Friday, August 31, 2012

in medias res: maths never sleeps.

so i woke up today at 5:30am.

it was partly by accident yet partly expected, due to jet lag. regardless of the cause, the moment when i saw the time on the alarm clock, i had one immediate thought:

if i get up now, then i might be able to Teχ up that lemma before my parents wake up and start talking at me.

yes, i know that i'm supposed to be on holiday. on the other hand, i'm trapped at my parents' house in the lost world of american suburbia, where there's little to do and few places to go.

at the same time, i'm still in the middle of a project that i had wanted to finish last week, but one thing led to another, the woman next to me on the flight across the atlantic wouldn't stop chatting at me [1] .. well, you get the point.

i don't know what i was thinking. i should have taken a proper holiday, away from both work and family, home and .. er, home.
often i dream about a little town in the mountains, reading a book left by a previous traveler in the only cafe in the area, where the owner whistles while dusting the mantle or stirring a pot of stew. probably i would write beginnings, endings, or random scenes of short stories .. but never quite finish them.

there would be mornings spent hiking through trails, with nothing more than sky and trees, dirt and dust and birds in branches. there would be evenings savored with a warm fire on the hearth, striking up random conversation with fellow travelers over a pint in the pub that doubles as a cafe during the day. [2]
i imagine that kind of holiday as something not easy forgotten, yet without any specific moment that's easily recalled. i imagine days and days, none more notable than the last, melting into each other into a diffuse feeling of contentment ..

.. not unlike the vague memories of a full and happy childhood.

i don't know how well i do without maths, even when i'd intend to set it aside for my own good. i guess some people just can't be helped.

[1] at some point in my life i learned that conversation can easily and often go one-way. i've become convinced that most people don't really listen to others; put another way, if you're thinking of how you're going to say something before the other person has finished talking, then you're no longer listening to the best of your capacity. i'm often guilty of this, myself.

[2] odd: when i imagine those scenes, i always see myself traveling alone. i don't mind people, but sometimes either my thoughts get in their way .. or perhaps that people get in the way of my thoughts.

Thursday, August 30, 2012

a rant on math ed: choose your words carefully.

from "the siege of academe" @ washingtonmonthly:

Minerva sprang from Nelson’s observation that higher education was increasingly a realm of mismatched supply and demand. Recent decades have been generally peaceful and prosperous on planet Earth. There are a lot more people with the desire and ability to pay for higher education than there used to be. Elite American schools are the unchallenged market leaders, which is why applications to Harvard have increased by double digits annually for years, with growing demand from China and other fast-developing economies.

what bothers me about this phenomenon is something more subtle.
what language is used here?
supply, demand, market leader ..

this is undoubtedly economics, the language of tit-for-tat.
i agree with the general opinion that education has value. what i don't agree with is that the value of an education is its sole defining characteristic.

i used to tell non-mathematical techies that my work is useless, in the sense that it has no direct applications or benefits to society .. that i can see, anyway. at the time i was trying to be (disarmingly) honest, compare my work with laboratory sciences.

on the other hand, like g.h.hardy i prefer it that way. i like the fact that somebody out there with the power to dole out resources realises that life and human understanding is more than just a matter of money and efficiency.

perhaps just to make my point, i'll start saying that my work is priceless [1], in the sense that you just can't measure how valuable or valueless it is!

Tuesday, August 28, 2012

military-grade mathematics problems, apparently.

interestingly enough, darpa has assembled a list of 23 mathematical challenges .. most of which focuses on stochastic and biological motivations.

as for the ones that seem passingly interesting .. to me, anyway ..

// warning: massive speculation follows //

Mathematical Challenge 10: Algorithmic Origami and Biology
Build a stronger mathematical theory for isometric and rigid embedding that can give insight into protein folding.

it seems that the convention is that isometric embeddings are defined as smooth, injective mappings between manifolds that preserve the (riemannian) metric [1]. on the other hand, the notion of embedding can be purely topological, so an isometric embedding should really be treated in the sense of metric geometry.

the problem sounds intriguing because it suggests to treat proteins like any other molecule .. which is to say, a discrete geometric object instead of something continuous, let alone smooth.

if they had intended someone to code all the symmetries and angular degrees of the various atoms and reduce it to numerical computation, then it would seem more like one of those big data kinds of problems. i don't think that's the point.

instead, i wonder if it has to do with some kind of "energy" .. in the sense that if a particular configuration of a protein is realised in physical space, then usually it's good for something. maybe it's even an optimal or close-to-optimal shape [2], so there would be some hope in finding a functional, some kind of "energy" for which the shape is minimal.

imagine that: a dirichlet problem for proteins!

Mathematical Challenge 15: The Geometry of Genome Space
What notion of distance is needed to incorporate biological utility?

i have no idea what this problem is even asking, but already i wonder what kind of geometry this kind of configuration space could obey. heck, it's not even clear to me what should be a point in this space: a gene? according to the wiki, they seem like finite subsequences that have been decided empirically to be distinct.

as a mathematician, that doesn't sound well-defined to me. \-:

despite the nomenclature of a "DNA sequence," could a sequence really encode all the geometry that serves biological function? what about genes that are related? now that i think about it, i have no idea what it means for two genes to be related.

all i imagine is a DNA sequence, strung out along a real line, and every relation between genes is like a wire connecting two points on the line. if the relation between genes could be quantified as more strongly or weakly correlated ..

.. then that sounds like a graph to me. i wonder if biologists treat DNA too one-dimensionally .. \-:

[1] every so often i run into a riemannian geometer, and suddenly i have to be careful to say "minimising geodesic" instead of 'geodesic' and "pointwise metric" instead of 'metric.' to be honest, sometimes i wish they'd start calling a riemannian metric a 'riemannian inner product' instead.

[2] .. which would only make sense if in the same medium or solvent, one sees essentially the same shapes all the time. it could be false from empirical evidence. i really don't know: according to the wiki, the configuration of a protein molecule could depend on the chemistry / structure of its surroundings, which only causes more complications ..

Monday, August 27, 2012

not a particularly flattering quote ..

"If the mathematicians are making fun of you for being too complicated, you know there's work to do."

upon reading that quote, i chuckled, then winced and sighed;
if you did the same, then i suggest revisiting the xkcd comic below.

as always, thanks randall!

Sunday, August 26, 2012

research, in medias res: reading, peace .. for a little while.

// initially written: thurs, 23 august 2012

yesterday and today i've been just reading and avoiding any LaTeX markup, which is its own kind of peace and quiet.

you can't really rush the process of reading, or learning in general:
at least, it doesn't work for me.

doing so is a split of priorities: the more one worries about how long a process takes, the less brainpower and memory is available to actually run the process effectively. [1]

i'm in the middle of finishing yet another preprint, though, so any peace i get is of a transient nature. it's hard to concentrate ..

.. but i think this is key for this point in my life. as for why ..

mildly relevant: a cost to critical thinking ..?

from "business guy to programmer - the good, the bad, and the ugly" @ crowdcademy.
  --- ✂ --- ---
I've also discovered that learning to code can have a big impact on your personality. Coding uses a lot of thinking patterns that I hadn't really used since my math and statistics classes in college, and even back then not in this intensity. As a result I've become more focused, more logical and smarter. But I've also become more detached from everyday life and less fun to hang out with.
  --- ✂ --- ---
after reading that, i imagined homer simpson shoving a crayon up his nose.

Wednesday, August 22, 2012

research, in medias res: insomnia.

// initially written: yesterday (tuesday, 21 august 2012)

last night i brushed my teeth,
drank a glass of water,
and went to bed ..
.. which, unfortunately, wasn't the end of the story.

hours later, i gave up, got out of bed, flipped on the light, went to the table, and jotted down some notes about the new project .. and in doing so, i ran into an error in yesterday's proof.

argh! i knew it, that something was wrong ..!
don't ask me why, but my body knew and couldn't rest until it was settled.

and so i worked until the coffee ran out, then until the sun rose, and just when i saw some progress .. it was time to go to work.

// added: today
the proof is patched: problem solved.

it turns out that the original argument was essentially correct .. only that i forgot to subdivide a set into subsets, where a previous condition becomes uniform on each subset.

now i feel cheated: i lost sleep over that?

more and more, i feel like my unconscious is constantly at work, running in the background, doing its best to catch up with the conscious, daytime version of me. i wonder if that's why i feel productive in the mornings:

is it really because at that point, my mind is fresh,
yet full of the ideas that my unconscious passed to me in sleep?

i have never woken up with the solution to a problem ..
.. but often enough, i wake up realising what doesn't work;
it's like the maths version of spider-sense or something.

the mind is a strange thing, i tell you .. mine, at any rate. \-:

Monday, August 20, 2012

research, in medias res: you know what's cool? (updated)

geοmetric measure theοry (gmt) is cool, really cool.
it's so cool that, at this moment, i think this stuff is cooler than elvis presley!

.. and, as you know, elvis is pretty cool.

man, i wish i could be a geοmetric measure theοrist ..

to elaborate, sometimes mathscinet is like wikipedia: you look up one thing [1], then after clicking on one of the references ..
.. then another ..
.. and another ..
.. maybe just one more ..
.. and pretty soon, you have a dozen tabs open.
so you start reading quickly. your short-term memory starts filling up quickly too, but before you're overwhelmed by the details, you realise that you've stumbled onto a mathematical thread, tracking the progress of a past line of inquiry:

what the questions were and what theorems came up,
who collaborated with whom,
what was never solved ..

[1] TIL that my newest result uses a certain construction from gmt from the 70-80's. apparently i'm not as innovative as i'd like to be .. but then again, that means that i can cite their work and not use up pages and pages to prove the same results! q-;

on (math) ed: brief thoughts about writing.

it doesn't seem fair to give alway all five points that the author below, davιd yοungberg, has made about online education .. but here is one of them:
4. Computers can't grade everything. MOOC's are feasible because a program grades all assignments. This works fine for answers that easily translate to machine language, but a machine can't grade an essay or a presentation. Papers are out of the question. But good communication is a valuable skill and one that's difficult to master. Fortunately there is a glut of Ph.D.'s in the liberal arts who can pick up the teaching in this area.
his other four points aren't bad, either, and the viewpoint is .. interesting, in that there is an undercurrent of marketing, of how to distinguish the good from the bad, the stars from the riff-raff.

see reason #2, then #5, and then re-read the first 2-3 sentences of the article.

at any rate, a common complaint about writing and literature .. one i've heard in school, anyway, and espoused for a while .. is the pain of subjectivity.

Saturday, August 18, 2012

epilogue2: consolations and comforts.

so .. as announced in an earlier post, yes: i got scooped. in fact, the other result was stronger than mine, which caused me no end of jealousy.

it took me countless hours of work and checking .. but i've submitted my own preprint and posted it on the arXiv. it includes a proof of the same (stronger) result, but by completely different methods ..

.. so i suppose that's worth something.

that said, i harbor no ill will towards that other author.  in fact, it's inspired me, led me to prove a new stronger version of one of my previous results .. but that's beside the point.
this other work is great stuff. i mean that honestly .. and as for why:

Tuesday, August 14, 2012

mildly mathematical: not your usual taxicab geometry.

absurdists and existentialists certainly have a way with words.

apparently sartre, like camus, had no objection to hone his descriptions with mathematical metaphors .. but mainly as a means of expressing cold, inhuman abstraction.

"Pure space suddenly appears. I imagine that if a triangle were to become aware of its position in space it would be frightened at seeing how accurately it was defined and yet how, at the same time, it was simply any triangle ..
.. My glance no longer lingers on facades seeking a house which, impossibly, would not be like every other house. It goes at once to the horizon and looks for the buildings which, hidden in mist, are nothing but volumes, nothing but the austere framework of the sky. If you know how to look at the two uneven rows of buildings that line the thoroughfare like cliffs, you are rewarded: they achieve their fulfillment below, at the end of the avenue, in simple harmonious lines, and a patch of sky flows between them."

~ from jean paul sartre's "manhattan: the great american desert"

i've always found mathematics comforting, myself. it was the one place which ever made any sense to me, an oasis in the deserts of philosophy.

Sunday, August 12, 2012

epilogue (or work-in-progress, part 5: the sequel)

my luck never ceases to fail.

i send off the preprint to a few colleagues, rest on my laurels for a little -- i.e. wait for their comments, if any -- and one day, while checking one particular post on the arXiv ..

.. i find out that i've been scooped!
more about this later .. but not now.

work-in-progress, part 4 of 4: triumph!.. i guess?

so this is the last of this series. maybe there will be others like it in the future, but i've told my piece for now .. maybe more than that.

honestly, this should be part 5; then again, why restrict ourselves to integers?

on a more serious note-- had this story ended otherwise, then i like to think that i'd still have told it .. but that's what i'd like to think. the truth is that i don't know. the most i can say is that i'd have written noticeably less, if only not to dwell too much on one failure and start something new right away.

some of you are students, graduate or otherwise, and perhaps this account just shows you that nothing ever goes smoothly, even years after the ph.d. .. but on occasion, that things move along well enough. (i could say the same about this blog as a whole.)

for those of you who know me, though: i still don't know why you trust me with anything ..

// initially written: mon, 30 july 2012 //

i patched the proofs and my old theorem's back from the grave. i feel pretty good, like not only have i dodged a bullet or two ..
.. but disarmed the bad guys, saved the day,
all the while muttering, "i'm getting too old for this sh-t" [1].. q-:
so if i may indulge in a little shameless gloating .. i feel like i really understand what's going on now, in the sense that my intuitions have become rigorous. along the way, i dare say that i've developed a few new techniques ..

.. and strangely enough, for Euclidean spaces!

it's either that, or i don't know the literature well enough so that i'm ignorant of previously similar constructions.

this has happened to me before; i thought that approximating Hölder functions by Lipschitz ones was novel,
but semmes has done it before, more elegantly; my colleague rοger's observations had preceded mine, too.

in particular, i am proud to say that some of these ideas involve geοmetric measure theοry. to me, this is a stroke of good fortune!

you see, this is a field that i have always admired; if things had gone differently, then perhaps i would have specialised in it as a student [2]. these things are hard to say, of course.

// initially written: thurs, 2 august 2012 //

you'd think that finishing a project would make me feel elated, but then you'd be giving me too much credit. i now feel empty, like i lost a worthy opponent.

at this point i'm reduced to typesetting, checking citations, and other minor corrections: hardly anything worth regaling anyone.

sometimes i feel like i've taken the ascetic ways too far: there is this common advice to not worry about the destination, but enjoy the journey there. now that this journey is over and i see the destination, i cringe.

that's it?

these last few mornings i've been scribbling on the same conference legal pad as i've done before, back when i thought i might lose those theorems. most of the time it's picking up loose threads from proofs, seeing where it takes me. i'm slowly collecting a small list of problems to work on next.

every so often, though, i try to construct a counter-example for the theorem again.   you never know ..

[1] suffice it to say that, as a hot-blooded american boy, i grew up on all sorts of action flicks: ah, the 80s .. (-:

[2] i feel like that about a lot of topics. for example, sometimes i think it would have been better if i had really focused on optιmal transpοrt or in stοchastic games, when i was younger and could pick ideas up more easily; maybe even banach space theory. for worse or for better, i'm firmly a metrιc-space guy now.
.. "if things had gone differently" .. that's a dangerous direction towards which to ponder.

Thursday, August 09, 2012

mildly relevant: this guy must have read my mind.

from: Work Less to Work Better: My Experiments with Shutdown Routines @ studyhacks:
This strategy worked fine for a while, keeping me engaged and happy, but then, in April, 2009, things took a turn toward the difficult. It was during this month that I accepted a postdoc position that would start in September.  This meant that I had to defend my thesis over the summer. [1] Suddenly the allure of tackling all new results began to wane.

Here’s a scenario that became common:

  • I would be working during the day on an important proof.
  • At some point in the late afternoon I would find a flaw.
  • A helpful voice in my head would point out that my whole future depended on finding a fix — without a fix, it argued, the thesis would crumble, I would be kicked out of graduate school and end up homeless, likely dying in a soup kitchen knife fight.
patching proofs is no stress-free matter;
it's nice to know that somebody else had the same weird fears that i did! (-:

there's one more bullet point, though:

Monday, August 06, 2012

work-in-progress, part π of 4: despair.

as i said before, this was the day when i learned there was something else in the preprint to patch. i didn't take it too well ..

// originally written: thurs, 26 july 2012 //

at some point i felt like giving up. i feel that way now. every time i try a patch i either find a hole in it later or i go back to another part of the manuscript and something else is wrong.

layers atop layers, patches upon patches,
water's still spilling in this leaky old boat.

i'm tired. it all seemed like such a good idea at the time, but it's gotten so ugly and i can't stand the thought of a simple idea — my simple idea — become this pockmarked thing:
bruised by technical details, carrying scars
from the surgical arguments that filled in earlier gaps.
i'm trying to make it simple: simple and clean. it's hard.

i can show you a highly technical statement that remains true, but it's not worth saying and wouldn't be worth writing down.

i feel like throwing away the result.

it was too much to hope for, anyway, and dangerous besides: it would have turned a perfectly good theorem into a tautology, sullied a very compelling proof that is quickly becoming a kind of template, a meta-argument.

it's late in the evening and i want to sleep, give it another go tomorrow.
things always look better in the morning, right?

i'm worried, though, that if i don't go to bed with a good idea, then i'll have nothing to try first thing in the morning. a bad start might ruin the effort, and i'll spend pages and pages, making desperate kinds of estimates and computations, draw many diagrams only to scratch them out after a few moments.

i want to sleep. i can't.

work-in-progress, 3 of 4: a suspicion or two, but generally proud.

this post is a little more cheerful, since by this point i start to see light at the end of the tunnel. in fact, at this point i thought i had fully patched the paper .. only to find, the next day, another gap in another part of the paper.

it never ends, does it?

no matter: the ending (that you'll later read) is near and this post gave signs of things to come. it is also a bit more technical, as you can tell right away. (to explain, all of my colleagues were away on holiday, so there was really nobody to share the details with.)

// initially written: tues, 24 july 2012 //

i'm a little suspicious now.

i saved one lemma with a new proof, and i think i may have patched the full one. for the latter case, i've narrowed down the technical issues to measure theory, so it should be a matter of checking standard facts ..

.. but i'm still suspicious:

i've never seen luzin's theorem used like this before, and i wonder if the statement is as general as i would like it to be.

added later: yes, the version of luzin's theorem
that i need is, in fact, true.
added even later: there's a more elegant approach,
so i don't need a luzin-type theorem after all!

part of me is seething, upset with myself for having been so stupid again.

how careless could i have been, to miss that error?
maybe i should collaborate more in this metric geometry stuff:
having another pair of eyes around has got to help, right?

something good .. or at least, something useful has come out of this, though. i know my proof much better than i did before, and i'm more aware of the subtleties involved.

moreover, i feel like i've become stronger with these techniques now.
to compare with rock climbing, it's like realising that, sometimes, you don't need cracks or holes in the rock to have a good grip.

if the rock is granite-based, then the surface is already rough with friction, i.e. a slab climb: you just have to be careful and not lose all the layers of skin on your fingertips, while doing so.
so i feel stronger and more able, and now i want to climb harder things ..!

in particular, part of me is proud. i honestly think that i invented (or at least, re-invented) a new construction ..

Saturday, August 04, 2012

mildly relevant: yeah, more (but interesting!) news.

as you can imagine, i waste a lot of time on the internet, reading news and pseudo-news. these are the slightly-mathematically-relevant links that i found recently.

so i've been a fan of jonah lehrer for a while, especially his book proust was a neuroscientist.  it made an interesting, humanistic take on an otherwise confusing discipline (to me).

it happened, however, that he was caught self-plagiarising [0] and making up quotations in his new (and now unavailable) book, called imagine: how creativity works.

anyway.. from "the deception ratchet" @ oscillatory thoughts (but as initially found off hacker news):
The part that is most relevant to the discussion at hand, however, is on the ethical "slippery slope" that I'm calling "deception ratcheting":
the blogger further cites the original article by tenbrunsel and messick, which can be found here.  the relevant excerpt is below.
The second component of the slippery slope problem is what we call the “induction” mechanism. Induction in mathematics is as follows. If a statement is true for N = 1, and if the statement for N + 1 is true assuming the truth of N, then the statement is true for all N. The way this works in organizations is similar. If what we were doing in the past is OK and our current practice is almost identical, then it too must be OK. This mechanism uses the past practices of an organization as a benchmark for evaluating new practices. If the past practices were ethical and acceptable, then practices that are similar and not too different are also acceptable. If each step away from ethical and acceptable practices is sufficiently small, small enough not to appear qualitatively different, then a series of these small steps can lead to a journey of unethical and illegal activities.

to be fair, this can be how errors propagate in academic literature, whether it be mathematics, physics, or economics.  just because an article has gone through peer review doesn't mean that it's been thoroughly checked, and one slip in one paper could mean that future papers that reference it will contain the same flaw ..

analogies aside, these authors just had to use mathematical induction to explain this, didn't they?

Wednesday, August 01, 2012

work-in-progress, part 2 of 4: lamentations.

there's more to the story, of course;
i wouldn't advertise four parts otherwise!

for those of you who are curious and want to know, right away .. yes: there's is a not unhappy ending. bear with me, though ..

// initially written: sat, 21 july 2012 //

when i think about it, there would have been an easy way to avoid all of this stress:
all i would have had to do [1] was recycle a talk from before,
instead of having insisted on discussing something completely new.
it would have been the safest bet,
the easiest approach .. but when have i ever been that wise?
then again, "easy" is overrated. q-:

stay long enough in this business, and you can hack your own habits.

over the years i've learned that i work well, but not necessarily better, under a deadline or some other constraint. this is especially true for ones that i cannot circumvent easily or without cost.

the reason, i think, is that there is neither room nor time for any debate:
when there is something that must be done,
in the here and now, then you do it [2];

it's that simple.
sometimes thinking is the enemy to progress;
other times, it is our closest, most valuable ally. [3]

[1] for a past conditional statement, i think it correct to use the pluperfect subjunctive mood, though i could be wrong. at any rate, who says that english is easy? q-:

[2] that said, i am terrible at keeping open-ended promises. (ask my collaborators and co-authors.) in fact, it's for close-to-the-same reason: when you stack your schedule and your mindspace with urgent tasks, then anything that can wait .. will wait.

[3] i get the feeling that most people don't have this problem. then again, this blog is called "the (frustrated) overanalyst" for a reason, you know ..

work-in-progress, part 1 of 4: confessions.

this is a 4-part series of posts that i wrote in the last two weeks, which might explain why i've not been updating very much lately.

// initially written: wed, 18 july 2012 //

so i found a gap in the proof of one of my "theorems" [1] and now i'm scrambling to fix it. one of the main lemmata still holds true and is interesting in its own right, so there's still a short note if this all goes to hell.

what bugs me most is that i gave a preliminary report about this result, about a month ago at a conference. though i didn't realise it at the time, this means that i was lying .. which bothers me to no end.

this is exactly why nobody should ever trust me ..!

[1] if it is not a rigorous proof, then the original statement is not a "theorem" but just a claim .. and possibly a false one. that's the whole point of the axiomatic method!