhere on a page, and organize the information later.

i may start with a paragraph of text,but being stuck, start drawing a diagram in one corner,then realise that i should estimate something, and write out a computation inanother corner,decide that, having understood what i mean to express mathematically, then i'll return to finishing the paragraph that i started earlier.

the reason why this is harder on a computer is twofold:

alphabetic languages are inherently linear, which means that you must impose an order on your thoughts .. at least, if you're unwilling to cut-&-paste or delete. not every thought or series of thoughts of mine is orderly, which makes typing out ideas hard.
the human eye is a wonder of processing power. perhaps in the near future, image processing will advance to such an extent that a computer can see and recognise images as well as a human .. but until then, if i doodle something, then my friend can see that i was thinking abot linear homotopy. a computer has no such flexibility.

at any rate, i found that most of my recent grant application was initially written in ink and longhand. for me, it was far more helpful to include an additional dimension's worth of freedom, if only to fully express myself.

[1] and, of course, believe in the church-turing hypothesis.

i say believe, because -- let's face it: you can never prove a statement like "everything computable is Turing computable." you can only disprove it, by constructing a fundamentally different architecture that is backwards compatible with standard digital software.

honestly, when i first heard the church-turing hypothesis, i was quite impressed. then again, i was quite young then and had no taste. as for now, i view it as the computer science version of a peano curve .. in the sense that a ticker-tape (i.e. a real line) can be mapped injectively to a 2-D microchip (i.e. the coordinate plane).in that sense, it's not clear to me if there is anything mathematically interesting in applied computer science. theoretical computer science, from what little i've seen, looks quite cool. any science that will take metric-space embedding problems seriously is a fine science, in my book!

[thumbs up]
well, at least it's better than intelligent design, which seems to me utterly untestable.

[2] i was visiting a colleague once, and stopped by departmental tea. like the hapless nerds that we are, we started talking about graphics via latex. one guy was complaining about this one package and i had no idea what it was, so i asked him to explain it to me (how i can download it, etc). when he asked me what i used, i told him "pstricks" and then he paled.

apparently i'm a masochist, by mathematical standards and otherwise.

bombast (or: i really need to finish this grant)

2011-09-28T08:12:00.000-04:00

i wrote this earlier in the morning, more because i wanted to get it out of my thoughts and move to more realistic matters.

as you can see, the wording is a bit .. grandiose.

"The path of research is not meant simply as one of self-discovery, but one where hard-won insights are shared with others, in hopes that the body of knowledge is further extended and refined.
in that sense we researchers seek a kind of diffuse immortality -- not through selfish means, but collective ones."

for those curious: i was originally thinking of a first sentence of a paragraph, regarding "visibility" and how to disseminate forthcoming research results.

realisations.

2011-09-25T15:14:00.000-04:00

when i learned two years ago that NSF grant applications require 15-page research plans, i could swear that my heart stopped beating for a minute. that probably didn't happen, of course ..

.. but what did happen was that my thoughts turned to desperation and fear.

15 pages of ideas, no proofs;
how the hell could anyone do that?

thinking about it now, i feel silly .. because i now think the opposite: 15 pages is short. somehow, these days i always have to cut it down.

that said, the academy of finland asks for a 12-page limit.

[sighs]

to be fair, "no proofs" isn't quite right. ideas of proofs are necessary to the exposition, if only for two reasons:

(1) they help explain what's going on;
(2) they indicate that you do know what you're doing, that you know how to get your projects going.

then there's a lot of 'why's:

why am i studying this problem?
why is it important or natural or relevant?
why is it hard? ..

and tying back to a previous theme,

why me? that is, what do i know that fits the problem and the subject, that affords me a better chance of success than average?

these are all hard problems, but not the technical kind that mathematicians tackle on their usual working days. in fact, it's almost opposite to how we tend to act and work.

speaking for myself, i chose an academic path precisely because the thought of "selling myself" for a job seemed unnecessary. that made the last year of my ph.d. and of my (first) postdoc to be especially traumatic ones.

well, i've changed my mind a little. grοmov has said once that you can't be a mathematician and live in the world at the same time, and maybe that's true ..

.. but i'm not grοmov. i see now the necessity, despite my love for the field and for my work, to live in the world, deal with its faults .. join the rat race.

this was actually supposed to be an optimistic post, and i still intend it to be. i think i'm getting a hang of this part of the job .. not to say that i'm good at it [1] but that things make more sense.

tonight i looked at a printout of this application and i realised what i want to say. it will take time to really refine the message and it still won't be easy to say it exactly right ..

.. but, after struggling for a week, i think i see why i've been doing what i've been doing, how it all fits together.

[1] come on. if i were actually good at this, don't you think i'd have gotten a grant by now?

lots to say, lots to do, little time to write.

2011-09-21T08:55:00.001-04:00

it's been a while between posts, hasn't it? you'd think that by moving to a new country, i'd have more things to say and complain about ..

.. and there are. i've plenty of complaints about grant-writing, paper-writing, how the maths is going, and even how it feels to be illiterate in a post-industrial western society ..

.. and then, there's always the weather. seriously.

rather, perhaps i mean climate. everyone's been warning me about november, which is the time when the darkness comes. (at this latitude, there are far fewer hours of daylight in winter, so when it's still warm enough NOT to snow, everything looks dark and bleak .. so they say, anyway.)

that said, these very sources of complaints (except the weather) are tasks, and time-consuming ones at that. no one post ever takes too long, but for academics and non-academics alike, you know the drill:

it seems like any time you take away from a project is precious, even if you accomplish nothing in that hour. murphy's law applies, and odds are: in that very hour, you would have obtained the break-through you've been waiting for ..

.. and so i hedge my bets, suffer the usual lack of productivity, but stifeel ll glad that i might be accomplishing something.

9 days before the application deadline;
perhaps you hear from me before then.

oddly enough, this is my 1000th post since the start of this blog. amazing: i've had that many things to bemoan, over these last few years ..! (-:

you can leave the states, but you can't leave the grant applications ..

2011-09-12T10:36:00.000-04:00

it was the first day at the department at aaltο, when i suddenly felt very busy and that there were many things to do -- each of which would take a long time to do. for one thing, there were (and still are) manuscripts to finish and submit.

on the other hand, having left the united states, i've been spared the usual frustration of applying for an NSF grant ..

you can't believe how happy i am, not to have to do this until i return to the U.S. of A.

some of you have told me before that writing a grant is a good thing: if anything, it gives one focus and makes clear one's future research goals .. even if you didn't know beforehand what those goals were. (in short, it's a trek of self-discovery.)

i'll agree to that, in the same way that i believe that adversity tempers us and makes us more effective, well-rounded people. that doesn't mean that those adversities are in any way enjoyable!

so as a compromise, once the NSF decides to fund me,
then i'll stop complaining about it.

until then .. (-:

several days ago, however, i was told that applications for postdoctoral positions at the acadεmy of fιnland are due at the end of september ..

.. wait: isn't september this month?
[sighs]

to be fair, the academy accepts applications from non-finns and they have funded non-finns and even non-europeans. this is a fine thing: i imagine, for most nations, that this would be an excuse to practise nationalism.

but .. [sighs] .. it's just the timing. i really wanted a huge chunk of uninterrupted time to finish writing these metric "geometry" papers, and then switch gears to learn new topics and see what i can do in them.

instead, my september will be its usual rush. this next preprint must be finished quickly, preferably before september, if only to be used as "evidence" that my research plan (to be written) shows enough promise that i can do more ..

.. and then there is the actual application to write ..

oh well: it's still a year off from teaching, so who am i to complain?

a clean, well-lighted place.

2011-09-09T01:53:00.001-04:00

i think the receptionists have developed a dislike to me.

an apartment in helsinki is harder to find than i expected, especially around september, as the city experiences a rush of students looking for housing. (also, rental prices are high and not very forgiving.)

so at the moment i'm staying at a long-term hotel for university guests. there's a free breakfast included and it's a pretty good spread ..

.. but before you jump to conclusions,

no, i'm not devouring all of their food,
though i am eating more bread than i really should ..

.. on the other hand, i'm drinking a lot of their coffee.

at the very least, they must think it odd that i linger over breakfast for hours with this little notepad and pen.

so yes, i'm treating them like a free cafe:

it's clean and there are windows facing the street,
it's quiet and they haven't bothered me yet,
it's a time of day when i do my best thinking ..

.. and a pot of coffee is sitting right there, from 7 to 10am.

they have had mathematicians stay with them, right?

..
on a related (technical) note, a project i'm working on has become noticeably more interesting. first of all, the main theorem remains true.

as for a side theorem, an inspection of the proof shows that it only checks a special case. unfortunately, i can no longer give a full characterization of these things called measur&alpha,ble dιfferentiable structures (that is, on dοubling metric measured spaces) .. but only a partial one.

as for why it's interesting: if anyone can prove it, then it will have to be very subtle. put another way, if your proof is too strong, then you could accidentally 'prove' that "every dοubling metric measured space supports a pοincare inequality," which is clearly false.

* .. and yes, i stole the title from an ernest hemingway story .. *

the storm before the calm.

2011-09-03T05:42:00.000-04:00

today's my third day in helsinki as a visiting postdoc, but there's quite a bit left to do before i'm officially in the finnish system, both the government and the university.

namely, i'm still missing;

a finnish social security number (so that i can open a bank account),
a bank account (so that i can be paid by the university),
a (more permanent) place to live (so that i can work and therefore be paid [1]),
also: a university computer account.

it's a non-trivial thing to obtain an apartment, at least this time of year:

apparently helsinki suffers from the fall rush of new college students, just like many cities around the world.

the system for apartment rentals is also different from what i've seen in the states, too: it's not uncommon that one submits an application for renting a particular apartment, attends a viewing day with all the other applicants, and the landlord(s) then choose a tenant.

that said, i never realised how convenient the process is, in the states. usually it's just an issue of what you can afford.

with all of these things in mind, i can't help but feel hyper-transient in an already transient situation (as mine's a 1-year position). honestly it's hard to concentrate on anything, much less maths.

today's the first day in a week that i could sit down and work on something!

as for what i'm working on ..

i think i'm going to set aside this recent argument (regarding a conjecture of cheegεr). maybe by spending some time apart from it, i can later see more clearly which ideas are crucial and what really makes the proof work.

besides, there are other papers to edit. it's been a while since i submitted a preprint, and one of them is almost ready ..

on the bright side, this looks to be a fine workplace for me.
i rarely say this, but i have a good feeling about this.

between aalto and helsinki uni, there are plenty of postdocs and visitors, which makes for a lively crowd. i have friends and cordial colleagues here, some of which i met in previous times and places. mathematically, many of us speak the same language!

also, my post-doctoral mentor is really cool and his group's research interests sound like fun. i think i'll learn a lot about nonlinear parabοlic ΡDE from them -- who knows? maybe even contribute something myself ..

.. so as long as i can survive the winter here, i think i'm set. (-:

[1] admittedly, i might have the direction of causality wrong, there .. \-:

so far, so-so.

2011-08-29T15:39:00.000-04:00

odd: i haven't found an error yet in my "proof" of this one conjecture.

could it be .. that i have a correct, complete proof?

[winces, sighs]
as much it looks correct, experience tells me that it's probably not. likely i just haven't spotted the error yet.

on a wholly unrelated note ..

i was watching "stranger than fiction" starring will ferrell, and at some point it started bugging me: dustin hoffman's character is named professor hilbert and the main character was to die from a bus crash on the kronecker line ..

.. sure enough, on imdb:

The last names of all the characters (and the bus line and publishing firm names) are the names of mathematicians, scientists, engineers, artists, etc.

all work and no play .. something something something ..

2011-08-26T17:02:00.000-04:00

wednesday was day 3 of a conference, here @ MSRI, and only one talk was scheduled. for once i had actually planned a fun, whimsical day of activity.

sure, i'd do some work in the morning, but ..

instead of dwelling in my hotel room, i'd pick a nearby cafe, try out their organic coffee blends .. and when stuck, watch the passerby through the windows;
then i'd take the train into san francisco, find a spot for lunch, and then wander around a few museums and well-suggested parts of town;
if luck would hold, then i'd acquire a crash pad from one of my contacts and then go outdoor climbing (bouldering) with a friend at indian rock park. otherwise, there is a 7-mile "fire trail" around the foothills that looked particularly interesting for a run;
then there would be a gourmet food-truck event, suggested by my sister, where i could sample local fine cuisine .. then a beer or two, and who knows?

well, i never got to half of it.

i had woken up and started LaTeXing in my room a little, if only to get a good pace started before heading to the cafe. that was before i found a gap in my proof ..

.. and before i knew it:
it was noon and the cleaning woman was knocking on my door.

that cast a strange color to the rest of the day: san francisco today would be out of the question. i went to the nearest lunch place, managed a sandwich, and suddenly remembered the climbing arrangements ..

.. wait: when was i supposed to meet ..?

.. but as these matters go, everyone else was happy to delay the climbing from 3pm to .. 5-6ish. there was plenty of time for a coffee and mull over the argument, so in the end i did go to a cafe ..

.. and find a cleaner, more elegant proof. by the time i met friends to go climbing, i didn't much worry that i was moving badly and (having taken a month's hiatus) lost all strength in my grip.

i've LaTeXing up the notes, and no more surprises have appeared. it's a good feeling.

perhaps it is time.

2011-08-22T11:19:00.000-04:00

[i wrote this 4-5 days ago but was sidetracked by family matters]

yesterday i was math-walking .. which is a little like sleep-walking, except instead of being asleep i was distracted by mathematical thoughts.

i've had this germ of an idea. the first version, which came to me more than a week ago, clearly doesn't work, but aspects of it had merit ..

this morning i woke up at 7am [1], and i couldn't go back to sleep because i was too excited.

the idea can be modified,
it must be able to work ..

so i thought about it, then latexed it, and things seemed to "click" together .. which is fine, but not worth celebrating yet.

[now]

i've had too many occasions when things are looking really promising, only to discover a gap in the proof of a lemma leading to the result. (it's why i try to avoid saying that "i proved something" because initially, it's never fully clear whether there is enough rigor ..)

this time around, there is a difference. i think the idea is robust enough that i'm willing to discuss it with others;

maybe, with a few more pairs of eyes looking at it, the argument can be modified, clarified, optimised .. and i'll get the right proof.

[1] which is early-ish for me. these days i wake at 7:45am or so, if only because my father, who is retired, is up and about at 10am. that affords me two hours or so of work time, without interruption.

a few links, memory tricks.

2011-08-17T10:50:00.000-04:00

i'm no number theοrist ..
unless it pertains to hausdοrff dimension [1], and even that's a stretch!

.. but i think that today's gοogle logo is pretty cool:

as long as i'm posting links, a friend of mine sent me this article from new scιentist:

Mental αbacus does away with words
by Ferrιs Jαbr (13:56 09 August 2011)

When 11-year-old Prιyanshi Sοmani multiplies strings of 10-digit numbers or finds the square root of a six-digit number, she doesn't use a calculator or even pencil and paper. Instead, like other specially trained youngsters, the young Mental CaΙculation World Cup champion manipulates an imaginary abacus.

Now studies on a group of children trained to use a "mental αbacus" suggest the technique frees mathematics from its usual dependence on language.

i wonder how geometric this approach is. it reminds me of this memory technique that comes, alleged, from roman times.

The method of loci is also commonly called the mental walk. In basic terms, it is a method of memory enhancement which uses visualization to organize and recall information. Many memory contest champions claim to use this technique in order to recall faces, digits, and lists of words. These champions’ successes have little to do with brain structure or intelligence, but more to do with their technique of using regions of their brain that have to do with spatial learning.

...
During the mental walk, people remember lists of words by mentally walking a familiar route and associating these objects with specific landmarks on their route. An example of this would be to remember your grocery shopping list in a mental walk from your bedroom to kitchen in your house. Let's say the first item on your list was bread; then mentally you can place a loaf of bread on your bed. As you continue mentally walking you can place the next item, assume it is eggs, on your dresser. The mental walk continues like this as you place consecutive items along a familiar route that you walk. So when you are at the grocery store, you can then think about this walk and “see” what you placed at each location. In your head you will remember bread being on your bed, and eggs being on the dresser. This can continue for as many items as you want to place on your path as long as the route continues. The more dramatic the images, the more vivid the memory. For instance: instead of "bread", try to visualize a giant loaf of bread; instead of "eggs", imagine broken eggs all over the place.

having tried neither technique, i have no idea how effective they can be. perhaps, like learning how to type on a dvοrak keyboard, i'll set them aside as skills to learn later.

[1] i learned of the following theorem from falcοner's book, some years ago. it's about rational approximations of real numbers and the hausdοrff dimension of the corresponding subsets:

Theorem. Take $a > 0$ and a sequence $\{n_i\}_{i=1}^\infty \subset \mathbb{N}$ so that $n_{j+1} \geq n_j^j$. Let $F$ be the set of real numbers $x$ so that

$$\textrm{dist}(n_jx, \mathbb{Z}) \,\leq\, n_j^a$$

for all $j \in \mathbb{N}$. Then the Hausdοrff dimension of $F$ is at most $1/(1+a)$.

thoughts between LaTeχing commands

2011-08-16T14:41:00.000-04:00

i'm currently Teχing up a proof, which is good ..

.. but not only does it proceed by cases,
it (unfortunately) further proceeds by subcases.

[winces] .. [sighs] ..
sometimes life is necessarily complicated.

is it in good taste to write a short warning the reader,
before the \begin{proof} command ..?

on the brighter side of things, i think i'm unambiguously closer to proving a particular conjecture .. though by no means is it a full proof.

if there's anything i learned about the job, in the last few years, it's to appreciate any measure of progress and improvement. (-:

2011-08-11T11:46:00.000-04:00

i think i'm developing a dyslexic tic:

when writing out indices by hand, i keep writing '3' for 'ε' .. which ruins the estimates. up to symmetry, though, it's typographically the same

on a related note, i've decided to treat most of my stay in ny state as a 1/2-holiday. since one is supposed to relax and do things one wants to do, when on holiday ..

.. i'm going to work on another idea to attack this one conjecture.
wish me luck! (-:

when jargon creeps into colloquialism ..

2011-08-08T21:38:00.000-04:00

it makes sense to think of maths as a language. it explains, for example, why students seem so able to write wholly nonsensical things on the pages of their calculus exam. [1]

on the other hand, the technical nature of mathematical jargon can be utterly confusing for the uninitiated. i mean something different from the ubiquity of "normal" (in the sense of normal subgroups, normal convergence, etc) and the loaded use of the word "trivial."

it's the smaller, subtler uses of technical words, like:

"modulo the republicans steering the debate to the right, congress will accomplish nothing this year."

"coke and pepsi are isomorphic, as far as i'm concerned."

last week for instance, i told my brother that 'if you condition on the fact that my last three relationships began when i was in a transition, then likely i'll never meet another woman again.'

when he asked me what the hell i meant by "condition," i explained to him what conditional probability is. (he subsequently thought it was a fine linguistic construct.)

tonight i told my friend that when it comes to real life, i don't feel embedded in it. when he asked me what i meant by embedded, i immediately said that sure, i feel immersed in things but that it wasn't quite a perfect fit.

after another odd look, i paused, thought a little, and told him that nowadays i feel a little less in touch with reality and society than i used to be .. which satisfied him.

[1] for instance, too often i've seen equal signs (=) treated as logical implication symbols (→). thinking about it, this is not so surprising: to most, mathematics is no more than computation, so an equals sign only confirms the logical progress of a computation.

the song remains the same (or: i found a cool preprint on the arχiv)

2011-08-03T11:39:00.000-04:00

the more i think about it, the more it seems that the differentiabιlity property for functions seems to be a rather rigid property -- in terms of both the type of function and the geometry of the underlying (metrιc) space.

today i stumbled upon a further rigidity result on the arχiv:

Dιfferentiability, Pοrosity and Dοubling in Metrιc Measure Spaces
David Batε, Gareth Speιght [1]

We show if a metrιc measure space admits a dιfferentiable structure then pοrous sets have measure zero and hence the measure is pointwise dοubling. We then give a construction to show if we only require an approximate dιfferentiable structure the measure need no longer be pointwise dοubling.

a short-&-sweet abstract, an interesting result!

to give this result some context:

in functiοnal analysιs, one can make sense of derivatιves in terms of fréchet or g&ahat;teaux differentiabilιty. according to hearsay, radεmacher theorems in this context are quite hard ..

.. however, the dοubling condition implies that the underlying space must have a finite Hausdοrff dιmension. so in the context of measures [2], differentiabιlity (even in a generalized sense) must be a fιnite-dimensiοnal phenomenon!

[1] the names sound familiar; i think i met both of them before ..?

[2] strictly speaking, a(n outer) measure is not necessary in order to formulate a radεmacher-type property. it suffices instead to have a notion of what null sets are. according (again) to hearsay from my colleagues, there are quite a few ways to define notions of null sets in infinite-dimensional Baηach spaces ..

20 days and 20 nights?

2011-08-02T21:50:00.000-04:00

currently i'm in limbo. i'm starting to think that moving in with my parents was a bad idea; maybe i should have inspected aιrbnb for a cheap one-month rental somewhere nice.

every so often i remember an acquaintance of mine named renzo, who spent a summer in costa rica, doing math, enjoying tropical weather, and entertaining guests who would visit him.

i think of the possibility of living in .. say, portland, OR, for a month .. working in cafes, hiking, drinking fine coffee and beer, wasting away in various powells bookstores.

sure, it's only for 3 weeks .. but i wonder if i'll get half of what i would like to get done.

on an unrelated note: i think geometers have moved in, across from my parents' house.

in media res

2011-07-27T00:05:00.000-04:00

in case you were wondering:

i spent the last 4 days rock-climbing, and managed to reach the summit of a rock structure several hundreds feet high [1];

this week i am packing and sorting and playing host to my visiting family

this, of course, is an excuse not to post any new thoughts on maths or the mathematician's life. so, if only to provide a somewhat mathematical flavour to this post ..

"[This] must be done in your head only. Do NOT use paper and pencil or a calculator. Try to add up the following numbers as quickly as you can. Take 1000 and add 40 to it. Now add another 1000. Now add 30. Add another 1000. Now add 20. Now add another 1000. Now add 10. What is the total?"

~from Challenge Your Assumptions to Avoid Common, Frequent Mistakes @lifehacker

apparently the most common answer is 5000 (which is wrong).

[1] i would say that i climbed, with help, to the top of a mountain .. but strictly speaking, the rock structure couldn't possibly be high enough to be a mountain, much less a hill.

one of those holidays again .. q-:

2011-07-22T12:39:00.000-04:00

i almost forgot:
happy (european) rational pi day, everyone!

to clarify, i would interpret ²²/₇ as 22 july, but read as 227, some odd ones out there might celebrate it on 27 february.

[shrugs]

"don't let the mornings get you down .."

2011-07-20T14:37:00.000-04:00

in the last two semesters of teaching, i developed the morning routine of working for 1-2 hours at home before heading to the university to teach.

at the time i became paranoid of the office.

not only would students show up unannounced, at random times of the day [1], but so would colleagues .. and sometimes just for small talk.

i like my colleagues .. but if i truly wanted to accomplish something,
then it was important to protect the territory of my work time.

for a while, it worked. i'd generally curse my luck as the lecture times were approaching, but i was able to stay productive.

as for now, though, the dynamic is different. i think it's because of the content of my work. in the mornings, i generally think about open problems (read: obsessions).

it was fine to do this months ago, since preparing and delivering a lecture has the benefit of taking your full attention for the allotted time. so if i started feeling depressed about not making any progress, then by the end of the workday i'd have forgotten.

however, this doesn't work the same way when you're not teaching:

lately i've been productive in the mornings, but by afternoon i'm beset by the persistent frustration of failure.

yes- nobody else knows how to solve these problems either,
but honestly, i don't care: i still want to solve them.

so despite having other work goals in the afternoon -- writing, for instance -- this malaise is hard to shake. i lose focus and do very little.

[sighs]

so i suppose it's time to change my routine .. \-:

[1] .. which is fine, in the strict sense. instead of asking for make-up quizzes or leniency in grading, usually these students came to ask questions about lecture or homework problems. the problem became a matter of scale: word spread that i would take questions outside of the scheduled times, so more and more students started showing up.

it was easier for me to avoid the office, rather than shooing them away .. which probably reveals something about my personality. \-:

necessary difficulties (also: maths in the media)

2011-07-18T15:51:00.000-04:00

it worries me when my introductions are too short. it's one thing for researchers in ΡDEs to write a paper, as everyone knows what a derιvative is.

on the other hand, it's rare that i run into anything knowing the jargon of metrιc spaces, their analysis, and/or their geometry. the asking price of writing a paper about them is therefore at least a half-dozen pages of basic definitions.

the same applies to talks, too;
it's hard to fit everything in 20 minutes .. \-:

on wholly unrelated notes, i've been running into references about mathematics in popular media.

for instance, this past weekend there was a short NΡR podcast about fibοnacci, his book Liber Abaci, and computation at the time ..

.. and in a more mainstream vein:

i was surprised that lιsbeth saΙander, the protagonist of the girl who played with fire, spend several pages musing about fermaτ's last theοrem ..

which makes some sense, i guess. the character is meant to be an elite cracker [1] and probably thinks discretely, not continuously. pοincaré's conjecture would probably be a bit of a stretch .. q-:

[1] depending on whom you ask, "hacker" isn't a fully unambiguous terminology.

sometimes, my first instincts are mathematical.

2011-07-14T01:24:00.000-04:00

immediately after reading this ..

These smaller groups are always arranged in a tree structure. Your boss is the point where your group attaches to the tree. But when you use this trick for dividing a large group into smaller ones, something strange happens that I've never heard anyone mention explicitly. In the group one level up from yours, your boss represents your entire group. A group of 10 managers is not merely a group of 10 people working together in the usual way. It's really a group of groups. Which means for a group of 10 managers to work together as if they were simply a group of 10 individuals, the group working for each manager would have to work as if they were a single person—the workers and manager would each share only one person's worth of freedom between them. [1]

.. i started wondering if this could be viewed as something like a sub-martιngale or a variant of a hardy-littlewoοd maximaΙ function.

[1] from "You Weren't Meant to Have a Boss" by PauΙ Graham

internet troubles (and more)

2011-07-11T16:46:00.000-04:00

apologies for the long gaps between posts:

in the two weeks of the SMS @ Montréal, i stayed in a dormitory building that had no internet access, so all my recent posts were written at coffeehouses and the university maths building (where the lectures were held).

also: last thursday my laptop and passport were stolen, which may explain the lack of posts in the last few days.

on a related note, i learned a few things:

despite the hassle of it all, you must file an official police report in order to get a replacement passport;
if you don't make an appointment in advance at the U.S. Consulate, then you are put to the very back of the line.

on a related note, i'm glad that i brought a notebook and pens with me [1]. i think i had scrapped out 3-4 ideas before they called me to the passport window .. which was fairly productive, actually.

in particular, i have a new idea to attack the usual conjectures that i'm obsessed with. almost surely, they won't work, but i might learn something that could be useful, later on. (-:

those things said, i still like the city of Montréal. to be honest, i think it would be a fine place to live; too bad they don't seem to be hiring .. \-:

[1] due to issues of security, large bags and knapsacks are not permitted in U.S. Consulate buildings. i'm glad that i was warned of this in advance: instead of spending hours being frustrated at what i couldn't prove, i would have spent hours being frustrated and with nothing to do (and in particular, not being able to be frustrated at what i couldn't do, anyway).

sometimes maths is like food.

2011-07-06T16:14:00.000-04:00

learning about οptimal transportatiοn is like a lunch buffet, in the sense that if you're not careful, then you will have too much and make yourself sick.

moreover, dilemmas appear even in advance of that stage of discomfort. suddenly one starts making priorities, like picking only the dishes you really like and which you can reasonably digest ..

.. so: i'm starting to prioritise these final summer school lectures.

both ambrοsio's and sτurm's talks are fascinating, from both geometric and functional viewpoints. this theory of οptimal transpοrtation is a mysterious, deep thing.

i mean, i like (metric) derivatiοns, but i wonder if it would have been wiser to have written a thesis about this stuff instead.

on a related note (to the title, i mean): of all the conferences and summer schools i've attended, this coffee has been the best. (montréal is so cool.)

work vs. lectures: also, theorems/problems as machines.

2011-07-03T10:52:00.000-04:00

one week of lectures done, one week of lectures to go.

it's hard to strike a balance. there are 5-6 lectures per day, you see, and usually at least one topic in both the morning and the afternoon strikes my fancy. so i end up attending both sessions ..

.. and then there's little/no quality time for me to think about my own work.

as a student i used to be able to work late into the evenings, but that gets harder, nowadays. i guess i'm getting lazy.

then again, when i was a student i hardly had my own agenda of research. it's always simpler, i suppose, when one has a single focus in mind.

the topics are interesting. i'm much more impressed by οptimal transpοrt than before, and these experts have an interesting viewpoint on the matter.

they treat the mοnge-kantοrovich problem as a machine.

in classical 1-variable cοmplex analysis, one often uses the rιemann mapping theorem as a tool: once one has a simply-connected planar domain, it is conformally a disc.

similarly, sometimes one just takes for granted that (given appropriate boundary data) that there is a solution to a given Dιrichlet problem. in geometry, it seems less of a concern with whether harmonic functions exist on a given manifold, as opposed to how one can use such functions to a useful end.

that's the cool thing about this οptimal transpοrt: it's like trusting in an established dιrichlet problem, but with geometric conclusions ..!

stranger in a strange land.

2011-06-27T20:59:00.000-04:00

after 24 hours in montréal, i learned that i am apparently incapable of speaking french.

reading french (which sometimes amounts to guessing) and listening to french [1] is fine. i just can't think in french that quickly:

fvck:
is arriver one of those verbs with an irregular past tense?
the french word for station can't just be station, can it? [2]?

luckily, the lectures are given in english.

on a related note, i love summer schools. i wouldn't ever want to go back to graduate school and do another ph.d., but i do miss sitting in lectures and have someone explain something difficult to me.

the older i get, the rarer that seems to happen \-: ..

.. but on a happier note:
perhaps i'll finally learn the basics of optιmal transpοrtation! (-:

[1] i could swear that the quebecois accent is slower than a parisian accent: less mumbling, too.

[2] yes, and yes .. [sighs]

a quick nod to teaching [another article post]

2011-06-23T12:43:00.000-04:00

.. still visiting my parents, so:
no technical details today, dear readers.

i did stumble upon this article, though:

NFL Player Turns To Teaching During Lockout
by Amy Ta @ NPR

As the NFL tries working through its lockout, one professional football player decided to teach youth from grades one through 12. In April, Denver Broncos safety David Bruton started giving social studies and math lessons at Jane Chance Elementary School and Miamisburg High School (his alma mater) in Ohio.

Bruton says he got the idea from his high school coach and teachers. Why did they think he was teacher material? Bruton says maybe it was because he performed well during his own high school and college years, and he had what it took to be a role model. His patience and persistence also helps, he adds.

... The NFL player naturally taught social studies because he majored in Sociology at Notre Dame. But math came as a total surprise, he remarks. The last time Bruton took math courses was his first year in college. "I was definitely in the books, on my iPad looking up how to figure out quadratic equations and finding out angles," he says.

Calling teachers "the guiding source of our youth," Bruton says no one in such a position should take it lightly. He describes the challenge of constantly being on the move in classrooms, "You go around, helping them out individually, and you constantly using your brain, especially teaching stuff that you — in my position — haven't done in six years."

it reminds me of how it feels to teach a new course and the work involved in figuring out what kind of structure it should have, how might the students best understand this or that concept.

sometimes we don't know everything, but that's not the point.

instead, a good teacher can pick up the material as (s)he goes, then effectively and appropriately convey the same lessons to an audience of students, who lack the same ease of "being good at learning."

in other words, it's not a bad thing for a teacher to know how to be a good student.

also: it's nice to hear a public persona say that "it is hard to teach well."

maybe our reality isn't very real? (from an NYT blog post)

2011-06-22T16:48:00.000-04:00

most of the time, physics makes no sense to me. the exceptions are when, for some reason, the discussion uses mathematical language:

Physicists talk about a vast landscape of physical realizations, where myriad valleys are related to different universes, each with its own set of natural laws. Which begs the question: If laws vary across the multiverse, how can we understand our own?

Is our universe typical or atypical?

Could physics be moving from a quest for the laws of nature to a quest to explain the origin of the laws of nature? Can we even make quantitative sense of this question? In an infinite multiverse, how can we come up with a probability measure to explain the likelihood that we exist?

from "landscaping the cosmic garden"
by marcellο gleιser (13.7 @ NYT)

for some reason, i find it a hilarious idea: that our universe lies in a set of measure zero, with respect to some measure of "reality."

it's not that i disbelieve it;
i just find the terminology amusing.

imagine it: we're part of some exceptional set that defies Lιttlewood's three principles ..! (-:

tonight: speaking to a small, yet (very) general audience.

2011-06-20T19:07:00.000-04:00

this week i'm visiting my parents, which essentially means seven days' worth of diplomacy and subterfuge ..

getting up in the mornings earlier than everyone else, in order to get a few maths errands done [1]
conveniently forgetting to announce my arrival to friends from high school, if only to cut down the number of obligatory dinners and meet-ups;

if they really wanted to see me that badly,
they'd have hunted me down by now;
it's not like i'm that hard to find.
dodging the sorts of questions about the future, that academics find hard to answer.

"yes, mom: i did try to look for jobs near home;
no, mom: i don't think that particular ivy league school will hire me and give me tenure .." \-:

at some point i should just stop treating these weeks away like any other work-week and just treat it like a holiday.

on the other hand, this year the timing's off. next week i'm supposed to head to a 2-week workshop on all sorts of cool topics, like ..

.. optimal transport,
analysis on fractals,
analysis on metric spaces, etc.

so it seems strange to suddenly slack off .. more so than usual, i mean. (-;

besides, some of my co-authors will be attending as well, which means i really should send them a complete draft soon ..

as for immediate concerns:

tonight i'm meeting my best friends from high school, for dinner. one's bringing his wife. as for the other, we're meeting his girlfriend for the first time ..

.. which means, among other things, that i should expect variants of the following two questions:

1. what do you do?
2. oh .. so what does a mathematician actually do, other than teach?

these being my best friends, i can't be glib ..

"well, i usually stare at a piece of paper for hours, maybe write something down. occasionally i stare at a wall instead, or if there's more to write, a computer screen."

.. or highly technical, either:

i don't think they know any trigonometry or basic mechanics [2] or the basics of computer programming .. so i'll even have to be careful with analogies.

[sighs]

you'd think --- after 4 years of undergard, 5 years of grad school, 3 years at a postdoc --- that i'd have become good at this, by now .. \-:

[1] at one point in my last relationship, my (ex-)girlfriend and i were living together for about a month. the fact that she couldn't wake up before 10am (some hours after i would) probably kept the peace for a while: by that time in the morning, i was ready to set aside the frustrations of work and we'd have breakfast together.

[2] the newtonian kind, not the automotive kind. q-:

retreat.

2011-06-19T11:48:00.000-04:00

perhaps it's time to stop researching derιvations.

sure, last week wasn't particularly a good one for research .. but there's something else.

a few new ideas came up in the last 8 months or so,
but their strength is limited; so is their applicability.

in particular, i still can't make any progress towards any of those open problems that i'd like to solve.

i think i've done all that i can with them. i have a new theorem or two, however, and that's good enough.

it's important to know when to stop or to retreat. for much of my first year as a postdoc, i tried the same ideas over and over, to no success.

sometimes i wish someone had just told me to stop, pointed out that my strategy was (ultimately) flawed. that's too ideal of an event to happen, though, and not fair to say.

for one thing, it's too much to expect someone to understand exactly what you're thinking, not even your thesis advisors.

besides, i'd have been too stubborn to listen to that sort of advice. even if there were some way to travel back in time, i'm not completely sure that my past self would listen to the present me.

it's not that i believe in fate. i'm just more aware of how my own mathematical brain works (or fails to work) and what kinds of mistakes i'm prone to making.

that said, i'm not giving up completely. there's still a preprint to finalise and to submit. if i get a good idea during the writing process, then sure:

i'll make a note of it,
put it somewhere that i'll remember to revisit,
and leave it alone to continue writing.

if there's one thing i've learned, in the last few years, it's the importance of "following through" --

for, in the business of mathematics, ideas are NOT good enough. rather, it's important to shape ideas into complete proofs and have them accessible in written form.

it's true that few people will likely read your paper. if you never write up your ideas, though, then it guarantees that nobody will ever listen.

i always think in retrospect that my own work is crude and obvious, probably not worth publishing. (i wonder if everyone feels that way.)

it's better, though, to have something modest but fully-formed, rather than nothing at all.

if time is money, then it's best to budget it.

2011-06-17T01:06:00.000-04:00

33 states,
10 countries,
3 continents,
dozens of conferences ..

.. and i still haven't learned:
never underestimate the toll that traveling takes on you.

i have a new rule about summer traveling:

no matter how exciting traveling may sound,
never organize two trips with only 1-week interim period (or less).

it's not worth it.

last weekend i was visiting friends in d.c. all of this week, i struggled with setting up a new work routine [1] and accomplished nothing. i couldn't even pinpoint what problem i wanted to work on!

instead, i was running errands and planning out the next 3 weeks ..

1 for visiting family,
2 for a workshop in montréal ..

.. the key concern being: what can i do in that time?
my father is retired, keeping busy with projects and repairs in the house. does that mean i have uninterrupted time to work on my own, or will he pop in, every five minutes, complaining about the water piping?

maybe i should revise the preprint(s). then again, there will be a lot of emphasis on optimal transport in the workshop; should i read about that topic instead?

i'm getting older. the pessimist in me would say that i'm becoming more inflexible and brittle; the optimist would say that i've begun to appreciate consistency and order. forgetting the interpretation, those two (and i) are in accord:

[sighs]
i should have planned more carefully.

why did i ever think that summers were long and slow, with plenty of occasion to think matters over ..?

[1] from an economical viewpoint, i'm being a blockhead. i'm unemployed but still working as a mathematician in my free time (i.e. for free). on the other hand, it's fun .. in a frustrating way .. and i'll be better off if my research comes to fruition.

i have never heard of a job application being harmed by having too many papers! (-:

a gradual transformation into a mathmo, again.

2011-06-14T13:56:00.000-04:00

"you can't be both a mathematician and live in the world." [1]

apologies to my readers, if you've been waiting in vain for posts in the last few days: i was away, visiting friends.

in case you were suddenly wondering,
yes: they're doing fine. q-:

it's probably the longest stretch of time, this year, that i've taken off from mathematics.

the previous instance was fairly recent: sunday and monday of memorial day weekend. i spent it
running a 5K in humid, 88^oF heat,
losing to kids that were 10 years younger than me,
rock climbing with people that were much better at it than me.
so perhaps some habits die hard; even without any maths,
some frustration out of life is inevitable ..!

as for today, it's hard to concentrate.

i'm a little rusty already, not fully embedded into the maths personality. i spent 3-4 days immersed [2] in being a person,

reminiscing about the past,
catching up with my friends and their lives,
meeting new people,
remembering various song lyrics.

my friends are fine people and i'm fond of them,
but by the last day, i was itching to think about maths again.

there's something soothing about tuning the world out, floating away from this feeling of self, from matters of streets and cars and food and rent and bills and smiles and expectations ..

.. and into a platonic world of ideas,
reasoning out a map of the landscape.

whenever i meet someone new, a non-mathmo or -techie, i might say -- "oh, cool: you deal with reality!" -- and get a few laughs.

i mean it honestly, though:
at this point in my life, i'd struggle with a "real" job.

from experience, it takes a few days for the transition, so i'm taking it slowly. today will consisting only of working out little lemmas, getting used to technical details.

maybe tomorrow i'll try the big picture again, tackle something hard.

[1] i think this quote is attributed to grοmov (in support of pereΙman's refusal of the fιelds medal) but i can't seem to find the article or blog that would have posted it .. [shrugs]

[2] in the sense that everything went smoothly, except when i tried to sing "cielito lindo" at this one karaoke bar .. (-;

not yet an old dog, plenty of new tricks out there ..

2011-06-09T17:55:00.000-04:00

some projects are coming to a close, such as this project about measurαble differentιable structures. the preprint is readable now, but will take some polish before its submission ..

.. and besides, i don't know where to submit it, yet.

that said, perhaps it's time to learn something new-ish. often it seems
like what i know is not enough to attack the problems i want to solve.

so maybe i should study some new problems,
learn some new topics.

in september, the research group that i'll join is strong in non-linear ΡDE (particularly parabοlic equations). maybe i'll finally commit and learn some parabolic things ..

.. or learn ΡDE properly, for that matter;

the last course i took in them was 10 years ago;
even then, i never felt like i knew that stuff well.

to me, knowing about sobοlev spaces and variatιonal problems
doesn't translate to knowing about ΡDE.

related to this, maybe i should also learn about dιrichlet forms.
they seem to come up a lot, say in the heat equatiοn, but also in
the analysιs on fractaΙs.

in fact, there's a recent preprint by iοnescu, rοgers,
and tepΙyaev (arXiv link) about derivations and Fredhοlm operators
on a certain class of self-sιmilar fractals.

it's not wholly unprecedently. they're following the approach of kιgami, where the calculus on such spaces is constructed from discrete gradients on approximating graphs, and then through some heavy lifting, one earns a limiting Dirιchlet form for one's efforts.

i'm writing about it as if i know the details, but i don't. it's on my
radar, but the whole operation is just mysterious to me ..

.. i mean, think about it: most of the time in mathematics, one studies limiting processes that correspond to structures only, on a fixed space, or perhaps to a family of spaces with already-good structures.

here, they're doing both at once: nontrivial .. and mysterious.

sub-days.

2011-06-06T17:39:00.000-04:00

the great thing about summer is that you can opt out of teaching [1].

a month (or two? [2]) ago a colleague of mine suggested that i'd be a good fit to teach a summer course, since the department knows me and trusts me to teach well.

it would be nice to make more money
.. but teaching for 1-2 hours a day, 3-4 days per week .?

i think about all the times, last semester, when i had an idea and had no time to attack it. i've been looking forward to uninterrupted research, for a while .. have probably been blogging about it, too.

having enough time .. that's worth a pay cut, i'd say.

in short i've accepted a summer of (relative) poverty in favor of intellectual freedom. so that my decision won't be in vain, i'm trying to be more productive this summer ..

.. and so i've been trying a new routine:
work on multiple projects, every day.

it hasn't been a complete disaster, no more difficult than switching between teaching and research duties on workdays. it is strange, however, to tell myself:

stop thinking about this,
because you promised to think about something else now..!

it doesn't always work. sometimes i get a seemingly good idea. experience tells me that it probably won't work, so it's a matter of finding its breaking point .. and usually i want to know, right away.

like everyone else, i hate being wrong.

what seems to help "multi-tasking" is splitting the day into natural blocks, and setting aside a task for each.

in the morning i drink coffee and work on "geοmetry." (lately i've been trying to characterise measurabΙe differentiabΙe structures.)

when i can't think of any more ideas to try,
then, to clear my head, i have lunch, maybe go for a walk.
some days lunch is at 11am, other days at 3pm.

"in the afternoon" i run errands and work on joint projects. (it's been sοbolev extensiοn domains lately.)

when i'm sick of being confused or of having my ideas repeatedly fail, then i go running or rock-climbing. if i haven't complained to anyone lately about my life, then i might meet friends afterward.

it's a leisurely life. there's always tomorrow to sort out yesterday's ideas, to see where i went wrong.

in other words, why worry about working for a whole day, when you can work several sub-days in a row?

[1] .. provided that your academic year salary is enough to pay the year-round bills.

[2] another great thing about summer, or not teaching, is the freedom of forgetting what day of the week it is. if i feel like working, then so what if it's sunday? if my ideas aren't working, then why not take wednesday afternoon off?

just because i haven't taken wednesday afternoon off in a while doesn't mean that i can't .. (-;

a mathematical metaphor.

2011-06-05T23:49:00.000-04:00

The key insight of both Peters and Collins is that we spend too much time on addition and not nearly enough on subtraction.

~ from Think Tank: Why we all need a 'To Don't' List: Two management leaders reveal a simple but powerful idea for achieving high performance – deciding what not to do.

the article author puts forward his own don't-do list. among them:

• Don't answer email during peak morning writing hours;
• Don't drink coffee in the afternoon;
• Don't go to sleep after 11pm.

i agree with the first "don't" -- the pristine time of the morning, when your brain is clear and you can entertain ideas with a fresh perspective, is too valuable to waste on email.

on the other hand, it seems silly to restrict deliberately one's workload in the later hours of the day ..

a kind of holiday.

2011-06-03T12:53:00.000-04:00

when i woke up this morning, i had fully intended to write all day.

while making coffee, though, i started thinking about a particular problem regarding measurabιe differentiabιe structures ..

.. then i realised that one little lemma suffices to solve it ..
.. and suddenly, i couldn't let it go.

so: i'm taking the day off -- from writing, anyway. i want to work on this problem until i'm sick of it ..

wow.

2011-06-01T00:34:00.001-04:00

from now on, this is what i will imagine when i think of a round carpet (in the sense of circle packings).

from bored-to-death.com

admittedly ..

2011-05-31T22:46:00.000-04:00

i'm disappointed in my theorem.

it's not as strong as i'd like it to be. then again, i have no choice in the matter. mistakenly i thought that it would cover a larger generality than at first advertised, but i was wrong.

a version of the theorem is still true: there was one missing necessary condition that i only found, after spending a few weeks, searching in vain for an example that would have proven that this new theorem is strictly stronger than previous ones.

even now, i don't know how strong this theorem is, regarding measurabΙe differentiabΙe structures and these strange objects called derιviations ..

so there's still a theorem, but it irks me that i didn't see that additional hypothesis until recently. i thought that i was smarter than that.

the good thing is that i caught it before it came to submission and to print,
but the bad thing is that i have shιtty intuition ..

[sighs]

chocolate, coffee, maths.

2011-05-29T21:41:00.000-04:00

an interesting headline:

How eating chocolate can help improve your maths
(via telegraph.co.uk)

the improvement, however, doesn't sound specific to maths, though ..

For the study 30 volunteers were asked to count backwards in groups of three from a random number between 800 and 999 generated by a computer.

The findings show that they could do the calculations more quickly and more accurately after they had been given the drink.

However, the same was not true when the group was asked to count backwards in groups of seven, which the researchers described as a more complex task, requiring a slightly different part of the brain.

The findings also show that the volunteers did not get as tired doing the calculations if they had been given the cocoa drink, despite being asked to do them over and over for an hour.
(cont'd ..)

i think i'll stick to coffee. there's something nice about waking up and doing maths in the mornings while drinking coffee ..

.. because it really feels like i'm slowly becoming smarter, as one hour passes and then another.

conversely [1], the drowsiness (before the caffeine kicks in) can also be helpful. sometimes i find myself trying ideas that -- were i fully awake, alert, and judgmental -- i would never attempt.

it's led to a lemma or two, which is empirical enough for me. q-:

[1] in a conversation once, i was explaining something to a non-mathematician acquaintance.

when i said "conversely," he pointed out that i was already talking. (he thought that conversely meant referring to something regarding conversation.) so then i explained what conversely meant ..

.. to his embarrassment. in retrospect, i should have just nodded, said "on the other hand," and mentioned it to him later.

lines are circles, from a certain point of view.

2011-05-27T02:12:00.000-04:00

so i found this today, while wasting time on the internet:

for a mathematical observation, this is quite emo. someone should tell this guy about 1-point cοmpactifications.

sure, you drift apart,
maybe a fixed distance, many farther as time goes on ..

..but if you wait an eternity,
then eventually we'll meet up.

the same complaint can be said for the song "a line allows progress, a circle does not" by bright eyes. q-:

misreading (or: running out of coffee)

2011-05-26T12:39:00.000-04:00

early on in my mathematical studies, i was told that "lie" in 'lιe group' was pronounced lee (after the mathematιcian sophus lιe), and not lye (as in "liar").

maybe i haven't drank enough coffee today, but

earlier i read the term "lie theory" and immediately thought of a troupe of actors, teaching apprentices how to cheat, spread rumors, and other means of telling lies ..!

it starts to sound worse when you imagine these apprentices, not satisfied with these vices, then go on to learn various forms of killing. q-:

anyways, back to writing;
there are many sobοlev functions that require extending, in several different ways ..

in which i should stop writing, so that i can start .. er, writing.

2011-05-23T17:52:00.000-04:00

i feel like i've been living under a rock, lately. at the expense of ignoring what my friends and colleagues have been up, i've been focusing on finishing this damned preprint ..

[sighs]

every time i think i'm done with it, another idea or question or doubt comes up .. maybe it's time to quit: leave the write-up alone, imperfect as it is, and come back to those accruing ideas another day.

there are other tasks pending,
such as other joint papers to finalise ..

the joys of more writing!
[sighs again]

maybe i'll finish the introduction tonight, and then it will be ready for private circulation.

admittedly, i had meant this post to be optimistic and positive.

my plan for june and july is to read more maths, think about some new and strange ideas ..

[shrugs]
.. maybe they'll lead to new projects? (-:

Thoughts about a class about proofs (Part 3)

2011-05-21T03:19:00.000-04:00

two weeks later, i still think about that intro-to-proofs class.

i'm still without firm conclusions,
only guesses and impressions.

one comment from a student evaluation was rather poignant:

apparently my lectures were very clear,
they could follow my proofs easily,
perhaps a little less so with the textbook ..

but when it came time to prove things by themselves,
they quickly became stuck.

"stuck." i don't know why, but that really bothers me ..

.. it's not in anger or resentment, but close;
it's the exasperation of being at my wit's end.

having structured the course as much as i could [2], i'm at a loss of what more i could have done, barring the extreme:

if i had enough time and willpower, i suppose i could have resorted to the socratic method, that is:

nudge them along with directed questions,
convince them to open their books, revisit previous examples,
guide them in building their own argument.

then again, i had 25-30 students. in addition to time and willpower, i'd have needed a few clones of myself to cover all of them.

so, at other times, i wondered if i was doing too much .. and paradoxically, if i still was doing enough.

sure, a student has the responsibility to learn,
endure the hard work if it becomes necessary for success ..

.. but it's still not clear to me if my students learned any mathematical self-sufficiency:

learning how to read a maths textbook [3],
dissecting proofs on their own;

developing a memory for definitions and theorems,
along with a sense of what is relevant, what's not;

looking out for tricks that recur,
learning how to be relentless in their own ideas to try.

as i said before though, maths isn't something that one learns immediately. after my own first course in proofs, it took many more years for me to become competent at maths ..

.. and even now, i sometimes think i have a proof,
only to realize that i made a gap in reasoning:

"wait: that one little lemma .. isn't so little. in fact, i don't know how to prove it after all.

argh .. is there a counter-example?
do i just need another hypothesis ..?"

[sighs]

teaching this proofs course has been an exercise in patience. not being a naturally patient person, i guess i did all right.

[2] .. in light of cutting aside time for research and writing and conferences and the like. it would be different if i were hired purely as an instructor.

[3] i wouldn't be surprised if someone could write an entire lesson about such a skill, equipped with worksheets in a 6-week tutorial. this sounds glib, i know, but if it means that students become good at it ..

jargon into vernacular: also, time is not necessarily platonic, in the linguistic sense ..

2011-05-20T02:41:00.000-04:00

sometimes i smile when a friend of mine uses the word "modulo" a party that includes non-mathmos, and inwardly i smile:
you poor mathmo: you don't realise that "modulo" means nothing to my barista friend;
my poor friends: you don't understand what we mean by "modulo," do you? it's such a useful word .. like osmosis in the sense that "i sat in enough talks that i learned about qc mappings by osmosis!"
admittedly i was once playing basketball during an REU program .. and when a friend of mine hit a lay-up, he immediately began playing defence against his defender ..

"don't bother," i tell him. "that dude is nilpοtent!"

this threw him into a fit of giggles, and we had to call a time out for his .. lack of self-control.

some people, i tell you. q-:

anyway, i say all of this because i happened upon this quote ..

"But perhaps most surprising is the team's suggestion that there is no "mapping" between concepts of time passage and movement through space."

.. which was from an article about how one amazon tribe has no independent sense of time .. in the sense, i suppose, that time is embedded in the way one thinks of a lifetime.

"To link number, time, tense, mood and space by a single causal relationship seems to me hopeless, based on the linguistic diversity that I know of," he told BBC News.

Dr Pica said the study "shows very interesting data" but argues quite simply that failing to show the space/time mapping does not refute the "mapping hypothesis".

Small societies like the Amondawa tend to use absolute terms for normal, spatial relations - for example, referring to a particular river location that everyone in the culture will know intimately rather than using generic words for river or riverbank.

These, Dr Pica argued, do not readily lend themselves to being co-opted in the description of time.

"When you have an absolute vocabulary - 'at the water', 'upstream', 'downstream' and so on, you just cannot use it for other domains, you cannot use the mapping hypothesis in this way," he said.

In other words, while the Amondawa may perceive themselves moving through time and spatial arrangements of events in time, the language may not necessarily reflect it in an obvious way.

"mappings," eh? perhaps it is a good thing, sometimes, to be a "geometric mapping theorist" ..!

writing annoyances.

2011-05-18T16:58:00.000-04:00

i've recently developed a dislike for the "mathematical we." you know what i mean:

"with these hypotheses in mind, we now proceed to the proof of the main result."

perhaps i should improve my writing skills. "we" is particularly convenient, though:

it avoids the passive voice, which i was trained in school to avoid. it's possible to use the pronoun "it," but the ensuing sentence becomes quite wordy. [1]
"one" (the neutral, impersonal, third-person pronoun) serves this purpose, but it sounds awkward.

[sighs]

anyways, back to writing.

[1] actually, the sentence -- "it's possible to use the pronoun 'it,' but the ensuing sentence becomes quite wordy." -- is itself an example of how awkward it is to avoid 'we' or 'one' ..! q-;

Thoughts about a class about proofs (Part 2)

2011-05-17T22:55:00.000-04:00

two weeks later, i still think about that intro-to-proofs class.

i'm still without firm conclusions,
only guesses and impressions.

when i was structuring the course, i added in two weeks of propositional logic and truth tables, because the syllabus i was given didn't include it. i was hesitant about this decision at first. it would mean that i'd lose 2 weeks of lectures.

looking back at it now, i wonder if i did enough.

there were a few lectures about "techniques of proof" --

for each type (direct, contradiction, contraposition, induction), i gave an example,

then in later lectures i gave more complicated examples that combined multiple types,

and then for the next two weeks, every time i proved something, i either
stated what kind of proof it would be;
verbally explained how one method of proof fit the situation better than another [1].

so yes, i think that's plenty to explain what a proof is ..

but when i think about it, though, i never gave a single lecture about problem solving. perhaps i regularly explained a strategy behind a proof, where it comes from, but i never formalised the approach.

to be honest, i don't know exactly how i would do such a thing.
it just seems like so much common sense, a few guiding principles.

[1] in one student's evaluation, there was a complaint that we didn't cover enough basic proof techniques. my best guess is that many students only wrote what i wrote on the board, and didn't record what i said in relation to it.

to be fair, i tried to write everything i spoke on the blackboard. that, of course, gets very tedious after a while. it probably explains why i used to be able to cover 5 pages of my own notes in a single 50-minute lecture; now i usually cover 4.

Thoughts about a class about proofs (Part 1)

2011-05-17T01:12:00.000-04:00

two weeks later, i still think about that intro-to-proofs class.

i'm still without firm conclusions,
only guesses and impressions.

i'd like to think that my students have begun to see proof as a means of discovering truth in mathematics, much like how scientists think in terms of experiments in order to make new discoveries.

we know the world is out there,
that it obeys certain laws and patterns,
but how do we find out what they are?

i should emphasise: begun. many students tried very hard, but they were not often successful [0].

however, i think it very important that they tried. many of them improved substantially.

when i was a graduate student and more cavalier, i used to say:
"you know what half a proof is? half of nothing."

my older self would now disagree. maths is not something one learns instantaneously, nor is much else in life. even a video game take days of obsession to figure out and win.

that younger self did always underestimate the value of progress .. to the point where he (the old me) was stunned, after some years, that he earned a ph.d.

if you think of maths as a language, then it takes a lot of time to become fluent at it. if you think of a maths ph.d. as "mastery" (which i discourage you from doing) then it takes quite a few years .. on top of having completed an undergraduate degree for its preparation.

so i thought of my students' proofs much as i did compositions in a first course in spanish: there are always a few made-up words, errors in grammar, a lack of style from a lack of experience .. and occasionally you read a sentence that sounds .. real, that you could hear someone actually saying ..

[0] even induction proofs were tricky to some.

mathematicians amongst mathematicians.

2011-05-14T21:38:00.000-04:00

thursday afternoon we're in a van, driving to a conference. i take out a paper and began to read.

one of my colleagues asks me, "janus, are you working?"

i grin. "refereeing a paper .."

.. which becomes a lively discussion on the annoyances of refereeing and reviewing papers.

today afternoon, we're back in the van, heading home. i'm editing a printout of my manuscript. a friend also takes out a printout, clearly formatted in amsart.

"say, are you editing a preprint?" i ask.

"actually, i'm refereeing this article," he replies ..

mathematicians are predictable: at one point during the ride back:

one of us was refereeing,
another was latexing,
another was reading a paper,
and i was editing that preprint that i've been constantly writing about here and there.

there have been some posts i've been meaning to write here, including:

thoughts on how the "intro to proofs" course went,
initial thoughts about leaving the country,
life after one postdoc,
how the conference went.

don't worry: i won't let the door hit me on the way out.

2011-05-12T02:25:00.000-04:00

little by little, reality sets in:

it started a few months ago, when i was told that the department extended an offer to a new postdoc in analysis.

huh. so they found my "successor."

about a month ago, one of the departmental staff asked me if i was still going to use my office after the end of finals week.

never mind that he probably does nothing except teach;
what's this about janus doing research?

to my credit, i said that i still had a few projects to finish with faculty, over the summer. the staff didn't bring it up again.

yesterday i received an email from university computing services. unless i contact the network admin, my once-university email will be defunct.

i didn't realise that server space is so expensive,
that the university was managing so many users that they can't spare one more?

i knew, walking into this job, that it would all be over in 3 years. i didn't expect a reward for services rendered ..

.. but it would have been nice to exit gracefully ..

"words, words, words .. what is the matter ..?"

2011-05-11T12:09:00.000-04:00

well, i was right not to predict 20 pages because,
for some reason, the preprint has now reached 25.

[sighs]

there's nothing wrong with 25, but i thought that this would be a short and easy paper .. well, "easy" by my own standards ..

that is: if i proved it,
then it can't be that hard. q:

admittedly, i added a table of contents, which is 3/4's of a page, and the references fill up another entire page.

it's still incomplete, so: back to work ..

this picture better be worth a thousand words ..!

2011-05-10T12:42:00.001-04:00

maybe it's good not to have too many goals or make many promises.

there are too many interesting ideas out there,
there's never not enough time,
and every co-author eventually becomes impatient.

in other news, i'm still working on this preprint about measurabΙe differentιable structurεs. i think i've created a monster.

to make sure that the theorem is non-trivially new, i've come up with a new example to which the extant theorems do not apply. explaining it is taking much more time and space than i thought, though ..

.. in fact, it's become a new, separate section of the paper!

conceptually, it's not hard; it's a self-similar construction. in fact, it looks like this:

i'm embarrassed to say how long it took me to make this simple image. at any rate, fractals are definitely the analyst's best friend ..! (-:

EDIT @13:19: i uploaded the wrong graphic; the actual set isn't supposed to be symmetric. argh ..

quietude

2011-05-08T21:48:00.000-04:00

it's been a quiet week here. thinking back on it, i should either have taken a few days off completely or worked proper, full days.

it was only some days ago that i realised that teaching forces natural breaks in the week, even the day. that said, i'm not used to working 8 hours at a clip, solely on research. instead i worked for most of the daylight, idled after 6pm. even then there were plenty of breaks.

some people are fine taking complete vacations for a week. maybe grad school trauma is still fresh in my mind, but i still can't do it. the last proper vacation i had, going on holiday with my girlfriend at the time:

we climbed in the mornings, to avoid the heat of the midday sun;
while stretched and napped (for hours!) i took out my notebook, listened to the sound of the waves, and work on math.

admittedly, it was one of the best vacations i ever took. the food wasn't bad, either; only in thailand can you have a decent papaya salad.

[sighs] i guess we mathematicians have quite a reputation ..

2011-05-03T21:50:00.000-04:00

.. even from colonial days:

"Thomas Godfrey, a self-taught mathematician, great in his way, and afterwards inventor of what is now called Hadley's Quadrant. But he knew little out of his way, and was not a pleasing companion; as, like most great mathematicians I have met with, he expected universal precision in everything said, or was for ever denying or distinguishing upon trifles, to the disturbances of all conversation .."

~ from the Autobiography of Benjamin Franklin.

"so you can teach: ok. can you do research?"

2011-04-29T21:15:00.000-04:00

I guess i'm a good teacher: everyone says so, faculty and students alike. there are certainly worse things to be told.

every time i'm told that my teaching is good, though, right away i think about how pleasant it would be if someone would compliment my research (genuinely).

this is not to say that i disrespect teaching. i like teaching to students who are engaged in the material, and over the years i find that i like explaining concepts to newcomers [1].

make no mistake about it;
teaching may be easy, but teaching well is hard.

maybe it's that i spend much more time researching than teaching (including preparations), and i would rather the response be proportional to this time allotment.

maybe i suffer more in getting good ideas, making them rigorous, and polishing them, so that i value critical responses of that end more than another. so instead of a proportionality of time, i weigh it as a proportionality of pain. \-:

maybe i just want to be good at everything, that i just want it all! q-:

[1] part of it, i suppose, lies in the challenge. can i explain measμre theory to someone who took calc ii? can i convince someone who doesn't know any math why there are infinitely many primes? can i motivate why baηach-alaοglu [2] is so cool?

[2] admittedly, it's probably my favorite theorem of the moment. baηach-alaοglu is like a get-out-of-jail-free card .. if you have the right "lawyer" (in the form of a reflexive space!)

a proof of least resistance ..

2011-04-26T15:01:00.000-04:00

for the final exam in my proofs class, there were five problems. i told my students what two of the questions were, but not exactly.

reproduce five definitions, which will come from certain sections of the textbook (and yes, this is the same kind of problem as before).
reconstruct one of two theorems: one of them was the Cantοr intersectiοn theorem. on the day i print out the exams, i'll flip a coin [1] which will decide which one appears.

as it happens, the proof of Cantοr's theorem that i gave in lecture was different than the one written in their textbooks, and i told the class that either one is valid for the exam .. and if you want, you can write your own proof.

[snickers ensued]

i thought my proof is intuitive, when i lectured it .. but i hadn't seen it in a while. for some reason, i thought it wouldn't be that different from the book proof ..

.. until i started grading the exams, and realised:

wtf? this thing is a monster!
why didn't i just give a nice, short proof ..?

.. ye gods, did i prove this on the fly?
i'm pretty sure that i wrote up notes for that lecture.

was i trying to be cute?

to explain, i was trying to motivate the theorem at the time, why anyone should believe that it's true [2]. so i indicated it in two steps:

nested intervals imply monotone sequences of the corresponding endpoints. if you take infιmum and suprεmum (respectively, of left- and right-endpoint sequences) then you get a possible interval;

(in other words, do what the picture tells you to do, but do so rigorously.)
as long as we can prove that this "limiting" interval is nonempty, we're done. this only happens if inf < sup ..

.. but that can't happen, if you treat inf and sup like limits. to do this rigorously, use an ε-closeness argument ..

it's a "geometric" proof, sure, but not the most efficient one. then again, it was the only way i could remember how to do it ..

.. so props to the students who actually proved it that way, on the exam!

on a related note, some students actually gave their own proofs .. distinctly different, too, which was quite cool. (-:

[1] as it happens, it landed heads.

[2] i spent an inordinate effort, all this term, trying to make some of the tricker theorems intuitive. as i told the class: "if you don't have an intuitive idea of why the statement is true, then odds are good that you won't be able to prove it." in other words,

(logic) + (intuition) = (maths).

the joy of writing (with no sarcasm meant, strangely enough)

2011-04-25T17:54:00.000-04:00

i don't know what i was thinking, some months ago. for some reason i thought that this one particular project would take only 15 pages ..

.. call it the optimism of beginnings.

already i'm on page 18,
there's still (at least) one technical lemma to add,
and it's missing an introduction.

[sighs]

i would guess now that 20 pages should be about it. based on past experience, however .. let's just say that i'll focus on writing, and it will take as much space as it will need.

this is not to say that the writing is going poorly or painfully. it's quite the opposite, actually, even fun. by putting all of this stuff into LaTeX, i feel like i know one thing particularly well. there's also something enjoyable about phrasing an idea in the exact words that you want them.

i might even throw in a problem or two, if only to reward anyone who will actually read the final version ..

.. with the understood disclaimer that nobody else might find the problem interesting, of course. q-:

i've always liked those articles with explicit conjectures and open questions in them: somehow, it's like "giving back to the community .."

in which i encounter a catchy title.

2011-04-24T15:00:00.000-04:00

i couldn't stop grinning when i saw this title/abstract from the cνgmt server:

(the) Hitchhiker's guide to the fractional Sobolev spaces
Eleοnora Di Nεzza - Giampierο Palatuccι - Enricο Valdinοci

These pages are for students and young researchers of all ages who may like to hitchhike their way from 1 to s ε (0,1). To wit, for anybody who, only endowed with some basic undergraduate analysis course (and knowing where his towel is), would like to pick up some quick, crash and essentially self-contained information on the fractional Sobolev spaces W^s,p.

maybe i'm just a nerd at heart. (-:

this is just as awesome, though, as when i learned that there was a "dark side" to the caΙculus of variatiοns.

american english vs. "mathematιcal english"

2011-04-24T14:38:00.000-04:00

there are some bits of mathematical english that i find myself using more and more, in everyday conversation, like:

"to this end .."
"it follows that .."
"note that .."
"consider the following ____" [1]

sometimes i wonder if others .. say, my students .. understand that i'm using these words in a slightly different sense than usual:

"Now consider the following linear transformatiοn," said the professor, who, upon walking to the left corner of the chalkboard, began to write at twice the speed as before.

The student paused, and considered the possibility of keeping his notes thorough and complete.

After a moment, he set his pen down and closed his notebook. "Nah," he said to himself.

i wonder how often that happens .. (-:

as for the word "indeed" -- how i use it, anyway -- it appears right after stating a claim, thereby signaling the start of a sub-proof .. but only when the claim is not stated separately like a Lεmma or Sublεmma. as an example:

"without loss, f_k also converges pointwise a.e. to f. indeed, the sequence {f_k} is norm-bounded in L^p(X) with p > 1, so a standard functiοnal analysιs argument, using the Baηach-Alaοglu theοrem and Mazμr's Lemma, implies that .."

to be honest, i don't know how i developed that habit .. maybe from the advisor? i probably picked up a lot of habits from him.

on a barely related note, i quite like old-fashioned words.

at some point i had wanted to give a lecture in which i'd state a claim and start its proof with the words "verily, we note that .." but i never got around to it, though. (-:

[1] after a while, reading maths papers feels a little formulaic. there are only so many instances of "such that" which i can bear. on the other hand, perhaps the formulaic jargon we have is good ..

.. because it means that non-native english speakers can present their ideas just as well, for everyone to read. it makes my life easier, at any rate: my french is rudimentary, my german essentially non-existent. then again, i wonder if i can read de giοrgi in his original italian ..?

every semester's end, i get a few more grey hairs ..

2011-04-21T21:31:00.000-04:00

the term papers are graded. that's done. i feel ..

.. much like how i felt when i came back from india, last year: tired, feverish, and jet-lagged from the ordeal of being stuck in a moving vehicle that circles the globe at unnatural speeds.

i was firmly without any motivation to accomplish anything .. yet fully aware that i had to return to work, right away: there were projects to write up, and an nsf grant application to prepare.

still .. somehow, i managed, after that.

the pace doesn't let up. tomorrow is a final for one class, so there is grading right away ..

[sighs]

i had these happy dreams of close-to-a-week's time, during finals' week, when i could hack through a complete draft of these recent results i have. (the process is more technical than i though, and there are many choices to make.) [1]

when i was a student, the end of the semester never seemed so bad. my own experiences were calm ones. sure, there was studying to do, but the days were mine to plan as i please ..

.. ah. to be young again. \-:

[1] no: it hasn't escaped me .. and yes: my dreams are about work. to clarify, i mean being able to work on the things that i want! q-:

ignorance can be bliss .. and confusion can be comforting.

2011-04-19T18:09:00.000-04:00

an excerpt from an NPR article:

That's the moment you realize you're separated from so much. That's your moment of understanding that you'll miss most of the music and the dancing and the art and the books and the films that there have ever been and ever will be, and right now, there's something being performed somewhere in the world that you're not seeing that you would love.

.. from "The Sad, Beautiful Fact That We're All Going To Miss Almost Everything" by Linda Hοlmes

huh.

i always thought of the enormity of literature, art, and culture to be liberating .. in that there will always be something to read or see or complain or rave about.

it would be frightening, to walk into a library and to have read, to have fully understood, everything on the shelves. several reasons come to mind:

what would i do then? twiddle my thumbs?
think of all the responsibility of having that much knowledge: being aware of so much, realising how much is beyond our ken, how this world could be, realistically and how far we are from achieving it ..

then again .. the author does point this out, too:

It's sad, but it's also ... great, really. Imagine if you'd seen everything good, or if you knew about everything good. Imagine if you really got to all the recordings and books and movies you're "supposed to see." Imagine you got through everybody's list, until everything you hadn't read didn't really need reading. That would imply that all the cultural value the world has managed to produce since a glob of primordial ooze first picked up a violin is so tiny and insignificant that a single human being can gobble all of it in one lifetime. That would make us failures, I think.

myself, i'll settle for a lack of boredom .. and leave the societal implications to others.

on a related note: at a conference, it worries me when i understand almost all of a talk or that very little about it confuses me ..

(not that it happens often)

.. because if i understand it that well, then i'm probably not learning anything new.

then again, maybe i've been a student, a non-expert, for too long. maybe i'm so used to being confused that i wouldn't know what to do if i were an expert at something ..

well, that didn't go well.

2011-04-18T13:18:00.000-04:00

there is a difference between:

being awake enough to write a lecture,
& being awake enough to give a lecture.

i felt half-asleep in my linear algebra course, this morning. i forgot to slow down the computations .. and when i turned around, every so often, to ask if there were questions ..

.. there were students writing furiously.
[sighs]

that's the odd thing:

my numerical computations are faster when i'm not fully awake; there's none of the usual neurosis to slow down the steps for the students ..

sometimes i wonder if my students are aware that ..

in a lecture, presenting a computation takes at least twice the time it takes to write it down yourself;
the closer one is to the board, the less one is aware of the big picture. it's much like how close one is to a kitty (xkcd).

2011-04-18T08:45:00.001-04:00

[while grading, last week thursday]:

argh! most of my students think that every bounded sequence of real numbers converges!?! .. is this some sort of temporary insanity, caused by quizzes?

[during lecture, last week friday]:

"before we go into the bοlzano-weιerstrass theorem, there are a few things i should clear up. first of all, what's wrong with the following proof?

"since the equation

$$1 = (-1)^{2n} = (-1)^n(-1)^n$$

"holds true for all $n \in \mathbb{N}$, it follows that

$$\lim_{n \to \infty} 1 &=& \left( \lim_{n \to \infty} (-1)^n \right) \left( \lim_{n \to \infty} (-1)^n \right).$$

"because the right hand side limits don't exist, it follows that constants sequences such as $x_n = 1$ are actually divergent, not convergent."

hopefully they got the point. \-:

on an unrelated note,

it was another busy weekend of technical details .. and other things;
these days I use the phrase "the following" a lot. i blame my writing habits.

paranoia, again (also: i love walks)

2011-04-14T11:35:00.000-04:00

i'm getting prone to paranoia.

last saturday i was writing up some of the technical parts of a manuscript, and at some point i stopped and rubbed my eyes.

then i looked at the top of the page, one more time.

i don't see it.
why should that weak-star limit exist?
..
..
.. this looks like nonsense ..

.. and then the paranoia kicked in.

for most of the day i was completely torn: while trying my best to work out a counter-example, i was simultaneously hoping that it wouldn't work [1]. it got to the point where, having promised to meet friends for a symphony, i paid the ticket and couldn't appreciate the performance.

forget brahms! he means nothing to me:
what about banach-alaοglu?

after the second part, i ditched my friends, went home, thought through thoughts that didn't work .. went to sleep fitfully ..

.. woke up on sunday, averted my eyes from the pages on my dinner table .. decided to go on a walk instead ..

.. and while sitting on a park bench, watching dogs and humans, waiting for rain .. wondering how it is convenient for the human to leash the dog and restrict its movement. as long as the leash is long enough, the dog is not unhappy.

suddenly, it hit me: leash .. not too short, not too long ..?
didn't i have this one other hypothesis? ..

.. so i bolted home and leafed through the pages of my notes:

sure enough, at the bottom of the last page before my paranoia:
based on this context, without loss of generality .. [2]

and then i sighed a great sigh, powered on my laptop, typed furiously for a few hours, and then called up a postdoc friend to have a beer.

epilogue: on monday i realised that the proof is even simpler. once one makes use of this hypothesis, you can argue essentially without weak-star limits ..

[sighs]

that said, if you're like me, then: never think and mark up LaTeχ at the same time.
also, get enough sleep and take an occasional walk.

on a related note: my students have term papers due tomorrow .. and rewrites are due in two more weeks' time ..

.. and i was really hoping to finish this manuscript before that.

[sighs]
plans change, i guess.

[1] i've heard of the word Schadenfreude before, so this could be a strange case of .. Selbstschadenfreude! this may be a personal contradiction, but not a mathematical one! q-:

[2] while typing later, i couldn't help but think about the usual twist endings to the show house, m.d.

easy for you to say, mr. cookie fortune.

2011-04-12T13:42:00.000-04:00

after today's lunch [1]

(on a related note, my lucky numbers are 34, 4, 12, 37, 32, and 33.)

[1] ma po tofu, if you were wondering. (-:

[old post] ask a simple question ..

2011-04-12T13:16:00.001-04:00

i thought i posted this last week .. .. but apparently i didn't, so here it is.

on wednesday a student asked me, in the middle of lecture,

"if cαuchy sequences on the real line are the same as convergent ones, then why are there two different notions?"

it's a reasonable question.
why the redundancy?

i think i gave an excessively blunt answer, though:

without cαuchy sequences, we wouldn't have real numbers.

so yes, i stretched the truth a little. there are lots of ways to construct real numbers ..

A proof, or not a proof?

2011-04-07T11:04:00.000-04:00

i'm relieved -- it turns out that my proof is wrong.
i should probably explain which one, though ..

last weekend at a coffeehouse, it occurred to me that a certain theorem in metrιc geοmetry [1] was exactly the right tool that i needed. i couldn't remember the proof though, and being without a computer (and mathscinet) i tried to re-prove it on my own.

by sunday, i had a good idea and thought it would be a new proof.

on monday, it occurred to me that the argument of proof is too strong: i worked through an example that would contradict(!) one of my results from this year ..

.. so i spent two days frantically checking my proof of that result:

lemma by lemma,
corollary after theorem ..
.. so far, so good!

admittedly, the proof is a little strange. (i might write about it sometime.)

yesterday i browsed through the proof of that geοmetric theοrem: the ideas are completely different from my naive ones, which is relieving ..

.. and this morning, i checked through that naive "good" idea from sunday. it doesn't work -- i see why, now -- which settles the issue:

yes, i'm an idiot;
then again, my earlier theorem is still a theorem.

as a bonus, i think i've found the right piece of evidence. that theorem (as discussed here and here) looks to be unambiguously original.

[1] it's Thm 14.2 of Cheegεr's 1999 GΛFΛ paper, regarding the rectifiabilιty of isometric images of a certain class of metric spaces. (the hypotheses are somewhat technical.)

they weren't kicking and screaming, but ..

2011-04-05T14:30:00.000-04:00

in my lectures yesterday i was too technical, having dragged my students through some gory computations:

in linear algebra, computing the 20th power of a 3 x 3 matrix, via diagonalιsation, took 20+ minutes.

(admittedly, i started wondering whether it would be faster just to have multiplied the matrix 20 times .. [1])

as for the proofs class, we spent essentially the whole lecturing proving that every positive number has a square root.

the proof involved two lemmas on the fly. at the time, i thought it would be rather artificial to prove them in advance. instead, i had hoped that by proving those results as they were needed, the students would get a better sense of problem-solving strategies.

by the end, i couldn't help but suspect that my students just wanted it to be over .. \-:

[sighs]

when i was an undergrad, i remember learning theorems that took a whole week of lectures to prove ..

[1] for a square matrix A, computing A²⁰ would only take 5 multiplications, since all one would need are the products A² = AA, A⁴ = (A)², A⁸ = (A⁴)², A¹⁶ = (A⁸)², and A²⁰ = A⁴A¹⁶.

now that i think about it, is 5 the minimal number of multiplications? combinatorιcs isn't my strong point, and right now i'd rather go back to working on a research question regarding rectifiabilιty of sets.

at any rate, perhaps A²³ is more motivating, since 23 is prime. q-:

the (Frustrated) Over-Analyst

short and sweet!

*winces*

waiting for the inevitable news ..

my digital self.

the more things stay the same, the more they change.

distractions, distractions ..

sorry, not the bachelor; also, jet lag does not make for good talks.

no, this blog has nothing to do with "the bachelor" / choosing another journal ..

not the website you were looking for?

new year's resolutions, of a mathematical nature.

mοnty hall as a meme.

holidays.

an update, 3-4 months in.

call me old-fashioned, but ..

"the horror comes in reality from the mathematical aspect of the event."

today i found out that i am an author.

for some of us, certain LaTeX symbols are un-natural.

in which some constructions are "natural" ..

on a wholly unrelated note ..

boredom is a luxury.

after & before: part two.

a little too easy (updated)

after & before: part one.

.. provided that ε ~ 5.6 x 107 ..

jyväskylä: then and now.

before, then, and after.

weekends, obsessions.

(-:

once a student, always a student.

(-:

gluttony.

on why some bibliography styles are easier on me .. even obscenely so.

where there is a mathematician, there is probably a cafe.

on why computers are not as complicated as you think.

bombast (or: i really need to finish this grant)

realisations.

lots to say, lots to do, little time to write.

you can leave the states, but you can't leave the grant applications ..

a clean, well-lighted place.

the storm before the calm.

so far, so-so.

all work and no play .. something something something ..

perhaps it is time.

a few links, memory tricks.

thoughts between LaTeχing commands

when jargon creeps into colloquialism ..

the song remains the same (or: i found a cool preprint on the arχiv)

20 days and 20 nights?

in media res

one of those holidays again .. q-:

"don't let the mornings get you down .."

necessary difficulties (also: maths in the media)

sometimes, my first instincts are mathematical.

internet troubles (and more)

sometimes maths is like food.

work vs. lectures: also, theorems/problems as machines.

stranger in a strange land.

a quick nod to teaching [another article post]

maybe our reality isn't very real? (from an NYT blog post)

tonight: speaking to a small, yet (very) general audience.

retreat.

if time is money, then it's best to budget it.

a gradual transformation into a mathmo, again.

not yet an old dog, plenty of new tricks out there ..

sub-days.

a mathematical metaphor.

a kind of holiday.

wow.

admittedly ..

chocolate, coffee, maths.

lines are circles, from a certain point of view.

misreading (or: running out of coffee)

in which i should stop writing, so that i can start .. er, writing.

Thoughts about a class about proofs (Part 3)

jargon into vernacular: also, time is not necessarily platonic, in the linguistic sense ..

writing annoyances.

Thoughts about a class about proofs (Part 2)

Thoughts about a class about proofs (Part 1)

mathematicians amongst mathematicians.

winces