first an end, next week's another beginning.

four years have passed since my ph.d.  you'd think i'd have become competent at judging what is good work, what is bad, and what work is worth pursuing.

i still don't have the hang of it .. not even close.  i depend on a lot of people for advice, i bounce ideas off of people whose opinions i trust (at least, when i can get a hold of them), and sometimes i try to sabotage my own work [1] by repeatedly asking:

is this actually interesting?
do i get excited when i work on it?

does this answer a question that i want to see answered?

it's an unfortunate fact that the answer is no, most of the time.

---

if you weaken the conditions enough for a definition, then you can always prove something.  the point is whether there's any sport or surprise in it.

there's also something to be said for clean statements.  i've thrown away dozens of pages of highly technical special cases to "theorems" that i wanted .. saving only the crucial pages that i need to rebuild everything else, if i need to.  (it still adds up to a lot of paper, though, which is why i seem immune to the risk aversion of throwing work away.)

on the other hand, this sounds elitist.  maybe i'm not being quite fair.

it's hard for me to appreciate, for example, an elementary proof to a hard-sounding result.  it feels like cheating, like you've been tricked .. like you should have known all along and how could i have been so stupid not to see it?

when someone shows me a seemingly effortless argument, it's hard for me to step outside myself and think how long it took to find and shape this argument into simplicity.

---

so i think i've come upon the end of another project: this time a short one.  maybe it will be a 15-page note, depending on how much detail i'll show.

i can't tell how interesting it is.  i like the result(s), but the proofs aren't that illuminating.  the techniques seem a little too simple.  it feels like the same game; part of it involves generalised calculus on the plane again, something that i know how to use with some competency.

i feel like i should take time off from it before writing it up, see it with new eyes.  that way, i can judge it more objectively, decide if it's worth writing up.

..
..

that said, i'm going on holiday this week.

i'm leaving the office and the country, joining up with an old friend to see new places.  there should be internet, where i'm going, so if any random maths thoughts come up ..

.. then stay tuned. (-:



[1] it might also be career sabotage.  it's not like i have many published papers or preprints. every successful project is rather crucial, so it's a fine line to walk .. \-:

mildly mathematical: the prisοner's dilemma, iterated.

i'm not usually one for discrete mathematics, but this title/abstract caught my attention.

Iterated Prisοner’s Dilemma contains strategies that dominate any evolutionary opponent (William H. Press and Freeman J. Dysοn)

Abstract:

The two-player Iterated Prisοner’s Dilemma game is a model for both sentient and evolutionary behaviors, especially including the emergence of cooperation. It is generally assumed that there exists no simple ultimatum strategy whereby one player can enforce a unilateral claim to an unfair share of rewards.

Here, we show that such strategies unexpectedly do exist. In particular, a player X who is witting of these strategies can (i) deterministically set her opponent Y’s score, independently of his strategy or response, or (ii) enforce an extortionate linear relation between her and his scores. Against such a player, an evolutionary player’s best response is to accede to the extortion. Only a player with a theory of mind about his opponent can do better, in which case Iterated Prisοner’s Dilemma is an Ultimatum Game.

what is odd for me is allowing a "theory of mind."  being a mathematician, if a mechanism isn't well-defined, then it is forbidden from use.  as long as we're allowing these vague matters into the discussion, however, it makes me wonder:

is Player X's extortion also an instance of mind?
say that both X and Y have the same extortion mechanism; will it be anymore effective?

of course, maybe i should just read the article.  i mean, it's only 5 pages long.  the cool thing is that the article is available @PNAS via open access [1] .. so anyone(!) can read it. (-:

(also, does anyone think it .. er, telling .. that in the abstract, Player X is male and Player Y is female?)



[1] on a related note, there is a whitehouse.gov petition to make all taxpayer-funded research available online for free.  if you think it will make a difference, then the link is here (but it requires an account and username).

we still remember ..

my grandfather passed away two years ago.  when my family gathered together, my sister lamented how grandpa was gone and we'll never see him again.

i then told her that he isn't really gone;
with a critical eyebrow, she asked me what in the world are you talking about?

so i told her that i don't believe in souls, but i believe in the essence of every person. [1]

you see, i told her, as long as we still talk about him, remember him, then he is not truly gone.  it's just that he'll never respond to us again, tell us his good advice for the events that will happen to us.

it is only when the last of us, who knew him, depart from this mortal plane that he will really be gone .. but, by then, it's not like we would be in any condition to lament this, would we?

---

so i have a point in this, even something slightly mathematical.

today my department held an excursion -- a boat trip, and it was a fine day for it.  as goes the tendency of such events, many colleagues and i got to talking ..

.. and soon we were telling stories about the advisor.

somehow talking about him convinces me (partially) that he is not really gone .. that yes, we all miss him and his ways .. but until he is really forgotten, he will never be truly gone.


[1] i think she thought i had gone crazy.  at that point, i seemed the only one who was not close to tears.  (the way i see it, grandpa had lived beyond the age of 80, which in his generation is doing quite well; he beat the average already!


moreover it was only a few months that he was in poor health, which is more than one can say with many senior citizens with a variety of ailments.  in that sense, i saw it as a good life: many years of health, and a mostly painless way to exit.)

They are alive and well somewhere;
The smallest sprout shows there is really no death;
And if ever there was, it led forward life, and does not wait at the end to arrest it,
And ceas’d the moment life appear’d.
And to die is different from what any one supposed, and luckier.

~ walt whitman, leaves of grass

from the arXiv: that doesn't seem "excellent" to me ..

as found in the arXiv preprint "a cantοr set with hyperbοlic complement" by soutο and stοver:

"... we say that a cοmpact οrientable $3$-manifοld is excellent if it is irreducιble, atorοidal, and acylindrιcal [1].  An excellent $3$-manifοld all of whose boundary components have negative Eulεr characteristιc is truly excellent .."

*blinks*

oh, come on; of all the names they could have come up with, they decided on "excellent"..? [2] (-:

the described properties rule out so many topolοgical obstructions that they could have called it an 'unobstructed manifοld.'

even calling it a 'no funny business manifοld' would have made more sense, to me. q-:

*sighs*

on a related note, irreducιbility actually refers to a condition on (embedded) spheres.  it sounds like a good name would be "aspherιcal" ..

.. but that, of course, already has a topolοgical meaning. \-:


[1] i couldn't find a wiki and not being an expert on manifοlds, decided not to add one.  a google search, however, suggests page 10 of this paper by mcmullen.

[2] to be fair, the terminology did not originate from these authors, but from an earlier paper by myers.

mildly mathematical: did he just call us "geeks" ..?

this headline has already gone through the rounds, but one quote makes me laugh:

-- -- ✂ -- --

Ancient Mayan workshop for astronomers discovered
By Malcοlm Rιtter | Associated Press – Thu, May 10, 2012

The room, a bit bigger than 6-feet square, is part of a large complex of Mayan ruins in the rain forest at Xultun in northeastern Guatemala. The walls also contain portraits of a seated king and some other figures, but it's clear those have no connection to the astronomical writings, the scientists said.
..
..
On an adjacent wall are numbers indicating four time spans from roughly 935 to 6,700 years. It's not clear what they represent, but maybe the scribes were doing calculations that combined observations from important astronomical events like the movements of Mars, Venus and the moon, the researchers said.

Why bother to do that? Maybe the scribes were "geeks ... who just got carried away with doing these kinds of computations and calculations and probably did them far beyond the needs of ordinary society," Aveni suggested.

-- -- ✂ -- --

i think that the anthropologist (or at least, the article's author) just suggested that every pure mathematician is a geek.

now i'm curious what was omitted in that ellipsis .. (-:

on (math) ed: outsourcing lesson plans?

admittedly, i wouldn't have thought of this:

-- -- ✂ -- --

“I had an insight that the materials teachers created night after night had monetary value, so I set out to create a marketplace called Teachers Pay Teachers,” Edelman told Mashable. “Teachers are now making a pretty significant supplemental income and creating higher quality materials.”

-- -- ✂ -- --

~ from "Kindergarten Teacher Earns $700,000 by Selling Lesson Plans Online" @ mashable

i have no doubt that good, well-tested teaching materials are valuable.  even something like an example or an analogy, one that makes a particular concept absolutely clear to students, is definitely worth sharing.

for example, i teach convergence and divergence of infinite series, by comparing it to hollywood teen movies:

say you're a character in a hollywood film about teenagers and high school.  for simplicity, let's assume you're not one of the cool people, but want to be one.

so it's friday and we hear that one of the cool kids is having a house party.  naturally, you want to go.  you and your equally not-exactly-cool friends show up to the right neighborhood, but there's a guy standing at the front door, letting kids that are "cool enough" in and keeping "uncool" kids out.  being shy and insecure, you don't know if we're cool enough to get in.  so what do we do?

if you see your really cool friend apollo walk to the door and get turned away, then you won't be able to get in either ..

if you see your really weird, lame friend hephaestus walk to the door and get ushered in, then come on! you're certainly cool enough to get in, too ..

.. but if apollo gets in and hephastus doesn't, then we still don't know if we're cool enough to get in. [1]

on the other hand, to actually sell your lessons ..?
..
..
it's not that different from writing and selling a textbook, i suppose.  the difference is probably akin to buying a music album vs. buying just the song you like.

i guess mine is more of an open-source or creative commons kind of mindset.  if it serves the public good and if i don't need the money, then why not make it publicly available so that everyone can use it?  a few of my colleagues make their lecture notes available on their websites; i do the same with my in-class quizzes and solution guides [2].

there's one more thing, but it has more to do with politics than teaching:

the common bias is that teachers don't work very hard.  i've heard the slurs that "they have the summers off" and "work only a few hours a day" and "since they are part of a union, they must be lazy" .. all of which miss the point entirely.

so if the public hears that "teachers don't actually write their own lessons," then that's only going to make it harder to argue on behalf of teachers and public education .. \-:

to be fair, in an ideal world, teachers wouldn't have to buy good lesson plans, because they would prepare the materials themselves.

on the other hand, in that same ideal world, a teacher's salary would be a livable one, their workload would actually fit into an 8-hour day, and teachers would have all the available resources and state funding they need to do their jobs well.

thinking about it now, the sale of teaching materials isn't a problem, so much as it's a symptom of a larger societal problem.




[1] this is just an analogy for logical implication, of course, but for some reason it helps a lot of my students .. probably because of the ridiculous nature of the example.  as a mechanism for memory, however, this isn't the worst of approaches: see #7 of this wikihow, for example.

[2] as for why i don't post my lecture notes:  most of the time, my notes are more of a script than a document.  those handful of pages are meant to remind me of what i want to say, what i want to stress, and what i can add if there's time remaining.  (it's rare that i intend for anyone else to actually read them.)

from the arXiv: more measurable dιfferentiable structures ..

apparently there's a new paper out about derιvations on metrιc measurε spaces:

-- ✂ -- --

On the relationship between derivatiοns and measurable differentιable structures on metrιc measure spaces

Andrea Schiοppa

(Submitted on 15 May 2012)

We investigate the relationship between measurable dιfferentiable structures on dοubling metrιc measure spaces and derivatiοns. We prove: [1] a decompοsition theorem for the mοdule of derivatiοns into free mοdules; [2] the existence of a measurable dιfferentiable structure assuming that one can control the pοintwise upper Lipschιtz constant of a function through derivatiοns; [3] an extension of a result of Keιth about the choice of chart functιons.

-- ✂ -- --


interesting.
i've only just browsed the paper, but it has some good insights:

1. the notion of "independence" for Lipschιtz functions, as observed in the Cheegεr and Keιth papers, also works for derivatiοns.  (the basic idea is that, if there were partial derivatιves that formed a kind of differentιable structure on the given space, then due to geometric constraints, there cannot be too many of them.)
   
   in particular, the technique doesn't require any kind of embeddιng into a Euclιdean space, in order to detect finite dimensiοnality towards a measurable differentιable structure.  (for some reason, the idea never occurred to me.)
   
   
2. there's a notion called a Lip-derivatiοn inequality, which is a two-sided inequality: roughly speaking, it requires that there are generalised differential operators that, when acting on Lipschιtz functions, are comparable to "slopes" of  the same functions (i.e. pοintwise upper Lipschιtz constants).
   
   in an earlier article it was used as a key ingredient for a characterization of measurable dιfferentiable structures, when the measure is dοubling.  as it turns out, this new paper shows that one side of the inequality comes "for free," simply by careful measure theory .. which is actually pretty cool.

.. and all this time, i thought that nobody was interested in derivatiοns .. (-:

mildly mathematical: a familiar face, and the maths of obesιty.

when i was an undergrad, my academic advisor (i.e. the one who signed my registration forms) advised me to go into applied maths .. and in particular, into mathematιcal biology. suffice it to say that i didn't listen to him.

one thing that i remember about him was that he was always a direct, no-nonsense kind of guy. as it happens, the NYT recently interviewed him about the mathematιcs of obesιty; it seems that his manner is much the same.

he also discusses some interesting conclusions.

-- ✂ -- --
..

Why would mathematιcs have the answer?

Because to do this experimentally would take years. You could find out much more quickly if you did the math.

Now, prior to my coming on staff, the institute had hired a mathematical physiologist, Kevιn Hall. Kevιn developed a model that could predict how your body composition changed in response to what you ate. He created a math model of a human being and then plugged in all the variables — height, weight, food intake, exercise. The model could predict what a person will weigh, given their body size and what they take in.

However, the model was complicated: hundreds of equations. Kevιn and I began working together to boil it down to one simple equation. That’s what applied mathematicιans do. We make things simple. Once we had it, the slimmed-down equation proved to be a useful platform for answering a host of questions.

What new information did your equation render?

That the conventional wisdom of 3,500 calories less is what it takes to lose a pound of weight is wrong. The body changes as you lose. Interestingly, we also found that the fatter you get, the easier it is to gain weight. An extra 10 calories a day puts more weight onto an obese person than on a thinner one.

Also, there’s a time constant that’s an important factor in weight loss. That’s because if you reduce your caloric intake, after a while, your body reaches equilibrιum. It actually takes about three years for a dieter to reach their new “steady state.” Our model predicts that if you eat 100 calories fewer a day, in three years you will, on average, lose 10 pounds — if you don’t cheat.

-- ✂ -- --

the article continues here.  for those interested, it also links to a "body weight simulator" java applet that realises their model, in online form.

mildly mathematical: another film about maths.

from an article in wired.co.uk, which i found off a friend's g+ post:

-- ✂ -- --

'Travellιng Salesman' movie considers the repercussions
if P equals NP

By Duncaη Geerε 26 April 12.

Mathematical puzzles don't often get to star in feature films, but P vs NP is the subject of an upcoming thriller from Timothy Lanzοne called "Travellιng Salesman".

The titles refers to the "travellιng salesman problem", which asks: given a list of cities and the distances between them all, what's the fastest and most efficient way a salesman can visit them once each, and return home? The problem is categorised as "NP", indicating that it's easy to check that a given answer satisfies the criteria.

[ article continues here ]

-- ✂ -- --

in case you do visit the film's website, be warned: they're already made a synopsis of the story available .. which suggests that this film is really trying for serious, dramatic undertones.  in other words:

now that we know what's going to happen,
we can concentrate on how and why.

it would be interesting if, given the setup of 4 mathematicians and a payout, whether some version of the prisoner's dilemma will appear.  more likely, though, it's going to boil down to a version of "12 angry men."

---

apparently the film premieres this june in philadelphia.  i'll probably wait until it comes out on dvd, though, then invite a few friends over and play a movie drinking game of ..

"drink each time someone commits a mathematical sin" .. (-:

article post: on aspects of work, research, and solitude.

[originally written 15 april 2012, in an effort to avoid doing my u.s. taxes]

---

this excerpt is actually about evolutiοnary biolοgy, not mathematιcs .. but reading it still gave me pause.

"Haldaηe never expanded his napkin calculations into a formal mathematical theory. That task fell to William Hamiltοn, a young graduate student at Unιversity Cοllege Londοn. He struggled for years on the project, often working late at night on a bench in Waterlοo Station, where the commuting crowds eased his loneliness."

from: "the paradox of altruism" by jοnah lehrεr (@wired).

that's eerily .. familiar.

after reading de bottοn's the art of travel, i once spent an afternoon at an airport, even though i wasn't traveling anywhere.  part of me wanted to experience the airport without the usual stress of traveling, just to confirm that it wasn't such an awful place in my mind.

another part of me was just tired of working alone in my apartment, all the time.

if you're willing to believe steve wozniak of apple computers,
then creativity requires solitude;

that may be true .. but it doesn't account for how hard it feels, sometimes.

so for hours i just worked on mathematics at an airport cafe in the ticketing area, occasionally looking up to watch people say goodbye to their companions and depart for their gates.  occasionally it was distracting: heart-warming and heart-breaking ..

.. but after a while i saw so much of it that i became de-sensitised and these scenes became normal.  i could almost guess how some goodbyes would unfold.

the mathematics became more interesting than the humans:
i couldn't predict how that stuff worked ..

..
..
on a slightly related note, i learned this the other day, from yahoo! news:

What they've found is that when all outside noise is removed from an enclosure, human hearing will do its best to find something to listen to.

In a room where almost 100% of sound is muted, people begin to hear things like their own heartbeat at a greatly amplified volume. As the minutes tick by in absolute quiet, the human mind begins to lose its grip, causing test subjects to hallucinate.

that's a little unnerving to know; this may be over-simplifying the message, but .. a perfect vacuum is toxic to the mind.  as humans, we need stimuli.

i wonder if that's why i seem to pick up good ideas whenever i take a break, go out for a run, a walk, or a coffee ..

---

epilogue. there's another interesting part in that wired article:

"“When I began reading Hamiltοn’s paper, my first response was that the equation was way too short. I thought, There’s no way it can be this easy. But then I reread the paper. And then I read it again. And that’s when I got jealous.” Wilsοn wanted to understand the altruism at work in the ant colony, and he became convinced that Hamilton had solved the problem first.

To further the cause of inclusive fitness, Wilsοn began writing about the idea in a series of influential articles and books, introducing the startling logic of Hamiltοn’s equation to biologists. “I really became an evangelist for the idea,” Wilson says. “And this was not an easy idea to sell ..""

what i find surprisingly reassuring is that this more established researcher put his ego out of the way, embraced the idea for its own value, and pushed it forward.

that's not the end of the story, though.  it takes a different turn, but you have the link now, so i'm not going to tell it ..