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.)

maτhematica1 pr0spe¢ting.

i suppose that many mathematicians state theorems in a similar way as how gold prospectors once staked claims in the california gold rush of 1849.

the average prospector was as able-bodied as his peers; if he staked a claim on a good piece of land, then it was only because of circumstance.

some theorems are just waiting there and anyone can prove them. it's a matter of who proves it and puts it in print first. there's no guarantee of worth; one just stumbles upon the claim, decides that it may be worth the time and effort, and starts digging.

yesterday i drew up a very special case of some ideas i mentioned before. the setting is not at all generic -- the hypotheses are strong -- but i worked through it because it was the only case i suspected i could do.

for now, it feels like a piece of land with the stream that is almost dry. the ground is mostly loose soil and no rockface: no real chance of gold.

at the very least, i found it first. i get to see how much (or little) it's worth.

if proving theorems is like prospecting,
then conjectures are like hidden treasure ..

.. only there is competition for where it is hidden,
and everyone draws a different map, where X marks the spot! (:

turkey eve → no math.

(thanksgiving holiday) + (girlfriend visiting) → ((no work done))

then again, everyone needs some time off. today i thought a little about this one vexing special case of an open problem from [AK]. i got nowhere, though, because i was ..

.. distracted.

---

as for this little problem, what helps is that i know what not to look for. for one thing, h@usd0rff dimeηsi0n [wiki] isn't a sophisticated enough tool for the job, because i have a few non-examples.

these same concrete examples suggest that the problem will involve subtleties in 9eometry (or more appropriately, some sort of 9e0metri¢ mea$ure the0ry). however:

1. so far i haven't found what this subtle property should be.
   
2. i have a guess, but i cannot formulate it in a sufficiently well-defined way as to test it.
   

this sounds like progress, but don't be fooled; i knew of these non-examples months ago. so i'm afraid that the first day of holiday will have to be unproductive.

maybe tomorrow will be different; maybe i'll do more reading and writing, instead of novel thinking.


[AK] from p.68 of @ppendix A of this artic1e from sprin9er1ink. some people refer to it as the "f1at ¢hain ¢onjecτure."

standing still; slow steps; a strange soothsayer [1]

since defending my thesis six months ago, i have gradually become less productive. this weekend i've not made much progress at all, and yesterday i successfully avoided negative progress:

i thought i had proved a lemma some days ago, but yesterday morning i realised that one step was wrong. after an undisclosed amount of time in a mild panic, i realised that the step was unnecessary and the lemma remains true from a weaker observation.

so i gained nothing but also lost nothing: non-negative progress!

---

i've also been browsing a few papers, trying to learn more about the sob01ev space W^1,n(M;N) for M and N manif01ds and n the dimension of M. like learning anything new, it's going slowly.

i repeatedly tell myself to be patient and that intuition and understand will come .. all in good time. i tell myself that my most recent experience has been a singular one: i was writing a thesis and concentrating on the same thing, day in and day out, for a year or more.

that's why the thesis stuff seems easy and this new stuff seems hard;
you've spoiled yourself: you've forgotten how hard it was, before.

these days i look at my thesis and i wonder why it took me so long to prove what i proved. i now think it is all obvious. maybe i'm right, but again: i'm biased and my opinion no longer counts.

i wonder how it looks to someone else.

---

on a lighter note, i ate at a "pan-asian diner" today for lunch and this was my cookie fortune:

the best angle from which to approach
any problem is the TRYangle.

ye gods, that's an awful pun;
to my discredit, i did chuckle a bit after reading it.


[1] my painful attempt at alliteration.

they understand! (on ρostd0cs being gh0sts)

ye gods, this particular ρHD comic is funny:

"The P0st-Do¢toral Sense, pt. 1"

sometimes, that's exactly how i feel;
there aren't so many p0stdocs here at ρitt than .. say, at mi¢higan.

random thought, in midst of research.

today is a day of handwritten notes and idle research thoughts. i've made the following trivial observation:

"nondifferentiability" is a long word to write out!

"m0re matter, with 1ess art." [1]

i feel like i've promised to read too many papers with colleagues.

as a result, i'm browsing through them,
getting the lay of those mathematical lands,
but only a little,
and i still get lost.

so from now on: fewer papers, more depth!


[1] one of 9ertrude's(?) lines from ham1et, in an early scene. the king's advisor, po1onius, has this tendency to ramble. so i suppose this is the $hakespearian way of saying, "get to the point."

also: a pleasant surprise.

this was strange:

after my afternoon lecture one of my students told me that i was his favorite prof of the semester, and he liked my lecturing style best. first i was suspicious of this, but it then occurred to me:

i don't know his name; he knows that i don't know his name.
this may not directly pertain to his grade after all.


to this, i wasn't sure what to say. i knew not to say,

but this isn't "real" math;
i never prove any theorems for you guys.

i'm just doing examples;
you can read similar ones from the book.


in the end i think i said thanks, and told him that he should be glad that he wasn't in my morning class instead.

about writing, again.

i remember being told by a colleague that f. 9ehring would always begin a talk by giving the definition of a qua$ic0nforma1 mapping. foolishly, while at michigan, i never actually asked fred if this was true.

for my own part, i suspect that each time i start a draft of a paper, i'll end up giving the definition of a Lips¢hitz mapping. [1]

on a related note, i'm finally going to write up those results that i proved ..

.. um ..
.. ahem ..
2-3 years ago ..

.. so yes, moving on: better late than never, i suppose, but it is certainly late:

1. i've already given two conference talks about the subject matter;
   
2. i don't think i'll be able to improve the results in the near future;
   
3. it never hurts to have another paper;
   
4. ..
   
   .. it's what the advisor would have wanted. i think he wanted to read the crappy draft/notes that i wrote, some years ago, but something would always come up.
   

anyways: enough regrets.
it's time for me to do what i promised to do, years ago.


[1] every so often i'm tempted to follow the old-school route and call them "Lips¢hiτzian" mappings. for some reason, however, i just .. can't.

good vs bad news (also: post #500 ..!)

the good news: i referred, some time ago, to a crazy idea.

as it happens, it's not such a crazy idea after all;
it will prove what i would like it to prove.

the bad news: someone thought of it first.

you see: to implement this no-longer-crazy idea, i needed this one result. as i browsed through the relevant paper, i found out one remark which encapsulates the spirit of that idea.

so in the form that i've thought it through, my idea is unoriginal and worthless and not worth publishing. it is a two-sentence observation [1] and adds nothing to the body of mathematical knowledge.

i could always try and think of a method of proof which uses independent methods, but ..

argh--
what's the point?

[1] this is meant in the same sense as when someone has a "one-line proof" of a particular result.

not my best friday, mathematically.

if i said that i was completely unproductive today, then that would be a lie. it would be true, however, that i was wholly unproductive while in the office.

so many mathematical things to do,
and i did none of them, today.

..

then again, that's not quite true .. not in spirit, anyway. i did teach today, though i think it went poorly. last night i got excited about this one "application" of stoke's the0rem [1] that i realised that i could explain.

put in layman's terms,

"the surface integral of the curl of "any" 3-dimensional vector field, over a 2-sphere, is zero." [2]

after i finished those few minutes of explanation, i noted that my students' eyes were more glazed than usual. oh well.

at any rate, i did do some sort of labor today. as for research, i did read a 3-page research note from 2003. from it, i learned something:

the more i think about it,
the less i really know about d0ub1ing measures.


[1] isn't it strange how badly math jargon translates, to non-mathematicians? for example, when someone in pure maths refers to an application, (s)he's probably not talking about physics or chemistry or anything related to the world.

similarly, when i say that i will estimate a certain quantity, i don't mean that i will give you a decimal number of a quantity that can actually be measured physically.

[2] i.e. sufficiently smooth vector fields in R³. sadly, to most calculus students, every function has a derivative.

thoughts ahead and abroad

i'm thinking of spending two months in europe, next summer. the dollar may be weak, but it would be nice to be away from it all for a little while ..

.. and, say, closer to a particular pretty german woman that i know, overseas. (:

already there have been announcements about summer schools in the czech republic and spain, about the topics in analysis that are familiar to me. in italy, there will be lectures on topics that are more applied (but still interesting).

in late summer, there will be an analysis/PDE meeting in sweden. probably there will be something in finland as well: the jyva$kyla summer schoo1.

i'm keeping a list of these meetings here.

500 and 1934

lately i have been reluctant to post here. one reason is rather superficial: this will be my 500th post on this blog, which is somewhat troubling.

have i procrastinated that much, over the years?
no wonder why i have so few papers and preprints.

another reason is that i have little to write about, other than the usual complaints about teaching and research and various aspects of academia [1].

today, however, i am in a good mood. i'm LaTeXing research notes and i don't have to teach for another .. 22 hours.

---

earlier i followed the advice of a colleague and looked up a splendid little paper of M¢Shane from 1934; it's the first discussion of extensions of 1ips¢hitz functions, which is now a very standard technique.

there is something soothing about reading papers that are at least 30 years old, but i cannot say why. it's like listening to bing crosby or frank sinatra; you cannot help but make a genuine smile.

those days were probably not simpler times, as nostalgia colors our judgment, but they still feel that way to me.



[1] it seems like the appropriate term has shifted to "acadème." i see it whenever i read about universities.

like the now-common spelling "élite" (say, for example, in time magazine), it makes me feel old and a bad speller. i guess the english language is changing, or the american perspective is truly becoming global.

sudden thoughts, upon waking .. [added: and before sleeping ]

upon opening my eyes this morning, i thought immediately of two things:

1. you know, it's cold in this apartment.
   did i turn the heat on, last night?
   
   
2. wait. if our metric space is the entire plane, then ..
   <insert unspecified singu1ar mea$ure here>
   
   .. shouldn't be doub1ing. so does it mean that ..
   <insert crazy idea here>
   
   .. will work?!?

mind you, this came to my mind without caffeine;
amazing!

as it is a teaching day today, i haven't had time to work out the details. subsequently, today i don't like teaching very much.

---

i had time to think about the details, after all. it's still not clear whether the crazy idea will work. on the bright side, is that it reduces to invoking the theorems of others and reading their work carefully.

on a related note,

1. sometimes i wish that a paper of a1berti, cs0rnyei, and prei$s were ready; it's the work in preparation that concerns sets on which Lips¢hitz functions are n0ndifferenτiab1e. at this point i really do want to know some details about how they do what they say that they do.
   
   
2. i know many mathematicians with very clear motivations. they are specialists, experts in their field and from careful, deep analysis, they prove good results. several words come to mind:
   
   centered,
   focused.
   
   i have no such qualities. maybe it is a happenstance of youth and having just started an independent, mathematical life, but it is hard to focus on one single subject and then to work deliberately.
   
   there are just so many interesting things to study. as a result, i seem to recall quite a few facts, but know how to prove very few things.
   
   the few theorems i have in mind have chimeric proofs; they are often formed from borrowing random ideas from different theories. the arguments would be valid enough, but for step 1 i need this theory and for step 2 i'll have to introduce notation and lemmas from another theory.
   
   a proof is a proof, but a good proof should explain itself. a good proof should adhere to morals. put that way, my proofs are immoral scrawlings!

a mild vow .. [NEW: EPILOGUE ADDED]

must .. stop .. talking, at seminars;
there is research and writing to do.

one last talk for the semester, and i will become a research obsessive;
that's a promise.

---

usually between my ca1cu1us lectures, i like to work on random research ideas. instead, today i spent the time jotting down notes for tomorrow's talk.

it's only now that i'm entertaining the thoughts that i would have had, in the morning.

---- ADDENDUM: 6 NOV 2008, 10PM ----

i like this seminar in my department and i like giving talks there; the audience is friendly. it's just that i don't want to talk too often.

so let me modify that vow:

when i have something new and interesting to say,
when i feel that i can say it succinctly and well,
then i will give another talk;

it will probably be late, next term.


as for these benchmarks, today wasn't one of my better talks. i began in media res and there were new faces in the audience, which i hadn't counted on.

this was part 2 of a talk, after all; i expected the audience to shrink, leaving only the hardy survivors that i didn't already disgust or confuse.

so my review of part 1 took longer than i'd have liked and as usual, i never reached the end of the argument that i wanted to convey. as a rough estimate, 1/4 or 1/3 of the talk occurred in the final 5 minutes.

i hate it when that happens: it's a disservice to anyone in the audience who's been patient enough to follow you for the first 45 minutes .. or in my case, 50. (i'm almost sure that i went overtime.)

a legacy from the advisor.

sometimes i wonder how the advisor once did it: he met with so many people and discussed all these different research problems with them. i'll never fully know what the advisor knew and thought about .. not that it's any of my business. [1]

sometimes, however, i felt there was a bigger picture involved, but we would be working with this one problem and seeing how it goes.

      fool that i was,
      i never thought to ask him explicitly:
      why? where does it come from?

that was why i liked the advisor's talks so much. that was when he'd explain the whole, what motivated him, and there was always a motivation-- what was known and what wasn't (but remains interesting).

he wrote much like he lectured. last week i had occasion to browse through one of his last articles, from bu11etin of the AM$ of this past year. there were many topics that i had seen, but not in concert like this. then there were new ideas there -- new to me, anyway -- and on the whole, i saw him in his article.

i didn't feel ignorant, while reading it;
i felt myself becoming more aware, about what it all means.

      it is nice to know, in this sense,
      that the advisor will never be truly gone.


so, enough apotheosis: i do have something slightly critical to say. i wish he told me about one particular open problem himself. i don't even know if he was the one who posed it or if it was j. ¢hee9er; the problem can be found in the bu11.am$ article, but in plain text. even there, one must be looking for it.

as for this problem, i had to learn it from others who asked me recently if my methods will work towards its solution. apparently i must have been among the only ones who didn't know about it ..

.. and i'm working on a special case, now.


[1] thinking about it, that would be creepy, to know all of that. i think of the opposite situation, where people are trying to determine all the research directions that i'm currently entertaining.

that would be big-br0therish! would that make me some winsτon?

unsolicited opinions on teaching.

earlier today i asked myself if i like teaching mathematics. that, of course, is a dangerous question, and i still don't have an answer. in fact, i'm treating it mathematically and handling special cases first.


i like teaching small classes. this is in contrast to how i am teaching, this term. in each section of multivariable calc, this fall, i have at least 70 students. when i administer a midterm exam, there are 150+ exams to grade.

then again, those are logistics. they are part of teaching, but there are other parts to talk about.

it's hard to say whether i am any better or worse at reading faces than other instructors. however, it is very hard to read the faces of a large crowd and determine their aggregate perspective about what you just said and ..

      whether they understood you,
      and if not, whether you should repeat what you said,
      or whether you should draw a better diagram.

also, it unnerves me to talk to a student who i recognise (because (s)he sits in the upper right area of the classroom) but where i don't know her/his name. maybe it's possible to learn 100+ new names, every term, but for me, it's not easy.

"mathematically, i'll need reinforcements."

all the second midterms have been graded,
part 2 of my seminar talk isn't until next thursday,
and i had all of today to think about research ..

.. so inevitably, today was unproductive.

rather, i thought off and on about one particular problem but couldn't get anywhere. i don't even know whether the answer should be yes or no, which irritates me.

i don't know why i thought it would be easy, because plenty of people have thought about it before. one special case does follows easily enough, but in general ..

.. argh.

the results that i've proven before, they are useless in the general case. i don't know enough-- not about this problem or about topics related to this problem. there's going to be a lot of reading involved, i think;

i won't be able just to waltz in, prove what's what, and saunter out. no: there's some hard work involved. even if i don't solve the problem, i think it may be worthwhile. i'll definitely learn something, get something out of this.

so ..
what papers to print out, first?

a year later: in memoriam to the advisor.

i haven't posted very much, this week. it's been busy with work and i've been preoccupied with the past: a year ago today, my dissertation advisor passed away.

he was a good man.

i can only say what i know from my interactions with him, but one thing that i always appreciated about him was his patience with me. he understood that i knew very little about the research area that i chose, when i chose him to be my advisor.

in our meetings, if i had never heard of something, then he would give me the general idea [1] and if we really needed those topics, where i would learn more, as necessary.

of course, the advisor had no fear of delving into the unknown:

• he asked me once to read mi1nor's paper on ex0tic $pheres, and report back to him after a week.
  
  
• another time we spent two weeks delving into bana¢h modules and other topics which hovered around 0perator al9e6ras.
  
  
• at some point in my graduate career, i could manage a half-competent conversation about harm0nic mappin9gs in the plane and the rad0-kne$er-ch0quet theorem.
  
  of course, i don't know anything about it now.
  

there is more to say about him, but the workday is not yet over, and i still have a talk to finish.


[1] well, except p0ntrja9in c1asses. as my differential topology and c0homo1ogy was quite poor, it would have taken a few hours for me to get the idea.

we didn't really need them anyway. instead, he told me about fibrε bund1es.

forget psychiatrists; get me a mathematician.

sometimes it's not easy being a mathematician [1]. we have no one to talk to about our work, except maths-savvy people .. most of whom are mathematicians.

i've written about this before, i know. as for why i mention it today, it came up while chatting with my girlfriend, yesterday and today.

---

she was talking about some issues she was having with her data sets, and how she has to account for small samples by means of this "bo0tstrappin9" technique [2].

in another chat she was talking about problems she was encountering, when using statistical software packages, because her population di$tributions were not normal di$tributions (or more familiarly, "bel1 curves").

---

as a (theoretical) mathematician, this terminology is compatible with my skill set. i might not get the nuances of certain social constructs in societies, but i can appreciate the annoyance of the statistical difficulties she's facing.

the thing is, this doesn't go both ways. unless i patiently explain it to her, she can't appreciate why i worry about non-smooth functions [3]. i can already imagine how badly such a conversation could go:

"well, an example of a function that's not smooth at a point is a corner, or a cusp."
"but those don't happen very often, do they?"
"no, they do." [π]
"so .. how often?"
"actually, the class of ¢ontinuous, n0where differentiab1e functions is of the fir$t categ0ry."
"wait, what category?"
"oh, sh-t. never mind."
"no, just explain it to me."
"never mind. it's not important."
"come on, tell me."
"put simply, there are a LOT of them."
"isn't that what you just said?"
"yeah. i did. so ... not important, see?"
"i'm not stupid. just explain it to me."

[30 minutes later, of explaining p0int-set topo10gy]

"wait. i didn't follow any of that."
".. see? it's not important .." [4]

she's right. she's not stupid, but she's not a trained mathematician. she's a social scientist. the paradigm is different.

for example, we mathematicians worry about theoretical pathologies. experimental scientists may understand very well the concept of a counter-example, but i don't think they conduct experiments which are supposed to fail -- not anymore, anyway. it simply costs too much. they think in terms of what they will learn from the results of the experiment, not if the experiment will work or not.

sometimes mathematics can be a lonely discipline. nobody really understands us.

---

as long as i'm rambling, what constantly irks me about explaining mathematics is the problem of analogy. maybe i will explain Rad0n mea$ures in terms of how mass is distributed. but if i do this, then inevitably i will be asked,

"so, wait: you study physics?"
[groaning ensues]

how do you avoid technical details in mathematical explanations to non-experts, if the analogies you make only confirm their bias that mathematics is only meant to be used as a tool, not as something to study in its own right?

this is why, on airplanes, i pretend to be someone else for a while, like a short-order cook or a computer programmer.


[1] this problem isn't particular to mathematicians, or even academics. anyone can be misunderstood.

[2] also called resampling, apparently. when she explained it to me, for some reason it made perfect sense and i think the idea is quite neat! the way i understand it, it corresponds to taking localised subsamples in the "neighborh0od" of a particular sample, to get around the issue of small data size. also: i mean neighborh0od in the mathematical sense.

[3] for those of you mathmos out there, yes: i mean sobo1ev spa¢es.

[π] to be fair, you could argue it both ways. in an even larger class of functions, smooth functions are dense by 1usin's the0rem and by the usual method of convo1ution.

[4] as a general rule, i avoid using terminology that either takes longer than 2 minutes to explain or cannot be described in pictures. so, no: never, ever mention 6aire categ0ry in conversations with non-experts.

huzzah! weird spe11ing → an0nymity!

just now i ran a search on "maτh j0bs w!ki" [1]. in the first 50 hits, there was no presence of this b1og or separate sub-pages from previous posts of this b1og.

it has finally become an0nynous with respect to rum0rs about j0bs in maτhemati¢s. in other words, i can rant and ramble about mathemati¢s in sec1usion again!

i have observed the same an0nymity when running a search on "maτh gr@d sτudent." once upon a time, i was in the first 5 hits. now i am nowhere near the top 50.

ah. life may not be good, but an0nymity certainly is!


[1] not exactly those alphanumeric strings, but you know what i mean. if i spelled them as words, then 90091ε would inevitably use their algos again and find this b10g. that would defeat the purpose of keeping this bl0g under the radar!

brief observations, while grading exams.

some of my students seem to dislike the second derivative test for two-variable functions; they don't use it at all. pity.

other students seem to like this second derivative test VERY much. in fact, they like it so much that they will even apply it to non-critical points!

ah, kids. q;

talks and (secret) seminars

after next week, i'll have given three talks this fall [0]. maybe i'll give more, but i doubt it. i'm older and wiser [1] than when i was a student. back then i'd average about 2-3 talks per term. i know better now than to push it.

then again, that doesn't mean that i'll stop everything.

---

recently i've been thinking about my late advisor.

two fall terms ago, he was teaching a course on ge0metric mea$ure theory and wanted to proceed to the theory on metri¢ spaces, after ambr0sio and kir¢hheim. course or not, he would still do it -- that's what he told me and others. so we began another seminar -- a secret seminar that was never listed on the U of M bulletin. we were off the radar.

on tuesdays we would meet in a lecture room and the advisor would say his piece. there were visitors sometimes, and they would give a series of guest lectures. some meetings were half-lecture, half-discussion. i learned a lot, then, about my own research.

i also realised how .. liberating it was not to present things so formally. in the U of M study seminar there evokes the same feeling, but it is nonetheless a seminar. those winter lectures felt different: informal, personal.

---

so i'm trying out this secret seminar thing, myself.

to two different mathematicians i've referred to this paper that i should know about -- heck, a friend of mine gave a talk about it before -- but having never read the thing, it would be pretty irresponsible of me to refer to it to others, right?

so now we are reading this paper together, and learning/relearning bits of analysis on metric spaces as we go. research-wise, i don't know where it will go; i don't need the results for any applications i have in mind, yet.

maybe some will come up; i don't know. usually i'm slow with thinking of questions, while i know of others at seminars, past and present, who start asking right away of "so from that, does it follow that ____?"

at any rate, it will be nice to revisit and to learn. sometimes i feel like i haven't escaped my thesis yet, and i'm looking to concentrate on something new for a while.



[0] this doesn't account for my tendency to ramble; when i mean "talk" this could mean a talk in two parts, each on different weeks.

[1] i mean this in the sense that, having made that many mistakes all those years ago, i'd be pretty daft if i made exactly the same ones now, wouldn't i?

if i were units of measurement, apparently i would not be english.

odd. somehow i became one of those "metri¢ people from michi9an." i would never have thought of myself this way, were it not for some comments i remember from today:

one colleague said that he wanted to learn about this "lips¢hitz world that i live in." he was referring to yet another talk that i'm giving next wee: fair enough.

another colleague remarked that it will be convenient to ask me questions about metri¢ spaces; this pertains to a paper that we are reading, on the sly (more about that, later).

still, that's eerie to hear. i can think of a half-dozen people who are better "references" than me, for those sorts of questions .. but yes, emails are annoying. it's easier to physically ask someone; i understand this ..

.. but ye gods: it's still eerie.

i'm always tempted to say that i don't know anything, but every time i say it, i mean it, but i don't think anyone ever believes me. they shouldn't, of course -- everyone knows something -- but that's not the point.

where's a bigger fish when you need one?

---

this also becomes an issue whenever i have to tell someone my mathematical classification. i balk at this.

• i can't say that i study metri¢ ge0metry, because people will assume that i study gr0mov hyperbo1ic groups or aleksandr0v spaces or something;
  
  
• i can't say that i study ge0metric fun¢tion theory, because people will assume that i know lots about qua$iconf0rmal mappings;
  
  
• i can't say that i study ge0metric mea$ure theory, because i don't know the proofs of the c1osure-c0mpactness theorems or the finer points of s1icing;
  

it's also not enough to say that i'm an analyst, sometimes. what's a postdoc to do?

an imp0ster, exp0sed.

a colleague of mine was giving a ta1k for the undergraduate math club tonight and invited me in advance. it sounded interesting enough, so i attended.

as it happened, my cover was blown. later in the talk, during the question period, the speaker exposed me as a postdoc. it wasn't malicious or anything; i just didn't expect it.

admittedly, i was curious how long i could blend in before someone would notice that this guy was using all this highfalutin mathematical language with the equally-out-of-place guy next to him.

(as it happened, i wasn't the only postdoc in the room,
and he brought with him a more interesting preprint to read than i did.)


odd. the imp0ster syndr0me is when a grad student feels out of place and undeserving of the "high academic status" (s)he has, in academia. i think, tonight, i felt the opposite feeling.

in which besτ-1aid p1ans ..

sometimes you just can't win.

---

this morning i woke up at 7-something to finish writing my lecture. by the time i reached the last page, it was a half-hour before class and the next bus would arrive too late into campus. so i half-ran, half-walked down fifth avenue to make it to class on time.

in the end, i never got to the part of the lecture that i took pains to write. everything else in the lesson took too long. so i lost an hour's sleep .. for nothing.

sometimes we get angry, and there is no one to blame but yourself. sometimes we get so angry that we cannot concentrate on research during those precious hours of the day when you can escape, not be a teacher for a few hours, and work out those ideas that mean much to you.

i can't do research when i'm angry.

---

later in the afternoon lecture i found places to skip a few details here and there -- the sorts of routine things that if the students were paying attention, they could finish on their own. it was a smoother lecture -- the afternoon lessons always are -- but i collected only four extra moments, despite these skips.

not enough time to discuss the last topic, i half-thought, half-sighed. so despite moving quickly through 46 minutes, i gave up, announced a change in the exam material, and finished early for the day.

the students were delighted, i think. they're within their rights to be.

as for myself, i'm not angry. then again, i'm not terribly happy with myself. i don't know why it bothers me so much, when i don't teach well. it just does.

at any rate, there is still research to do ..

unexpected consequences.

the older i get, the more suspicious i become of the axi0m of ¢hoice.

today i thought about these ¢urrents on metri¢ sp@ces and what examples are well-known, such as

(f,π) → ∫_Ω f det[Dπ] dx

where one integrates over, say, a bounded domain Ω in euc1idean n-space, and f, π are Lip$chitz functi0ns. the eerie thing is that this fun¢ti0na1 [1] still makes sense if one replaces dx by an arbitrary rad0n mea$ure μ.

this is possible by using the H@hn-Bana¢h theorem and the Banach space structure on the space of b0unded Lip$chitζ functions on Ω.

so something survives out of Radema¢her's The0rem, and we have not accounted at all for the geometry of the mea$ure μ.

i guess i just find that creepy.


[1] careful: i said fun¢ti0na1, not ¢urrent.

thoughts at the end of the day.

i've just written a 1ess0n p1an. now that i think about it, i've spent the last few minutes coming up with an interesting example ..

.. that is, i find it interesting.

this probably doesn't bode well for class tomorrow, since "interesting" may mean something far different for instructors/mathematicians than it does for students.

i'll sleep on it and fix it later.

---

lately i've tried juggling two research projects at the same time. it's not really working, and i never get very far with either project. i don't know how my colleagues do it, how they have the energy ..

i've decided for now that teaching a calcνlus c1ass is like giving a very sketchy talk. one don't prove much of anything ..

unless it's an honors course,
or if you want to gain the ire of your students.

instead, one gives rough ideas as to where this or that formula comes from, why one shouldn't be surprised at certain outcomes, and where to be careful. (also, examples are very useful, in order to emphasise certain points.)

i've heard it said that "a calculνs course is not a mathematιcs course.

whether you agree or disagree, a ca1cu1us course isn't a theoretical maths course, just as $tats 101 isn't a theoretical statistιcs course.

one's studies must begin somewhere, you know.

---

i've never taught a theoretical maths course before.
i'm curious what it's like ..

.. but not so curious as to teach one, right away. there are papers to write and more research to do, after all!

second helpings, anyone?

as promised, my talk slides from the AM$ $e¢ti0na1 @ Wes1eyan Univer$ity are available on my homepage in PDF form.

[to get there, click on the "My Webpage" link on my Blogger profile]

¢aveat empt0r: i follow the minimalist approach of talk slides. this means that there is enough on the screen so that i can remember the salient points ..

.. well, most of the time ..

but i tend to avoid excessive text.
it is, after all, a talk and NOT a paper.

speaking of which, there is a preprint forthcoming, but not yet ready for the public.

the day after (or: the conference is over).

i made it home at midnight, last night. one more conference is over, and the next one won't be until march. hopefully i will have new results to talk about, by then.

some tips to remember:

1. carry a water bottle around. i forget how much coffee i drink at conference, and how few my visits to water fountains.
   
   
2. talk more research with people. i talked a fair deal of math, but this was often the result of asking "what is your research about?" it was worthwhile to ask, and in the future i'll also keep asking, just to know more about my colleagues.
   
   however, i suspect that to do well in the forthcoming years, i might have to become more mercenary.
   

as for what went well,

• i learned about the ana1ysis on fra¢ta1s, as studied by several researchers in the northeast. these guys don't mess around. perhaps i'll say more about this, another day.
  
  
• i saw old friends from michigan and old acquaintances from previous conferences. often these meetings have a feeling of a reunion, about them.
  
  
• my talk went smoothly --
  
  
  though i went over; then again, so did many others
  
  -- and i received good comments and questions about it. nonetheless, it would be nice to talk about something new.
  
  
• everyone is fascinated by my laptop, desdemona. apparently the a$us e³ p¢ is a crowd-winner.

siblin9 riva1ry (of a mathemati¢al nature)

whenever a conference runs parallel sessions, the audience and speakers are inevitably at odds. for example, i'm slated to talk on sunday at 9am ..

.. [groans inwardly] ..

.. and so is M@rie $nipes, my mathematical sister. [1] the session organizers couldn't have known, of course, but ..

.. [groans inwardly again]

i was looking forward to attending her talk, too. i only remember a little of what she spoke about, in her last talk, and i'm curious what new results have come out.

life is ever against us. there may be some of you who will participate in this conference -- it's the AMS $ecti0na1 @ Wes1eyan Univ -- and you may be facing the dilemma of choosing between me, M, and sleeping in ..

..come on- it's 9am on a sunday, conference or no conference..

.. so i will make it easier: GO TO HER TALK. she's a wonderful expositor and her pictures are better than mine. i recommend her talks, no matter what time of day it is slated.

if you absolutely want to know what i'll talk about, it will be the same content as in the Heinonen Mem0ria1 C0nference in May 2OO8 and the Rea1 Ana1ysi$ Ex¢han9e $ymp0sium in June 2OO8.

if you missed those, i'll post my slides online (after the conference) in PDF form. there's also a preprint coming, i promise. besides, if you really wanted to know that badly, then you probably know me personally, so just ask ..

.. and go to her talk. thank you.


[1] this means that we have (or rather, had) the same advisor.

in which 90091ε sees all.

you have got to be kidding me. as of now, this bl0g is hit #1 for "m@th 9rad student" under g0o9le sear¢h.

in efforts to hide from g009le,
i am hereby obscuring any terminology related to mathemati¢s or b10gs..

for there must be better references and websites for that sort of thing, other than this bl0g. besides, i'm a postd0c now, not a student ..

.. but does 90091e notice? no. q;

a quiet moment, after another day is done.

my mind seems clouded. there are some lingering thoughts in the directions of my thesis research, but there is little time to entertain them.

i'm not griping about teaching. i'm not exactly griping, either. life's not bad. this is just an attempt at understanding where i am.

i'm at a new position at a familiar university, working with a new colleague who's smarter AND faster than me. i guess i'm too used to working with an advisor who waits or slows down for me to catch up. i forget that i'm supposed to be something like an equal to faculty members that i know, to stand toe to toe and work just as well.

this will take some getting used to. i will have to become better.

there seems little time for anything more than the daily grind. i haven't been sleeping very much lately.

the last edit i will make (of this manuscript).

some tasks will always be easier than others. for example,

• criticizing others or yourself is easy, but rectifying inaccuracies or imprecisions is less easy;
  
  
• jotting down comments by hand is easy, but fixing the comments in front of a screen is less easy.
  

i'm in the home stretch. i'm this close to a finished, readable manuscript.

in which the library section (QA614) is compared to the produce section of a market.

for me, going to the maths library is like going to the supermarket. i was browsing through the differenτia1 top0lo9y section -- QA 614 or so -- and everything looked good.

i even picked up this book by de Rh@m, called Differenτiab1e Manif01ds. turning to the table of contents, guess what?

chapter iii discusses ¢urrents, including D^k and D' and boundary currents and everything.

to think: "differenτiable manif01ds" sounds like such an innocuous title; in this slim volume he goes on to discuss h0mol09y and harm0nic forms.

---

so i'll hold on to this little book for a little while, and see what ¢urrents were like, before the federer-f1emin9 theory. in the meanwhile, there are a few errands of mine, in differenτial top01ogy.

in which, while teaching, the theorist begs to come out.

this morning i lectured about the method of la9ran9e mu1tip1iers. after finishing up one example of a function of two variables, one student asked:

"so what does the λ mean, anyway?
i mean, why does it show up?"

i did my best not to gape and not to reveal my glee. ye gods, he actually wants to understand! i can explain!

of course, not all students want to understand, so i kept the exposition short. but i felt very happy to explain.


this happiness is a dangerous thing, and for me, it must be kept in check. for example, when discussing local extrema and the second derivative test (using the He$$ian matrix) i was sorely tempted to launch into a mini-lecture about eigenvalues and princip1e ¢urvatures.

but that would have been very, very wrong for a calculus iii class. still, i was tempted, because .. well, ¢urvature is really, really cool. ask anybody. q:

reluctant workday; thoughts on a research shift.

i'm short on motivation today. this morning i woke up late, around 10:15am [1], and i told myself that i would work for a few hours in exchange for my morning cup of coffee.

a fair trade: i ended up thinking about something i was asked to think about, over an email from last night ..

.. so yes, sometimes i do take requests,
but it depends from whom ..

.. and i got as far as i could [2] without cracking open one particular book: hirs¢h's differentia1 t0polo9y. it's a good book, but like i said: i can't seem to work today.

two further thoughts come to mind:

1. i blame this lassitude on yesterday and the day before: thursday i was driven to finish this manuscript, and i did. on friday night i was also up late -- until 2am -- reading a printout of that manuscript and jotting down comments, in efforts to have something tolerable by monday.
   
   so perhaps my mathematical battery's gone dead and needs a recharge.
   
   
2. this looks to be a transition in my mathematical life.
   
   i've spent the last two years of my life studying one particular topic -- derivati0ns and ¢urrents on metri¢ spaces -- and maybe i'll write a paper or two about some of the more interesting thoughts i've had; maybe more, if i'm lucky.
   
   now i'm in a new research group and they think differently. different topics have importance, such as s0bo1ev spaces in various contexts. i once told a friend of mine to think about sobo1ev spaces as a tool and not as a primary object of study -- he was thinking about giving a talk on the subject -- but i think i will change my mind now. i've always been fond of those function spaces, but now, with new company, i think i will study them again with renewed curiosity.
   
   perhaps, back then, i was simply asking the wrong questions, or i didn't have the right perspective.
   

strange, how a place makes you rethink who you are. i never thought of myself as a "ge0metric" person, but now i seem more inclined in that direction than say, ana1ysis of PDE.


[1] some years ago, i would have called that waking up early. odd how things change when one gets older!

[2] i thought of an idea, then realised its weakness, then realised that i was needlessly adhering to one way of doing things. so i came up with a different idea, and over lunch, realised that i was thinking too 2-dimensionally and relying too much on my illustrations. i think this second idea might work; if worse comes to worse, i'll see if i can find something in hirs¢h.

like day and night .. or rather, morning and afternoon.

my poor morning-section students. they always get the unpolished version of that day's lecture. today i made all sorts of mistakes, and not just computational ones. in one spot i wrote "maximum" instead of "minimum" and that must have confused everyone ..

.. until one student summoned up the courage to ask. he still asked timidly, though, or perhaps uncertainly.

more than that, my morning students don't get benefits like

• my sudden realisation that it would have been easier to explain the topics in this order instead of that;
  
  
• more succinct, "all you need to know" versions of my rambling algorithms, as well as systematic principles in place of what would have been clever tricks;
  
  
• my complete wakefulness .. because no matter how much coffee you drink, it's not quite the same as afternoon wakefulness.
  

today's morning lecture was particularly unfriendly.

last night i kept working on my manuscript of a paper, and i was dead set on finishing all the writing. (the editing, however, is a different matter.) i actually succeeded at this goal ..

.. but only to learn that by the time i was done, it was 3am, not 1am.

waking up at 7:30am, i didn't feel human and i didn't feel like an automaton. instead, i felt like one, big ache, unfortunately manifested in human form. it didn't help that i went wall-climbing yesterday and woke up sore all over.


despite this, even if i could, i wouldn't have done anything differently. i'm glad that my poorly written, 34-page manuscript is done and that i can proceed to edit it now. it would have been nice to have completed it earlier ..

.. but as someone told me already, this week, time always works against you.

a mathematician by any other name ..?

sometimes i miss being just Janus [1]. on most days and in most correspondences, i'm Dr. So-and-So. \:


[1] real names withheld, for some semblance of anonymity.

in which i am humbled by what i don't know or am unaware of.

it just goes to show you that curiosity can harm as well as help. i was writing earlier and there was a citation that i wanted to look up, but my physical copy was at home. so i went on mathscinet ..

.. and was distracted by the number of papers who referenced it.

soon i was clicking away, a dozen tabs open, downloading this PDF and reading that abstract. i have come to several conclusions:

• the internet is dangerous, but admittedly, it is good for research.
  
  
• my radar doesn't work very well; i am terribly (mathematically) uncultured and naive. there seem to be dozens and dozens of papers that i should know about and have browsed (it's too much to demand someone to read them all), just to have a sense of what is known and what is currently being investigated.
  
      in other words, how did i miss all of these?
  
  
• i'm never going to have time to read even a fraction of these .. [sighs]
  

while quasi-procrastinating [1] i also learned not to rely entirely on the arXiv

.. though i did find an interesting abstract there today;
more about it, at some point ..

instead, i have bookmarked sub-pages of homepages of various mathematicians. so yes, i am stalking them, or at least, their research output.

if you're like me and have few (if any) good ideas of your own, it helps to know where to look for some.

---

on the plus side, my draft is almost done. there is probably something else i'm forgetting, but i have one last proof to write!


[1] in other words, doing things that are mildly important, like looking up conferences and new papers by big shots.

the calm after the storm .. and other idle bits [twice-edited]

all is right in the world again,
and none of my students are especially panicked.

this week, only one of my students has attended my office hours. (;

---

added 17:22 -
it was 2 or 3 years ago, back in ann arbor, when the einstein bros bagels on state street closed down. i felt like a guilty customer, back then, because i may have bankrupted them. it was a fine place to dwell in the summer: good air-conditioning, free coffee refills, and free wifi.

it must have been the free refills. back then, i drank more coffee than i should have. i still do, but less so.


6 or 7 years ago, there was this one coffee stand in the engineering building at pitt where i would buy a quick bite to eat, between classes. it's still there ..

.. but in the form of an EINSTEIN BROS BAGELS!!!!! it's not as grandiose as, say, an espresso royale caffe (at the corner of south and east u aves, in ann arbor town), but i was just happy that the coffee stand didn't perish under renovations.

perhaps, i will work there in the afternoons. the windows are large and the cafe tables are probably empty when the engineers have had their fill of lunch and coffee.

---

added 18:24 -
wow.

having just opened up a PDF copy of c. vi11ani's draft of a book, 0ptimal Transp0rt, 0ld and New [link], one thing startles me.

it's 998 pages long. i know there is a lot to say about 0ptimal transp0rtati0n, but i didn't know that it was that much!

just when you think the writing is done .. dilemma.

by my estimate, i have two subsections to write, then a full-scale edit, and then i will have a preprint. let's hope it will be ready for the AMS secti0nal in october ..

.. that is, if anyone bothers to come to my talk and even then, if they bother to ask about my work.

inevitably i'm always scheduled against a more popular figure in a parallel session and the masses goes to his (or her) talk. a few colleague-friends stick around sometimes, if only to be nice.

you'd think that, upon seeing the finish line to be so near, this would be cause for comfort. then again .. come on.

if you've been reading this blog for a while, then you'd know:
there is never any cause for comfort or joy. q:

on friday i vowed to leave the grading in the office ..
(which, oddly enough, was a vow that i actually kept)

.. and worked on writing. when i tired of writing, i thought about where these ideas were going, what i want to prove next. that usually means a game of "let's pretend we know a lot and see what happens" and degenerates into depression, once the game stops and reality sets in.

then again, what's life without a little risk?

so i've started a new batch of research notes -- a kind of mathematical diary, if you will -- pertaining to this one c0njecture that i've been obsessing about. some months ago i told my mathematical sibs about an idea of mine which is related to other things. at the time it convinced them, but oddly enough, not me ..

.. partly because they didn't check the details and thereby failed to realise what might go wrong.

i don't blame them; i'm loathe to check the details of the ideas of others, too, mostly because i'm lazy. if i were left unchecked and i thought harder and past their ideas, it would also feel like stealing.

now it seems like the idea works. so what's the rub? my soon-to-be-preprint is supposed to be a special case of the conjecture.

i've proven all these lemmata just so the machinery runs the way that i want it to, in that one little case.

there would be similar machinery involved if this newish idea works -- similar, but sufficiently different that the proofs i've written already would require small but consistent modifications ..

.. and probably 2 more pages of some generalities. argh.

so, the dilemma: if i don't do some rewriting, then i have to reprove little tweaks of everything in the next paper as well as the newer machinery i need to run the newish idea.

i'm bleeding pages, already;
29 pages and counting, for the current writeup.

on the other hand, this is assuming that the idea really does work, and that i'm not having delusions of grandeur again. so unless i'm absolutely sure, why risk changing something that's perfectly fine?

..

i hate to say this, but sometimes i wish i could ask my late advisor for .. well, advice. i wished i asked him more often, when he was still alive.

i have learned a lesson today about writing exams. unless you really mean to, never ask your students to graph anything. it amounts to grading artwork on lesser merits.

[sighs]

on the plus side, apparently the exam i wrote wasn't up to my usual standards of diffculty; for my 40-point test, there were quite a few marks in the mid-30s.

either i'm getting soft, or the students here know how to compute (as opposed to the TI-8? clutching masses that i remember at the U of M).

in fact, there is a no-calculator policy for exams. huzzah!!!!
(you have no idea how happy i was to hear about that.)

at any rate, let's hope that the students don't get too cocky. i don't mind if they don't listen to me in class (as long as they're quiet), but next comes optimizations, lagrange multipliers, and everything that comes with integration. the last thing i need is to settle an unexpected panic on a second midterm.

until then -- a load of grading off my shoulders. teaching-wise, it's smooth sailing until mid- to late october.

one month in, and what has happened.

there's nothing like usurping a departmental computer when nobody's around, and using it for your own personal ends ..

.. such as blogging.

i miss blogging, even though i probably do as much of it as i used to. if i had to gauge now vs. before, then it seems that these entries are more urgent and repetitive. i feel like i keep on reiterating the same goals and fail to meet them, week after week.

now that i think about it, it's a month into my postdoc. so far,

• no new directions of research taken; i've thought about similar problems which are related to my thesis, but nothing serious. rather: i've proven nothing too interesting, but i want to reach something interesting. this may sound over-confident, but i think i can. [1]
  
  
• no papers finished. in email correspondence i've assured friends, who are venturing into the market, that they are better prepared now than i was, then: i was a man without preprints, only a research statement, and for some reason, someone still wanted to hire me.
  
  funny world, that.
  
  i suppose it is good that i can soothe the anxieties of friends, but it also works the other way: ye gods, everyone seems to have papers and preprints. i have a job now, but in three years ..?
  
  
• i've fallen ill and only now do i feel normal again.
  
  
• .. and yes, there is something positive on this list. i'm slowly getting to know my research group, visitors included. today i met a finnish analyst whom i've only before known by hearsay. it is nice to know that he exists and seems a friendly fellow.
  
  
• related to that, i'm slowly feeling like less of a visitor and part of this faculty.
  
  just today i was invited to sit through a pre-dissertation meeting for a newly-made candidate. it was interesting work and i could even understand some of it. for the experts, it was a mixture of su6riemannian 9e0metry and ma$$ tran$p0rtati0n problems: good stuff.
  
  

every weekend i hope for a productive weekend. sometimes it happens; other times i can only say that i visited the library.


[1] this could also be my one-track mind: i've not even one paper about my thesis work ready, yet. in my own mind it feels like i shouldn't work on anything too new until at least that paper is done. it tests my patience, of course ..

.. but one works faster when one has something to do and would rather do something else.

busy, busy, busy: no time for goals.

a few days ago i learned that TAs here are not expected to help grade midterm exams. so that leaves me with 150+ calculus 3 exams to grade. i'm 1/4 done already, but that's because i graded the vector operations problem first.

the rest will be gory;
i have a bad feeling about this.

today is also part 2 of an expository talk that i promised to give. that won't be so bad, though: i've given it before and it's mostly fun and picture driven, with many happy colors.

---

i just wish that at some point, i can get back to writing. odds are that once i get to writing, i'll start complaining about how annoying that process is ..

.. but for now, the grass is truly greener in terms of motivation. if i finish grading these exams, then it only means that upon returning them to students, the complaints will start. if i finish a draft, then at least it's next a gamble to see if it can be accepted into print, by some journal.

it would nice to flesh out new research as well. (ye gods, the backlog is growing.)

a forecast of an AMS meeting; (future) thoughts about writing

i just checked the schedules for the AM$ Se¢tional Meeting at Midd1et0wn, CT for october. indeed, i was told before that pekk@ k0ske1a will be giving an invited address, but apparently there are 2, even 3, special sessions that will be of particular interest to me.

"analysis on metric measure spaces and on fractals" - it seems that there is much excitement about analysis on laaks0 spaces and on $ierpinski ¢arpets, at least in the research centers of c0rnell and uc0nn. i know these spaces, but nothing about their analysis, so it will be good to learn some.

i wonder if they will be disappointed if i tell them of we@ver's example: there exist no nontrivial derivati0ns on the standard $ierpinski ¢arpet. \:

"geometric function theory and geometry" - this seems to be a mi¢higan hyperbolic geometry sub-conference in disguise. i recognise at least half the names of the speakers: some whom i've met, and some from hearsay, including the m@th j0bs wik!.

L: if you're reading this, i'm counting "qua$iregu1ar" as a hyperb01ic word. q:

"geometric group theory and topology" - i'm less familiar with this kind of thing. however, ome talks will discuss this "1amp1ighter gr0up" that i've heard about, in the work of na0r and collaborators. maybe i'll go just to learn more about it ..

.. then again, i remember once attending a talk because the title included the term "kahler manifold" and i wanted to know what such a manifold was. i went to the talk and listened, and the speaker never defined what it was.

so perhaps, in light of 20-minute presentations, i shouldn't hope for too much. maybe it's better that i stick to what i think i know.

at any rate, there will be many interesting talks.

the last time this happened, very few people attended my talk. maybe the same will happen again, but that could be good. if i give a bad talk, at least nobody would be there to listen to it.

---

as for other matters, my students have an exam on wednesday, and i think my lectures have scared them enough that they are worried. this, of course, means more emails than i'd rather deal with.

speaking of which, i have a review class to prepare for tomorrow. i hate review classes: either nobody has any questions or there are floods of questions because the students are panicking. besides, as long as the student goes over notes and the textbook and the homework and the department sanctioned practice problems, what is there to worry about?

it's not like this is a qual or a prelim or something, least of all a thesis defense. q:

on top of that, part 2 of my expository talk is thursday. it's going to be the fun, picture-drawing part, but it will take its own share of preparation.

on the plus side, i finished most the introduction to a draft of a paper today; what remains is the part about "notation and terminology" and the last section. i've begun to plan for the future already: what the next paper will contain ..

.. assuming, of course, that the details work out.

i've been running into the problem, lately, that i don't have anything interesting enough to make more papers. i mean: it would be nice if i could prove this or that, which would round out the discussion nicely, but can i prove those things?

most of the time: no. that leaves this little auxiliary, good-for-nothing lemmas -- the very results i could prove because i avoid the hard bits. you can't (or rather, shouldn't) make papers of those things.

then again, publish or perish. we'll see how long my principles last. \:

cutting my losses (unhappily)

i've too many things on my plate and not enough time, so i think i'll apply for the NSF resear¢h grants next year and not this year.

maybe it was the time i lost at the hospital, but more likely it was how i used up my summertime. if i had spent the month of august 2008 at pittsburgh, i might have spent more time working on a good application and have been ready by now.

might have,
could have,
should have:

these are silly words. we do or we don't: those should be the only words that matter.

so instead i suppose i should really get down to writing papers and furthering my fledgling research. after all, you can't write a good application if you don't have anything to put on it.

there's that, and i have to prepare an exam to my calculus students, this coming wednesday. somehow i have a very bad feeling about this.

---

so yes: another year and i'm bowing out of the competition. last year it was the NSF postd0ctoral fe11owships; it was bad timing and i wouldn't have known who my mentor/supervisor would be.

still, it irks me. maybe i'll just remember how indignant i feel, and maybe i'll finally do it right, next year. everyone needs his motivations, so maybe i'll just hate myself for a while.

priorities.

i was discharged this morning and i made it to cover my office hour today. after helping out my students, this occurred to me:

i don't do as much listening as i used to do. the student would say something, try to ask the right question to express his confusion, and all i could think was how do i explain that it's this way?

maybe this is a sign of efficiency. if i listened closely to all of my students, then i would be a commendable teacher, but i would also have no time for anything else.

it still bothers me;
then again, so do my unwritten articles and grant proposals and talks. \-:

still stuck.

sometimes life is unfair.

i'm stuck in this hospital with no ssh, no sftp, and no LaTeX, but emails from students just keep on coming ..

[sighs]

also: it's hard to think about maths while in hospital rooms. there are many machines around and a lot of beeping. on the plus side, i'm glad that i brought a ton of scratch paper.

in which i am an invalid, and technologically trapped.

it's been about 48 hours since i last saw my laptop. it's not missing because i know where it is: on my office desk, charging. i can't remember whether it's on or not.

if i had my laptop, i'd have blogged sooner, i'd have popped open a LaTeX GUI program and worked on my draft of a research article. i'd have uploaded a review sheet to my students so that they could better study for next week's exam.

but no: i'm stuck in a d@mned h0spital.

i've been here since monday afternoon, and fool that i am, owning an a$us eee p¢ which is very easy to carry around, and i leave it in the office.

admittedly, i thought i'd just see a doctor, get a prescription, and make it back to my office hour. it's not dangerous and not life-threatening and i'm doing well, but it was more serious than i thought. maybe they will let me out by tomorrow afternoon ..

.. but part two of my seminar talk for tomorrow has been preemptively canceled. one less worry, i guess, but i was looking forward to preparing it after my treatments today.

what's good is that this hospital has a library for patients and their visitors, and in this library they have computers which connect to the internet.

the lobbies all have wifi, too. it used to be tempting; now it is taunting.

---

it's easy to feel useless. i just didn't realise how much i depended on my personalised computer, with sftp and ssh and LaTeX and so on.

on the plus side, i started an introduction for the paper draft. it's written in longhand, in bits and pieces and revisions, and after today it might be ready for LaTeX.

i suppose that's something. without the right tools, it's a bit challenging to be productive.

i think they mean statisticians, not mathematicians.

i ran into the following article after a friend of mine posted it on facebook. apparently it's called "Mathematicians are new masters of the universe."

an excerpt:

These IBM spies are members of a new breed of mathematical whiz kids who are collecting data on almost everything we do and using it to get us to work harder, spend more, vote for a particular candidate, and even choose a particular partner.

They are the subject of "The Numerati" (Houghton Mifflin, $26), a book by BusinessWeek writer Stephen Baker.

In the 20th century, industrial companies used stopwatches and clipboards to measure productivity. In the post-industrial age, companies need only monitor our computer keystrokes to turn office workers into what Baker calls "data serfs."

it sounds like statistics to me.

maybe the writer is referring to applied mathematicians, which would fit. as for opinions about this sort of thing -- data mining, i mean -- for now i have none.

maybe i should be more specific and start calling myself a theorist. heck, i'd settle for being called a teacher because among other things, that's who i am ..

otherwise people might think that i'm one of these people, looking over their shoulder or their web bookmarks. \:

(ethni¢) c0mmunicati0n breakd0wn.

today is one of those days when i really, really, really wish that i could ¢ommunicate in mandarin.

just earlier, one prof walked up and just started speaking. it was hard to find the most tactful moment to tell him that no, i couldn't understand him.

[sighs]

sometimes it is a burden, being born and raised as an american.

---

on the plus side, the prof still wants to talk math, at some point. i also find it surprising, because he attended my talk yesterday.

if i had gone to my own talk and listened, i wonder if i (the audience member) would bother talking to me (the speaker). \:

the talk: after-math

well, part one of the talk is over. i think i essentially confused the audience, or bored them: whichever comes more naturally. thinking back on it, i tried to do too much in one session.

however, it is helpful to have experts in the audience.

hopefully part two will go more smoothly. there are more gory details, but the setting is much more concrete.

a visit to the library; talk paranoia continues.

i meant to post this earlier, but there is something very wonderful about having the mathematics library two floors down from my office. i don't know exactly how many times, while in ann arbor, i thought about checking out a mathematics book from the library and decided that it wasn't worth the walk.

of course, it would help if my university ID were working and the librarian didn't have to swipe my card through the reader, again and again and again, to no avail. in the end she manually checked my books out -- that is, wrote down the library of congress numbers, book titles, authors of the books on a special form, and then gave me the books.

at first i didn't realise that this was possible, until it occurred to me how they must have done transactions in the days before barcode scanners. i've lived with these fine electronic devices all my life, you see; i've never seen anyone check out a book "manually."

it made me feel like a suspicious person, like a visiting scholar instead of a postdoc. i've been getting that feeling lately, but not as often anymore; i think this means that i'm wearing out my welcome.

oh well. at least i have the books.

---

i've been getting paranoid about how little background i have about the topic i'm presenting tomorrow, in a talk. having checked out these books by väisä1ä and by ah1f0rs, i learned several things.

1. either it's been too long and i've forgotten, or that i never learned these things properly, but these books are really good;
   
   
2. i wasn't paranoid; i was right. i do have a paltry background and am the wrong person to give this talk.
   

since it's too late to cancel, i guess i have to build a talk from the notes i've jotted down from the last few days. there's still the challenge of giving a talk that is both (partly) accessible to students yet interesting enough to experts.

i've seen so many talks that were successful in this dual approach. it makes me wonder, now, how they do it.

a (work) holiday in spain ..

it seems like spain is the place to be, next summer,
and specifically: barcelona.

i wonder if i can convince anyone to pay for my trip.

---

anyways, either back to work or off to sleep;
i haven't decided yet.

a non-teaching day (i.e. now i have time to worry about a talk)

you know, at first i thought that this talk would be much the same as one i've given before. then i thought about my audience of yesteryear and my audience to come, and now i've realised something.

our analysis group is a coalition of 7+ profs and spans many different areas, among them

several complex variables,
banach space theory,
harmonic analysis,
analysis on metric spaces,
and analysis of differential equations (both ordinary and partial).

in short, i'm no longer in "the qua$i-w0r1d."

worse yet, now i'm supposed to be an "expert" on quasic0nforma1 mappings. of all people .. me?!?

i suppose it's the usual stereotype: if you're an analyst from the U of M and trained under so-and-so, then "you must be an expert on this-and-that." i know that nobody expects me to be as capable as my advisor, but ..

.. this should be interesting.

oh well: if i am really, truly a mathematical fake, then at least my colleagues should know earlier than later. q;

---

on the plus side: browsing through this paper again, i've forgotten how cool it is.

i've forgotten how wonderful it is to work in euclidean space and use old, familiar friends like the Riem@nn Mappin9 Theorem or harm0nic mea$ure or linear transf0rmations of space.

reading these proofs, you get the feeling of cleverness, from someone who is classically trained and knows the right tools for the job.

of course, i say this because today is tuesday, and i don't have class to teach. there are still many things to juggle:

there's the usual research and writing, of course.

at some point i should attempt a grant proposal. in the same spirit, our research group has to do its part to secure funding for the department. so i'm to give a brief description of my research.

there are travel plans (read: headaches) to make for an october conference. in my mind i want to have a readable draft of a paper by then, and talk about the results. why do i get the feeling that i won't make it?

[sighs]

well, the work piles up. i'd better get to it.

the usual complaints.

i've lately had this suspicion that i'm doing something .. not wrong, but inefficiently. all my time seems to be slipping away, and my research-related to-do list grows by the day. perhaps it used to be funny, but not anymore.

---

students flooded my office hour today. their homework is due tomorrow and wednesday. i find that to be a bad omen.

i would keep ranting about teaching, but it's getting old and i don't want to think about it until tomorrow afternoon. suffice to say that it's draining -- the lectures, the email, the students and the requests. sometimes it's not even an issue of time, but of the mental space it takes up.

maybe i should stop worrying. a friend of mine once argued that when the teacher doesn't teach so well, the students will work a little harder. i wonder if i should try that out.

argh: just now, another email. sometimes i hate myself. i told the student to email me, otherwise i'd forget. now i can't.

---

if i counted right, it was 45 minutes. today i thought about new research for a total of 45 minutes, before succumbing to hunger and buying/eating a small chickpea curry with rice from one of the campus food trucks.

i know some of you will say: "45 minutes? Ha! I haven't had time to do research in weeks!" so, sorry: it still bothers me and i want to be productive. if i had larger blocks of time, i'd be writing in efforts to finish up a draft of that Paper Which Refuses To Be Finished (and Which Once Took The Form of a Thesis).

others of you will say: "Janus, you are doing something wrong. If you don't have time for research, then you are spending too much time on other things." i wonder about that, a lot .. or maybe as often as i have time to worry about such matters.

most of tomorrow is wide open. apart from exercise and one appointment with a student, i have the day to scholarly pursuits ..

.. like writing that talk that i'm supposed to give on thursday. oh well. it's a worthwhile endeavor. at some point i have to squeeze in time for research and writing and, as a new challenge ..

.. applying for a grant.

TGIF

the one friday that i miss colloquium becomes the friday night in which i'm attending dinner with the colloquium speaker.

argh.

he seems nice, though .. even offered to show me his slides at some point, next week.

---

as for why i missed it, i was running around, doing teaching errands of questionable importance. after 1 1/2 weeks into the semester, i think i have a newfound respect for administrators and people who interact with people for a living.

i also have a newfound wariness of scale: both my lectures are larger in population than how i've taught before. one has around 80 students.

emails alone are bothersome enough. they come every day. i used to have this rule of "no email or internet before noon" in efforts to get more research done, but i wonder how well i can stick to it ..

.. especially as the semester moves forward. in a few weeks, i'm suppose to have written and administered my first midterm.

ye gods: how does anyone get any research done?

---

at some point i'd like to blog about research-related ideas again. in the meanwhile, dear reader(s), i suppose you have to settle for my gripes about teaching.

the less0n p1an that i can't seem to write.

one thing is going to bother me about teaching tomorrow.

we're covering the curvature of vector-valued functions r(t) and their associated curves, and there is a formula for curvature k(t) only in terms of the first and second derivatives r'(t) and r"(t), respectively.

it's a useful formula. you'd otherwise be stuck computing the unit tangent vector and its derivative (w.r.t. the parameter t) which would involve the length of the tangent vector and messy square roots.

the proof -- a clever computation -- isn't terribly intuitive but makes sense, once you make an observation or two. so i'm going to skip it; there's too much to cover, anyway.

however, i hate giving out formulas without explaining them: if you understand why a formula is true, then you're more likely to remember it correctly and less likely to make mistakes with it.

the converse isn't necessarily true, of course.

however, i'd rather not turn my calculus course into a "memorize and use these formulas" sort of course. there is that bias already in mathematics, just as how history is "just names and dates."

the trouble is, i can't find an intuitive way to explain this formula, other than the proof. at some point i should decide something .. and this lesson plan is taking longer than i'd like: there's still some research that i want to do, tonight, and of course ..

.. NSF grant proposals are in a month.
arghhhhhh. i hate myself already.

so i was right and it's official. i'm slated to talk next week. the odd bit, though, is that i gave this same (or similar) talk 3 1/2 years ago ..

.. as you can guess, it will be an expository talk. somehow i get the feeling that i'll be talking quite often, this term. \: