interlude: while on holiday ..

today i thought about the hei$enber9 9r0up while my traveling companion read her book. i then remembered that i don't know enough about 1ips¢hitz fun¢ti0ns on that metri¢ $pace.

also, i constantly forget: mathematics is not a mode to which one can switch, with the press of a mental button .. not for me, at least, not unless i'm obsessed about a particular problem.

instead, i have lately been concerned with obtaining cold things to drink, how to climb outdoors, and whether or not the mosquitoes will eat me alive.

see you in 2009.

i'm currently in japan (for 15 more minutes), and i will be in thailand until january. over a 12-hour flight today, i realised that an idea i had to prove one case of a conjecture can't possibly work.

this made for a very annoying rest-of-flight.

i might get some math done in that time, but more likely i will be rock-climbing and spending time with a beautiful german woman. so until then: happy holidays!

on strategies and un-wisdom.

there is a reason why my brainstorming sessions naturally last at most 3 hours (and often 2 hours). admittedly, the ebb and flow of ideas follows my caffeine levels.

there is also the inclination towards bad or crazy ideas: the longer i think about a problem, the more i want to solve it and hence, the more likely i will try crazy ideas that shouldn't work.

this morning i thought about the cases i know and whether it would be fruitful to push the idea further. in the end, having no other ideas, i did, and i completed my thoughts on paper:

"try this, maybe that.
if there's any justice in the world,
then this property should follow,
and by adapting someone else's theory ..

.. it might,
just might,
give us enough leverage that we can prove something close to what we want."

in other words, some sessions i spend, forming strategies and conditional mathematics. indeed, sometimes it is easier to edit a paper than to write one; similarly, it can also be easier to adjust a bad plan than to think of a perfect plan from scratch ..

.. well, at least for mere mortals like me. maybe you, my readers and colleagues, fare better in the world of ideas. \-: as for why it's helpful, sometimes you need an idea on a rainy day, and you'd rather just check details, so this makes portable the research process.

---

as for why all of this came to mind: just earlier i was running. as i was scaling the top of a hill, i suddenly thought:

wait. that's idiotic;
that can't possibly work.

angrily, i made it to the hilltop at a good speed but as i was cruising downhill ..

oh, wait. there is that property left.
maybe it might pan out.

so, a word to the wise, from the unwise: conditional mathematics is useful, only to a point. if you cannot see far, you might just make yourself go crazy with doubts.

retrospe¢tive: p0stdoc, semester 1

i wrote this last week but haven't gotten around to posting it until now. lately the end of the semester has caused me to lose any ability to focus on anything. research is in the back seat, especially today.

my students have their final and i have grading to do: both are arduous tasks. maybe after this weekend, i will be able to think again.

---

when i was a graduate student i met many postdocs. while there, my research group was one among many, and until recently, at any given point that group would have 2-3 postdocs.

thinking about it, it was a minor aspiration of mine to become one. as a student, i simply could not imagine being tenured or tenure-track faculty. there would be too many strikes against me.

i'm irresponsible; i know it. i don't know how to do research. besides, how will i ever know enough (and know it well) in order to teach a graduate course? could i possibly be a mentor to others and not inadvertently ruin their lives?

on the other hand, i could imagine being a postdoc when i was ready for it. it seemed to be something within reach. it looked like the good life, or at least, a good enough life: traveling here and being invited there, collaborating with good people, writing papers.

to this day, i don't know if i was ready to become a postdoc this year .. but i know i was definitely, unambiguously ready to finish graduate school. maybe i wouldn't make it as a researcher, and maybe i'll crash and burn. but ye gods, i want to try.

---

it's almost the end of my first term as a postdoc. i've made my first-timer mistakes and i already have some regrets. i've not done very much research, i've not gotten back to my collaborators (yet), and i've written less than i would like.

a year ago, i think i would have viewed it as a failure. a year, however, will change you.

it changed me: it took me a whole thesis before i could understand that worthwhile goals take time -- enough time to cause a mental itch or two -- and that we can only work a day at a time.

i've told graduate students this for encouragement: there's no reason why things in life should ever work out for the best, but for some reason, they do. working hard doesn't guarantee success, but it is important to keep working: of all those little claims and sub-lemmas and thrown-out mistakes that we collect, over months and months of work, some of them will be useful. you won't know until later, but they will be.

the best part is that it's not a lie -- not to me, anyway.

as for why it's not a failure,

i'm learning new things. i never thought that i'd study things related to non1inear e1asti¢ity or think about st0chastic games, much less learn them from people who are happy to tell me about them.

i may not have written much, but i have a better feeling of what i want to write. it will take time to put it all down into latex and to revise it into something that won't disgust me and others. a spring semester won't be enough time, but then again, who in life ever has "enough time?"

lastly: i feel useful, albeit a slight fraud. i know people who really study the ana1y$is on metri¢ $pa¢es, and i don't think i'm one of them. on the other hand, apparently my opinion is in the minority. sometimes a fraud can be a good thing: if i'm supposed to know things about metric spaces, then it means that i should learn more about them and become better ..

.. and maybe, just maybe, i will.

mathematιcs is tγranny, not demοcracy ..

.. and in some sense, that is a good thing;

i've just graded 5 homeworks. if mathematics were governed democratically (by my students) then given any smooth curve, tangent vectοrs at a point would be ιnvariant under a change of parametrizatiοn.

[winces]

oh well. at least it's not going to be a landslιde victory.

imagine, though, if true/false questions were really propositions to be voted on, and not propositions to prove! q:



EDIT: never mind. it was a landslide.

in which i think about un-research articles (and refer to j.l. borges)

it is hard to be productive.

yesterday morning and today, i jotted down a few ideas and worked with them a little, but it is hard to follow them through. none of them looks especially promising, so it's not clear if each is a waste of time. experience tells me that none of them will work, but one of them could possibly point me in the right direction ..

.. which means more nascent ideas. experience also tells me that none of those second-order ideas will be especially promising either ..

.. and even if this program does terminate, i may only end up with a counter-example. i may learn that it was, after all, a hopeless end. i would have learned something, but it's not like there is a journal which accepts un-articles and stupid ideas, right?!?

---

sometimes i wish that i would get good ideas more often. then again, that luckier, hypothetical me probably wouldn't notice that additional success, and he/i would probably wish for more good ideas.

so yes: even my hypothetical selves are ungrateful bastards .. which now makes me paranoid in a way which is reminiscent of jorge luis borges:

what if a wish did come true, and i am getting more ideas than usual? what if i am a realised form of a hypothetical, ungrateful bastard?!?

it's best not to offend the mathematical wish gods, then. i'm getting back to work.

i call it the "finishing disease."

some hours ago i gave my last lecture for this year, 2008.

i'm glad of it, but the last few classes leave a bad taste in my mouth. i have never understood the purpose of review classes; there are of course some topics with which most students have difficulties, but is it really purposeful that students sit passively in class, watching yet another example and copy down the details?

my opinion is that the students are better served when their role is active, not passive, and when they take their own initiative and work out examples of their own, decide for themselves how the steps should go. if i could have replaced these three review classes (three of them!!!) with three office hours, then i would have.

regardless of opinion, the teaching is done. that should count for something.

---

the end of a semester fills me with malaise. i think of what i wanted to do and what remains, what i did not do. more strange feelings:

i feel the urge to do something, now that my lectures are over. now i have time to read and write and think ..

.. but then again, it's been a long, first semester as a postdoc. i'm tired. my holidays are full of worry. when i try to sit down and think, nothing comes.

i will call this the "finishing disease."



oddly enough, i have some positive things to say, too. more, in another post.

on mentoring (read: NOT ME).

(at some point i'll write less about teaching and more about research, but as the semester is ending, teaching pains become more severe than research pains.)

i know many mathematicians who are nurturing and who promote young people to pursue mathematics. on the other hand, i suspect that my manner unconsciously discourages students to go any further in mathematics.

as for why, i have a few hunches, but they'll have to wait.
for now, i have a review class to teach.

[continued from earlier] maybe i shouldn't count teaching calculus as either nurturing or discouraging. i mean, it's .. calculus.

let me rephrase that: a calculus class is not the sort of place where you run into many excited, curious minds .. at least not with a curiosity towards maths. this is not to say that it is some sort of pur9at0ry or prison, either.

in fact, it's more like a long flight .. say, twice as long as the battery life of your laptop. there's nothing you can do as a passenger but wait out the trip, hope for the best, and if possible, make good use of your newfound "free" time. [1]

there's an apathy at work: a lot of academic majors require calculus and to pursue their own disciplines, students have no choice but to sit through the lectures and the problems with funny symbols and diagrams.

admittedly, i understand this.

as an undergraduate, for my own major i had to take a physics course. it wasn't bad and i do think physics is mildly interesting .. but nonetheless, i wouldn't have taken the course if i weren't forced to take it. in fact, i was perfectly happy to take (english) literature and sociology courses instead of physics courses.

but i digress.

as for why i discourage my students, my teaching manner bifurcates. i feel like whenever i explain something, then i receive one of two reactions from the crowd of student faces:

1. ye gods, this is boring. you just do this and that. didn't we go over this stuff earlier? i know how to take a gradient, already!
   
   
2. i have no idea what he's talking about. he seems to be speaking english, but .. what just happened? ¢rap: notes! i have to write this down, somehow.

as for why, i have my suspicions.

i hate starting a topic from scratch; it's like a "cold boot" for a computer, and for me it's disorienting. if mathematicians understand anything about the human condition, then above all else they understand what confusion feels like.

so i like to start a new lecture by reviewing a little of last lecture or a familiar topic from before: thus, reaction 1.

of course, "familiar" is a relative word. my students might remember that the computations will go roughly this or that way; to them, those computations are "familiar." [2]

remembering, however, a geometric fact and then using to explain this fancy new formula, may not be familiar. if they only remembered, then everything would make sense. instead, it has all suddenly become gibberish. hence, reaction 2.

i referred before to the demographic of my students, but i should clarify something.

where i was a ph.d. student, we would never get to teach any mathematics majors. they were tracked early to honors classes, leaving us graduate student instructors to teach the students who had no particular enjoyment for math and wanted their course requirement out of the way.

here at my postdoc institution, the culture is different. humble beginnings are fine. some of my students have told me that they are considering the mathematics major.

whether they will follow it through, i don't think i'll be of much help.

i'm too young, too pessimistic;
i even ask myself why i do mathematics, sometimes.

i don't think i have enough hope, now, to share any with the next generation.

[1] ah, who am i kidding? flights are their own share of he11.

[2] from experience, students love equals signs; everything is equal to everything else. they will say "equals" when they really mean "thus."

sometimes i fear that if i don't write a fact down as an equation, then half of my students may never remember it.

briefly, about teaching (yesterday)

i will now make a teaching vow. unless i know that i will be the one writing the final exam, i will never again permit formula sheets on exams. you think that you're being a nice guy by making the lives of your students simpler, but that's not the case.

it just means that when the formula sheets are gone, then you become an "unreasonable" teacher, the bad guy. since the students have never had to learn without these formula sheets, they don't know that they can do without them. it seems like too much work.

such formula sheets are crutches, anyway;
i should never have used them.

---

i forget this, every time, until that moment while in lecture when everyone is deathly silent: never, ever suggest that there is no sure-fire way to solve a particular problem.

this time around, it involved showing that a mu1tivariate 1imit exists or does not exist. in textbooks the problems always boil down to tricks -- how one chooses certain dire¢tions to obtain different directiona1 1imits.

i give suggestions and general principles. they are still panicked that there is no hard and fast rule to "how do you know when you should try to show that the 1imit does exi$t?"

i guess students are really that insecure about their reasoning skills. perhaps they've never been responsible for "thinking their way out of a problem," which is what we mathematicians do for a living, day in and day out.


(in the past, i've panicked students when discussing c0mparison tests for infiniτe series and methods of inte9ration for riem@nn integrals.)