interlude: a list of challenge problems.

yes, i did promise that i'd talk more about how math is going for me lately .. but i didn't say right away. q:

in the meantime, i stumbled upon this post from ars mathematica about 23 problems that D@RP@ (Defense @dvanced Research Projects @gency) has challenged the world to solve.

[PDF link]

they're rather applied sorts of problems and a few which are biological in nature, but one includes the Riemann hypothesis.

in transit.

back in ann arbor .. essentially, anyway.

i started typing this post on a michigan flyer bus, which is a shuttle from detroit-metro airport to ann arbor, detroit, and perhaps east lansing: i can't remember. after reading that it went from AA to DTW, my interest vanished like a smooth test function outside its support.

at any rate, it offers free wireless internet and if i heard correctly, free bottles of water (i haven't checked the cooler up front, yet). rather pleasant, actually.

as for any math done .. not much: more to come, about that.

mathematics, jobs, research meetings, and holidays.

in retrospect, perhaps i should have attended the AMS joint meetings this year, after all. (so L, if you're reading this: yes, you were right.)

a few schools (those with tenure-track openings) have written me to ask if i will go, which i will not. the question is:

are meetings and interviews there really not mandatory?

and then, the deeper question follows, which i have been consistently avoiding:

where do i want to go?

as for my usual response,

it's a dangerous thing to want something.

---

on the plus side, my third letter of recommendation appeared on mathjobs today. (:

but to add to the minus side, i haven't gotten back to thesis-writing yet, though i'm still on holiday and visiting family. i promised a postdoc friend i'd talk math with him on friday, so at least that should keep me honest; his questions will further keep me on my toes!

i did think about a little mathematics, today: one of my old lines of argument. i don't believe that it's wrong, but when does anyone truly believe that they are wrong?

time's up: now, the holidays.

something's not quite right with a recent proof i had in mind. there's a gap in the argument and i can't seem to patch it. i still believe that the result is true .. just not quite a proof yet.

tomorrow's a trip home to see my parents. i decided on three days off, which isn't much anyway because day one is for travelling. tomorrow is:

8:30am airport shuttle,
10:00am flight,
.. then noon,

and life begins again:
lunch and family,
but not mathematics.

day three is christmas tuesday. i'm not religious and my family's not religious.. but somehow it's christmas. you shouldn't do math on christmas, should you?

at least not on christmas morning?
even if you don't have any presents?

..at least, i don't think i'll have any presents..

mathematics and holidays.

it's hard to concentrate now. in retrospect it would have been better to have taken a holiday this week and then go back to work next week.

then again, that would have been uncharitable to my family. i'm visiting them next week and though we're not religious, it is christmas-time and we have our own little rituals.

here's the paradox:

in the week or two before visiting my parents, i never feel like doing any math ..unless it's going REALLY well, of course.. and when i'm there, well..

.. there's nothing to do.

we mope around at home, albeit in familial togetherness. it's too far and unsafe to walk anywhere, because highways partition that suburbia; we don't have cable or high-speed internet, either. with six people under one roof, it's hard to have any solitary time, to let one's thoughts wander.

it's almost a counter-reaction:

there's nothing else to do, so let's do math.

meanwhile, i'm blogging and twiddling my thumbs in procrastination.

i've already scanned through the "headlines" (i.e. title/abstract/author lines) of recent posts on the arXiv, as well as the maths preprint servers at jyväskylä, helsinki, pisa (SNS), leipzig (Max Planck), and prague (KMA Charles Uni). i've done my "window shopping" for conferences in 2008, despite the fact that i'll be too busy to attend any between now and may.

then again, just because you can't 'afford' certain things, it doesn't mean that you can't look longingly at the sale exhibits.

every so often i'm tempted by honesty and i peruse a rough first draft of section two of chapter one of what-will-be-the-thesis.

maybe i'll find the willpower later, later today. maybe it's not so bad, to take a break now, and get back to it. that's the beauty of academia, right: to set one's own hours, for best creativity and effort?

i keep telling that to myself, but remain unconvinced.

all a mystery..

i would say that geometric measure theory is one of my mathematical interests. it would be too late to say otherwise, anyway: i've already listed it as one of my 'research interests' on job applications.

i think i would like gmt much more,
if only i understood any of it.

mathematician's block: the curse of notation.

yesterday and today didn't feel much like work. i've been trying to understand a "proof" that i wrote two months ago.

the good news: it still looks like a proof. i haven't found any flaws or counterexamples yet.

the bad news: i can't seem to write it well.

you see, there's some funny induction in the argument. each time i write it down, the notation obscures everything. the frustrating part is that the case of dimensions 2, 3 ..even 4.. they work out by hand.

so it's possible that it will work out nicely. i just can't seem to find it out.

in memoriam.

as friday turned into saturday i began to think of this weekend as memorial weekend .. not memorial day in the calendar sense, but a memorial day for the advisor.

he died on october 30th. since then, every morning i've woken up, never sure if it would be a bad day. "bad" doesn't mean a bad math day; i don't bother worrying about such things. instead, "bad" means that the mathematics reminds me

• that he died, and that i will never see him again or share in his insights;
  
  
• that he would want me to carry on the work, improve intuitions into ideas and arrange ideas into proofs;
  
  
• that there are more qualified minds than mine. i cannot do justice to what could become an amazing problem or a beautiful theory, if only it were put into the right hands ..

.. but there is no one else. only i was there, in that very time and place, and if the advisor's ideas mean anything, then there is no real choice in the matter.

i have to see them through, but that only makes it harder.

---

i've written before about this weekend, and about the nature of funereal gatherings. one sees family and friends and for that, one is happy. on the other hand, none of us have any illusions about why we are there.

a few friends wrote me early, suggesting that we talk math in the time before the memorial service. the thought troubled me, at first, but then it occurred to me:

with such a critical mass of analysts and geometers, the advisor would have laughed at us if a few theorems didn't come out of that many days of conversation.

so we talked and laughed. we posed a few problems, and we might have proven a theorem: we'll know once one of us decides to write down the proof.

life goes on, i guess. this is not to say it's easy, but it goes on.

about jobs, and the law of conservation of anxiety.

it's slightly disturbing to apply for a job, then to see that the job opening has been re-posted.

it's my inferiority complex at work, of course: i become inclined to suspect that their applicants are not "prestigious enough"

      ..among them, me..

and that they want someone smarter, better looking (at least on paper), and more published.

---

i'm becoming less worried about applications, but the law of conservation of anxiety persists: i'm more worried about writing/finishing a thesis, and this makes up for it!

two anxieties.

1. in my (forthcoming) thesis, i will have a theorem which is relevant to measurable co-tangent bundles on a certain class [1] of metric measure spaces.
   
   one requires two other theorems for this application. i've been working on writing a good proof of the first theorem, which reduces to a lemma.
   
   so far, the proof of the lemma takes 2 pages (standard amsart format) and i think when it's done, it will be 3 or more pages. possibly 4.
   
   i have a bad feeling about this.
   
   
   
2. i am ambivalent about this weekend. on one hand, i will see many friends and familiar faces. on the other hand, the reason is a tragic one.

[1] too technical to say, here; those of you who know me well will probably know the exact hypotheses.

endgame: time to write.

it's time. aside from a few last tasks with applications, i will start writing my thesis.

it's scary. i don't think i've ever followed through with writing anything of length; the joint paper about Grushin spaces doesn't count, because my co-author helped me tremendously.

i'm flying solo for this one, to some degree. the new adviser [1] will give his comments and suggestions and criticisms, but ultimately i have to write it.

oh well. it had to happen, at some point.

also, what i wrote before may be kaput (an earlier post); i have the feeling that there will be much reworking, as i write.



[1] it's strange to have a new adviser, if only because i'm afraid of replacing and forgetting about the first adviser. it's not disloyalty, but .. it makes me uneasy.

also, kevin: if you are reading this, i suppose in some sense we are brothers now, and xian qian is a little sister to us both.

endings and beginnings.

i feel like an "old" graduate student, even though it's been only..

(winces)
..5 years.

i know quite a few light-hearted sixth-years, and last year i remember a really mellow N-year, where N ≥ 7. they never struck me as "old."

then again, i do recall him saying that he would be happy to be a grad student forever, in ann arbor. i wonder if he really meant it.

i don't know why i so easily feel "old," meaning tired, irritable, worried, resigned, cynical, prone to ranting and giving unsolicited advice of a pessimistic nature.

that's the reason why i feel "old" recently. in the last 24 hours, i've told a few younger kids (i.e. 1st- or 2nd-years) to go home and get some rest.

in retrospect, i realise now that my words are futile. when i was their age, i wouldn't have listened to what i'm saying now.

now that i think about it, i still don't listen to my own advice.

it could just be the end of semester which causes all of this, especially with thanksgiving now over. there is not much time left, and the end is near.

i've seen too many baggy eyes on young faces, and i've encountered too many students with colds or the flu. everyone who's teaching this semester knows exactly how many classes are left to teach.

maybe everyone is tired. i'm tired. i know what comes next in the months to follow: writing feverishly to finish a thesis by the deadlines, sending job applications and worrying about never getting any job offers..

..no matter how many friends, many of which are wiser and cleverer than me, say otherwise..

..and i know that to be tired now means to be very tired later. still, i don't know what to do about it, other than to press on.

back on the job, and typing problems.

i thought about taking the day off today .. but there are too many cover letters left. if i push it, maybe i can finish them ALL by end of thanksgiving weekend ..

.. and maybe, just maybe, i can be a mathematician again.

---

on an unrelated noted, today is a very annoying day for typing, because

1. my left pinky finger has a cracked nail, after a basketball game. to be honest, i can't remember how i injured it; it was only after the game that i noticed that it was bleeding and the nail jutted out at a slight angle.
   
   recall: on a QWERTY keyboard, the pinky does the a's, so the band-aid is messing up my usual typing flow.
   
   
2. there's a cut on my right pointer finger, exactly at the spot which presses buttons. a turkey rib bone jabbed back as i was preparing the turkey for roasting, yesterday.
   
   that finger hits the keys: u, y, h, j, n, and m, which is enough to be annoying. what's really annoying, though, is trying to move the mouse pointer on the touchpad, when the d@mned band-aid gets in the way of touch sensitivity!
   

it should make for an interesting day of typing cover letters .. \:

job compatibility dilemma.

suppose that

1. i've written one joint paper in the subject of analysis of PDE,
   
2. but haven't studied PDE in a few years,
   
3. and the next slew of papers that i'll (hopefully) write will be metric-geometrical in nature;
   

is it fair to say that my research interests include PDE?


the problem: if i say that i am a PDE guy, then people imagine all sorts of things, like

• numerical methods and simulations,
  
• the KdV equation and wave propagation,
  
• curvature flows on manifolds,
  
• actual applications in the real world,
  

and so on. none of these things describe me.

for the same reason, unless i am in finland i almost never refer to what i do as "geometric analysis," because it would sound like i study PDE on Riemannian manifolds, or Ricci flows.


argh.

everyone wants a PDE guy or an algebraic geometer or a mathematical biologist, or perhaps a banach spaces/convex geometer.

idle foreboding.

it's occurred to me that i won't be doing any "new" math for a while. i hope that soon all these job applications will be done, but after that is writing a thesis ..

      .. which i am looking forward to doing,
      if only because i want to see this thing through.

but this thesis will consist of ideas that i've already thought about, and the proofs won't be new in spirit. instead, they will be codified with as much rigor as i (and the committee) can tolerate.

oh well. i guess i'm whining. i mean, do i want a ph.d. or not? \:

jackpot! ..perhaps?

i don't know how it happened, but i have just discovered job listing webpages for jobs in the U.K., in Canada, and even in Australia.

i feel like a silly american, but i'm just crazy enough and disloyal enough to apply "anywhere" at this point! q:

how to fit inside a department? (the job hunt)

today, however, i nearly fell into self-defeat.

it began well. at the apartment this morning, i jotted down ideas which, when combined with good transitional flow, might make a tolerable teaching statement. i set it aside and thought to go to the office

      (where there is internet)

and start looking up information about various mathematics departments, in efforts to prepare for cover letters.

i never realised how specialised my work is, and how hard it can be to fit research interests into another department. i began my search by asking,

      "does anyone in this/that department study the analysis or geometry of metric spaces?"

and realising how few there were, i thought about what i could say i know:

      a little geometric measure theory
      a little PDE,
      a little sub-Riemannian geometry..

and that was it. there weren't so many departments for those, either. so i began to ask myself,

      "what could i realistically LEARN during a postdoc, in order to do research with people in this/that department ..?"

note the emphasis on the word 'realistically,' so now you see: self-defeat.

---

it took a lot of self-convincing that an application consists of research (and teaching) statements, vitae, publication lists, recommendation letters, and cover letters -- that's it.

i can try and make the best packets i can possibly make, and maybe (just maybe) i might look good on paper, and good enough that someone will hire me.

then again, maybe everyone else is feeling equally inferior.

trip, and opinions about theses.

i spent the weekend in cincinnati, which i would like to call my "mathematical vacation" from the job hunt. it was both productive and great fun. i met with "mathematical family," and i dare say:

      i think i see the beginnings of a collaboration.

.. but first things first:

      apply for jobs,
      then thesis,
      then collaborations.

---

thinking about it, i may have become a mathematical hermit in recent time.

it's only recently that i've discussed research and problems with others, perhaps because of paranoia and worries ..

      (to which my hold remains firm)

.. that if i talk about them, then someone will say something intelligent and useful, and then it becomes a joint project. that is a fine thing; it makes for better mathematics but then again, it's your thesis work.

maybe i'm old-fashioned about this sort of thing, but i always believed that a thesis should be your own, original work ..with some suggestions from an adviser, of course.. but essentially your own work.

on what "productive" means.

i guess i haven't posted in a while. much has happened, but i'd rather not talk about it now. those who should know the events of the last few weeks: at this point they already know.

life has been hard, and it's going to be hard for a while.

---

today i felt productive, until i realised that i've misused the word "productive" lately.

i spent last week writing drafts of research statements and jotted scraps of what i think about teaching onto sundry scattered pages. i would have said that those days were not productive, even though they were. they serve the purpose of making me more eligible for jobs that i want.

i spent today doing some mathematics, thinking through several theories of others and if they do fit together well. so today, a math day, felt productive.

so maybe i should have said that i felt stimulated or pleasantly challenged instead. it's the same fallacy of which i am commonly guilty: when i say that i feel "old," what i really mean is that i feel tired or slow or forgetful.

at any rate, the job search presses on, and work beckons.

writer's block

i haven't felt productive in a while. i have two "statements" to write for job applications: a teaching statement and a research statement. all i can think of is

why does my research matter?
well, it doesn't.
rather, it doesn't matter very much.
very few things in life actually matter much at all.

why is it interesting?
well, why is anything interesting?
sometimes it bores me too, but not enough that i give it up.

what do i like about teaching?
um
..
still thinking.

well, sometimes research drives me nuts,
and it's nice to do something else for a little while.

i used to have no problem writing about mathematics. after all, how often have you readers been subject to my math-y rants? maybe it's the non-voluntary nature of this task, to write about math for mathematicians that is trying my wit and patience.

i don't know.

sometimes i feel like i've had far too few conversations about my work with other mathematicians. at this point, it might as well be a foreign language to them, and it's my own fault, really. i don't think anyone understood my talk last month and besides, i didn't get to anything interesting.

---

i'm constantly astonished at some mathematicians i know, and how much they love their mathematics. they will talk about it all the time, even at social events. forgetting social convention or etiquette, this indicates something.

it really matters to them: what they study, what they learn, what they want to know.

i wonder what that says about me.

timing and readiness.

someone once told me that to graduate in spring, it's best to have the math done in that preceding october.

i wonder when in october he meant. i'm still doing math, still working out what the thesis might look like .. not that i'm complaining, but the risk of not getting a job is, say ..

.. good motivation. \;


i just feel unprepared. i guess that's all.

isolation, and "translating" mathematics.

maybe there is a down-side to working at home most of the time. today i met with the adviser, and i felt like someone stumbling out of a cave, squinting at sunlight and rambling nonsense.

(never mind that today was actually cloudy.)

there is a drawback to working in isolation for too long a time. it takes longer to explain, because you have to remember how to communicate with others, how to find that conceptual framework which is compatible for your mind and their minds.


in other words, if math is a language, then it takes longer to translate what we want to say to each other.

me, i doubt i'm easy to understand; there's also the possibility that i'm wrong about something and it wouldn't make sense anyway. \:

having a look.

just to learn the enormity of what is soon to come, i looked through the ams job listings for postdocs, snobbishly chose schools which sounded nice, and came up with ..

.. 23.

wow. even choosing snobbishly, there are already 23. i think i'm in for a very long fall term ..

proto-(seminar webpage).

in an act of wasting time, i updated the SAnS (student analysis seminar) webpage. this is what it looks like now.

as you can see, there is a cute google.calendar thingie for playing around. other than that, it has no purpose: only for fun.

ah well. back to work.

the cycle of reading a paper, and subsequent research.

[first day, browsing paper]: "hmm. that is interesting. i wonder if.."
[second day]: "that's odd. why don't they just do this-&-that?"

..
..
[Nth day, rereading paper]: "oh."
[later, on Nth day]: "huh. so if not, then i wonder if, instead .."

when will people ever learn?

i think my plan of working away from the office has backfired.

because my time in the department is a rare thing, i think people are more inclined to stop by and talk to me when they see me.

this does not bode well. the point was to consistently disappoint people in a form of negative reinforcement; time and again, they would stop by my office, note my absence, sigh to themselves,

alas, janus is not there! woe is me.
i shall ne'er again call on him at his office.

and develop the habit of never seeking me out. so after a while, nobody would think to look for me, and i could return to the office, unsought and having all this time to work.

at any rate, that was the idea.

argh. LaTeX.

not much work done today. i thought to completely rewrite something that might someday become a research article, but for a silly reason:

it feels like it's taken a life of its own, and a complex one at that. every time i try to change one thing, i feel like i need to edit a dozen other items in those 22 odd pages ..

(for those keeping score at home: among others, i'm using the package fullpage and have set linespread to 1.6.)

.. and those pages don't even contain the full proof of a main lemma/theorem, yet, which is proven in three stages. only one stage is TeX'ed, so far.


at any rate, i have 9 pages, five of which were cut-and-pasted from that previous writeup. in retrospect i shouldn't have bothered; i didn't even TeX up any new proofs!

there must be something wrong with me: i am incapable of being productive, with a computer.

careful hope.

the conspiracy makes more sense now, though not complete sense. maybe i should say that there's hope.

it's been a frustrating few days. it finally feels like i'm done with "fixing an old error" [1] and rehashing old things, polishing them into good definitions and valid techniques in proofs. this stuff is in the same direction, but it feels new.

i don't know its shape yet, what form it will take. i suppose, and i toy with some nascent ideas of others, see how they fit together.

it's a familiar sort of frustration, but one i haven't felt in a long time. i mean the frustration of creativity and trying new ideas from old.

i don't think i'm making sense, but i'm feeling better. i think there's hope, and what's more, i'm willing to hope.


[1] actually, i never really fixed it in the literal sense. i acknowledged it, threw away what i thought was true, and kept the corollaries by using other techniques. in short, the ship sunk, i cannibalized the hull and cargoes, made a little raft, and found another ship ..

.. sailing for parts unknown, but interesting.

conspiracy

technically, it takes several people to form a conspiracy. if you have a crazy idea and nobody else knows it, that it's your secret, crazy idea, then it's not a conspiracy.

last week the adviser suggested that i think in a certain direction, about some ideas some other mathematicians have been talking about in the last few years.

after a week of "surveillance," i met with the adviser again, and told him a crazy idea or two. he listened and didn't disagree.

so now, we have a conspiracy on our hands. i wonder if it will work out.

the after-math.

well .. i've given better talks, and i think the second one confused and horrified the audience more than anything else,

but at least they're over now,
and i can go back to work;

it's what i've been telling myself, anyways.

flattery will get you everywhere.

man, sometimes i love reading paul graham:

      Outside of math there's a limit to how far you can push words; in fact, it would not be a bad definition of math to call it the study of terms that have precise meanings.

but sometimes..

      [4] Philosophy is like math's ne'er-do-well brother. It was born when Plato and Aristotle looked at the works of their predecessors and said in effect "why can't you be more like your brother?" Russell was still saying the same thing 2300 years later.

Math is the precise half of the most abstract ideas, and philosophy the imprecise half. It's probably inevitable that philosophy will suffer by comparison, because there's no lower bound to its precision. Bad math is merely boring, whereas bad philosophy is nonsense. And yet there are some good ideas in the imprecise half.

ouch. i may have said similar things myself, but usually when i'm angry or trying to exhibit wry wit. myself, i tend to take the historical perspective and view math as a subset of philosophy.

it has an added bonus, too: if mathematics is philosophy, then there is no reason why it should be useful, except to prepare the mind for other processes of thought. if it is useful, fine, but that's not why we do it.

next week's song and dance.

well, this should be ..interesting, for me. below, i copied a few day/time/titles from this week's U of M mathematics seminars bulletin..

(you can download it here)

.. and among them include:

Monday, September 24
3:10-4:00pm Student Analysis Seminar --- janus geminus (UM) Fractals and Measure Theory --- 3866 EH

...
Wednesday, September 26
3:10-4:00pm Geometric Function Theory Seminar --- janus geminus (UM) Derivations and Currents on Metric Spaces --- 4096 EH

i guess my time has come, after all. i'm speaking at the `grown-up' seminar. before i was mildly petrified at the thought, but i think i'm okay with it now.

two talks in one week..

the last time it happened to me, i spoke one monday, was stranded at various airports (due to inclement weather) the next day, made it to the conference by the third day and then spoke again. flying back that weekend, i resumed the talk at U of M that next monday.

i'm surprised it didn't turn into disaster.

maybe i make much of this. after all, a visiting prof here (r. barnard) has given three talks in 8 days, and a fellow student is also doing double-duty:

Tuesday, September 25
3:10-4:00pm Student Seminar on Representation Theory/Lie Theory --- Marc Krawitz (UM) TBA --- Room TBA

...
Friday, September 28
3:10-4:00pm Student Geometry/Topology --- Marc Krawitz (UM) The fundamental group of a compact semisimple Lie group is finite --- 3096 EH

---

as i've been telling everyone, "i'm paying the piper early."

a few days ago i ran a search for job listings on the AMS website, and realised how soon some application deadlines are.

i don't think i have my act together for the NSF postdoc fellowship. it's too early, i haven't asked for recommendations yet .. and how can someone know who their "supervising scientist" will be, before the postdoc begins?

well, maybe next year.

at any rate, i gave myself this advice: better to talk now, so that i don't have to talk later..

..@ U of M, i mean;
there's another talk coming, in november.

that should be great fun!

.. because i envision days of frustrating fiddly application stuff, days of frustrations in trying to write up math, and more days of frustration with creating new math (which might not want to be created).

progress, measured by pages.

maybe i misjudge: it's easier to say that i've been more productive on non-computer mornings and days, because i am a compulsive "jotter" ..

"writer" is not quite the right word, because i feel it should mean someone who composes and edits texts which will eventually be complete and contain deep ideas. i haven't proven myself capable of the word, and that is the truth.

i can say that my daily quota is 3-5 sides of paper (possibly from the recycling bin) full of notes, but this includes smallish diagrams, attempts at proof, recapitulation [1] of yesterday's work (and days before, if i'm still trying to prove the same thing!), and then there is some extra exposition (in order to remember the intuition, and how to proceed).

plus, i'm wordy, which confuses everyone, including me.

but count the pages: over a week, that's at least 15 pages, or even 30! it's very good for self-esteem, i say!


in contrast, if you manage another 1/3 of a page of LaTeX on a computer, it seems to tell you little: it does not count the volume of words which has been cut and pasted and deleted. often only the nonsilly part is what survives the "save" command, and one does not keep much extraneous data.

(so let me tell you: when working things out "by hand," i often write silly things!)

moreover, your computer remains the same size: mass or volume, it doesn't matter the measurement. it is not like a pile of paper notes, over a week, which forms a pile ..neat or messy, it weighs more. though there is no reason to measure progress by physical size, we humans still do it anyway.

i know i do.




[1] let me disspell illusions; i learned this word from the adviser. i may be wordy, but i'm not that eloquent!

argggggggh.

sometimes i hate computers. they are very useful, but they have this way of making me unproductive while, at the same time, making me appear productive. while on the keyboard, i even feel productive.

ah, the deception.

in hopes that it would make me more accountable mathematically and professionally, today i sought to work on writeups of my research, but in the end i only fiddled and tweaked existing text and formulae and whatnot.

i tried to write up proofs, but ended up in the land of frustrating technical details which should be done thoughtfully and by hand, and not quickly in hopes that one can return to typing.

"should" is the operative word: i should have. but i fumbled around and tried to be clever and think without writing anything down; in the end i achieved very little and pressed the <delete> button a lot.

argh.

---

i missed a friend's talk, because of a scheduling error. rushing out of east hall, i hurried to rackham graduate school, scaled four levels of staircases ..

.. only to find an empty 4th floor;
so much for mentoring training,
and it was too late to hurry back to east hall,
and hear about geometric group theory.

argggh.

---

earlier i found another error in reasoning in my argument of an implication, and i had previously sent it to the adviser (yesterday?). just now, i've fixed it, so now i have to warn him.

just once, i wish that i could write something up properly.
argggggggh.

then again, i suppose it's good news,
that the proof can be fixed with no trouble,

but i wonder if, one day,
i'll run into an(other) error which i cannot fix.

serving two masters.

i guess i think of it more like "guild master" and not "slave-master," which means that i could be an apprentice. maybe postdocs are journeymen, then.

anyways: the adviser is well enough to tolerate my mathematics again, and recently i've begun talking with another member of my thesis committee about my research and its state of affairs.

i spoke with the adviser yesterday,
and i've spoken with the committee member just earlier, today.

it's been interesting, as well as a little difficult. i'm learning how to explain what i've done and what i think to someone "new," from the very beginning. i guess this will make me more accountable, or at least, more understandable.

one person understanding you is a fluke;
two people understanding you is a coincidence ..
so let me get back to you,
when i can convince a third person to listen to me .. q:


yeah.. so maybe there is hope.

doubts and demons.

sometimes i wonder if i should be a mathematician.

sometimes it feels like i'm not cut out for this. i'm too error-prone and too neurotic, too often like this namesake, a "frustrated over-analyst." too often i worry about my work, wonder if i've bungled another proof, made another error, and convinced myself of something wholly erroneous or unfounded.

sometimes i believe i can't do anything right, and sometimes that belief is justified when i find another error or inconsistency in something i've written months ago, an idea that i've run off with and imagined grand, shining towers ..

.. only the foundations are faulty and though imagined, the towers nonetheless fall into a collapse.

---

for some reason today, i feared there was an error in one of my proofs, and i read through my write-up. sure enough, i found one: a flaw.

i thought about it last night, lost my nerve and couldn't sleep well, a half-sleep. i "woke up" this morning at 8am, jotted something down ..

.. and it seems like it works. there might be something wrong with how exactly to say it, but it's clearer now, and the result is still true ..


.. i think,
but i've been wrong before.

it's a terrible thing when a mathematician loses his nerve. at the moment, i can't bear to look at it anymore, but i've been told that i'm finishing soon ..

.. and there's a lot of work to be done,
if that's really the case ..

.. and i haven't even gotten anywhere near writing up the schoenflies stuff (read: salvage from the first thesis problem) or doing professional things, like cv's and teaching/research statements and fellowship grants and the like!

---

i don't know anything; i don't think so, at least.

all these hours, all these days,
is this really research that i've been doing?
valid, careful arguments?

what if it's all wrong, and i've simply not found the one error that i cannot fix? what's to say that i haven't convinced myself of something stupid once again and wasted days or weeks or ..ye gods.. months?


when you don't believe in an afterlife, then the time you have remaining is what matters most. i don't mean moments (because one has to get on with living life and not watching clocks) but i mean longer times that you can forget but still count, like days and months.

yet i'm wasting all this time ..

epilogue to "me, the morning misanthrope"

a few days ago i woke early again (ca. 7:30am) and without really thinking about it, made a pot of coffee and moved to brush my teeth. it was then that i realised that i would have to work at home ..

.. because sometimes, the coffee tells you where to work.

walking into the bedroom, there remained the detritus of which i spoke before, lying upon the desk. it was my only option: i cannot work on the couch in the living room or couches in general, because i cannot write well while juggling pages atop a sturdy plane, in turn atop my lap ..

(and then, there are pens to worry about!)

even without caffeine, the solution became obvious:
  i took an empty cardboard box,
  shoved everything on top of my desk into the box,
  put the box on top of my bed, [1]
  pulled out my pens and pages and other work materials,

voilà: problem solved.

whistling happily to myself, i headed to the kitchen to pour myself a coffee, ready for a productive morning .. and indeed, it was. (:

[1] that wasn't a mistake. you see, if there is a box on the bed, then i'd have to move it out of the way in the event that i would get back in bed. trusting to laziness, it assured one less means of procrastination.

me, the morning misanthrope [from 4 Sept '07]

in the last year, fall and spring terms, some have pointed out to me that i am difficult to find and never in my office. but recently, my bedroom desk is a mess.

saying that is not quite a non-sequitur.
let me explain.

---

right now, my bedroom is in disarray. my desk is cluttered with moving-apartment debris (like old newspaper), knickknacks, airplane ticket stubs and receipts, and pocket change in a few currencies.

this is no good for the "up-and-at-em" morning mathematician, whom i strive to be. after years of suspicion, i am convinced: mornings are great for getting things done, especially mathematical research things and especially over a cup of coffee or three.

so it doesn't matter that the morning was a dental appointment; seeing such a desk, i'd have biked to the department in hopes of more productive environs and getting a little work done .. well, after visiting the espresso royale first.

but after all, i have an office: personal space, right?
"surely i can work there?"

8am - noon:

i faced queues at the dentist and horrid drilling because of my bad teeth. eventually, they fix what they can, and i slink away, a nonexample to proper dental hygiene.

1-2:30pm:

a postdoc friend peers wisely through the crack between door hinges, spies me at my desk, and asks, "lunch?" [1] and hungrily, i say yes. coffee follows, with another postdoc and a member of my committee. the weather is good, but we complain about it anyway.

3:15-5pm:

fellow student analysts arrive at the office; my officemate betrays my secret, and points behind the door. we are three and we plan an organizational meeting for a student seminar. one asks me what i know about this and that, and i rant without order or purpose for a while.

5-6pm:

there is a welcome back reception, 2nd floor of east hall [2]. i pretend to be a person, and exhibit something resembling normal conversation. apparently i am terrible at remembering female first-year students' names, and when reminded, either suspect incorrectly that they are german or ask strange spelling questions.

7ish-..

i realise i haven't gone to the gym for a while, so i go ..

.. blah blah blah, yakity schmakity:
the rest isn't worth telling.

so i must seize back my effective mathematical mornings, at ANY cost. maybe i am a misanthrope at times, but sometimes one needs peace and quiet to do something on paper that one won't immediately throw away, the next day.

i already miss the caribou coffee, near my former apartment; it was always there if my apartment wasn't quite right for mathematics in the morning, and none of my colleagues were ever there ..

.. well, except for one, when he was visiting for a month. but we ended up talking math then, so it doesn't count.


i've been told that my department is a rare find, that it is social and people actually talk kindly to one another. that's good: i agree, and my colleagues are fine people.

but i hope they are not dismayed if i'm not there, in the mornings.



[1] well, technically his question was longer, but it amounts to the same thing.

[2] i.e. free food and wine. also, there is the phenomenon of cliques, so being social is actually an option.

i guess we analysts can be helpful.

after the meeting.

so as announced previously and elsewhere, i saw the adviser yesterday, for the first time in .. months, and it went reasonably well.

talking to him made me feel ..not "older"
..but perhaps more sobered and "grown-up."

in the last month or two i've felt like an independent researcher, albeit not a very good one; older blog posts are evidence of this.

but i've thought and carried on my research,
minded my errors and flaws in reasoning,
  cursed my follies, then moved on,

reduced to cases, realised there was rigor missing,
succumbed to intuition and pictures,
  occasionally inspired, occasionally deceived.

expecting to discuss it with the adviser
..but never quite sure when, next.


well, apparently my recent ideas are not so crackpot.

---

it was good to see the adviser again.

it's obvious, but i'll say it:
things are not going to be the same,

.. but they will be close enough, for comfort.
he will do what he can, and so will i.

strike 1, 2, ...

i had high hopes for these past two weeks of traveling. among various goals, i had hoped to

1. meet the authors of papers that i've read, and discuss some ideas with them;
   
2. see the adviser again, and ask for advice;
   
3. see a postdoc acquaintance and talk some math.
   

so i was unsuccessful in all of these, but i think i realise why.

---

for 1: (X).
at a conference, everyone's out to meet someone, whether you already know them or not. the more important you are, the more likely you will be busy with co-authors and ambitious youngsters. here, tact is useless against persistence and insistence. so i learned that i have neither of those latter two qualities, or not enough of them.

so i did meet some of those authors/mathematicians, but never discussed much math with them. since it was formulated as an AND statement, it's safe to call this a failure; hence, the (X).

in some sense, conference social dynamics are a little like high school dynamics or prison dynamics. in the first few days, critical cliques form and they are nigh-impossible to break.

good luck, if you want to meet someone on day 3 of a conference (unless it's the last day and everyone's reveling). the acquaintances you make are fun and friendly, and the big shots are always busy.


for 2: (X)
i have the worst luck in the world.

the fellow grad students in my research group and the fellow (some former) students i know from other universities, they chose appropriately and attended a summer school, which happens to be the same town where the adviser was. as an untimely choice, i pick the wrong conference to attend, and become a stranger in a strange land, for a while.

others pay their visits, and then one week passes: the helsinki conference. i have an extra day to make the trip north to see the adviser -- but as it happens, the adviser's not taking any more visitors and will be flying out (by med-evac plane) to michigan, before conference end.

well, the adviser is now in MI and so am i. i guess he'll get back to me, or i'll get back to him; he has plenty of treatment and recovery to come, and there's no sense in my interrupting that.


for 3: (X).
i did see him, but we ended up talking only a little. i don't have any ideas ready, and at the stage they sat then, they weren't worth mentioning in discussion ..

.. especially in light of my fellow students (same research group) who have better, more disseminate ideas. time is short, and chances of meetings are few. let the best ideas win, i say; i'll get back to him, or anyone, when i have better (or good enough) ideas.

---

oh well. i guess conference resolutions are a bit like new year's resolutions. maybe i shouldn't have made any.

i still don't know if it was worth crossing the pond, this time around. i'm getting worse at traveling and i'm not getting any smarter or any good ideas.

sometimes i don't know anymore.

analogy, of a talk.

i heard curtis mcmullen talk today, but an acquaintance of mine overslept and missed it. when he asked me what i thought of it, i said,

  imagine someone showing you artwork, in the form of photographs. imagine also that you are extremely colorblind .. maybe you can only see a few shades of green, and you know for a fact that the photographs are in full color and fine resolution.

looking through the exhibit, sometimes the lighting is dark, and half the shots are during sunset hours, where green is impossible to see. but in the other half of the shots, what you see in green is amazing, brilliant, and insightful.

so you ask yourself: what would these look like, if you could see all the spectrum, if you had sharp eyesight, and if you could appreciate the artistic motifs and cultural references.

imagine how beautiful that could really be ..

that's what mcmullen's talk was like, i told him.

quick post: an idle self-observation.

lately i've been giving a good deal of unsolicited and unqualified advice, but it is in my nature, when meeting new and younger mathematics students.

they know me as a pessimist: i told someone yesterday that

    relax, and make time to enjoy yourself while in math grad school, because there is no reason to believe that hard work will pay off when you expect or want it to pay off.

writing, in longhand, against the clock.

i am finally being somewhat responsible. i'm writing a talk for next week .. although the flight there is tomorrow, and i should have done this earlier.

then again, it's a 20-minute talk; i'll barely have time to introduce the basic notions properly. also, more intelligent and important people will also be talking at the same time ..

(contributed talks are in parallel sessions)

.. so it's a gamble:
will i really have an audience?

it is no matter: the honorable thing is to prepare well and give a good presentation, regardless of audience.

---

i don't trust LaTeX for first drafts of talks.

the format is too slick, and every time i've drafted in LaTeX first, i run absurdly overtime.

so i'm writing a brief first draft, in longhand. it will tell me how long it takes to explain something, and i might actually plan my time well, for once.

also: pencils are very useful for this sort of thing. they provide the "Delete / Copy / Cut / Paste" options that are so tempting on LaTeX, and on computers in general!

the problem: i'm rapidly using up the paltry eraser on these mechanical pencils. i wonder if the manufacturers accounted for this, in order to force demand their way?

---

anyways, enough blogging.
there is plenty of writing to do!

how high can it go?

this is troubling:

      i've been checking the proof of a particular claim,
in order to assure that it is a theorem.
it involves Lipschitz (continuous) functions.

      every time i edit the writeup,
i spot an error, but that's not the trouble;

      so far, i can always fix them ..
      .. but the Lipschitz constant gets bigger, every time. ):


EDIT (9 august 2007): never mind.
i found an error that i can't resolve.

nature: 1,   janus: 0.

struggling against the calm; the storm soon comes.

after re-reading my previous math posts, i've decided that i sound more like a "neurotic over-analyst" than a "frustrated over-analyst."

i can't seem to concentrate on work today.

i broke my rule of "no computer/internet before noon" and i've put the radio on. it was intended originally to be a motivation in making today's worksession a little lighter, and to foster productivity.

the music has become a distraction, though, and i don't know if i have the willpower anymore to be disciplined and productive today.

this will be a busy august ..in terms of annoying trifles of life, i mean, such as moving apartments and promises to meet with incoming students. that sort of thing can cripple creative (mathematical) thought.

up, down, up .. again?!?

funny how life works. after writing yesterday's blog post last night, i left to return a few dvds to the local video store. i was walking back, thinking about what i've proven ..

.. and i realised where my flaw in reasoning lies. it's not a proof; there's a misstep.

i knew it.
i knew there was an error.
there are no such things as happy endings.

arggggh.. oh well.
i'll just let it go ..

.. but, of course, i didn't.

---

i dwelt on the obstruction for a while, reading what i wrote and shaking my head. i tried not to think about it, but thought about it anyway. i read a magazine or two but had no interest in it, and when i set down the pages i thought about the flaw again.

sometimes it's no use. i had suspected i was wrong, hoped i wasn't, and in either cased i'd have been an emotional wreck. for a moment i wondered whether one should take a psychiatric evaluation in order to qualify to be a mathematician.

the frustration can eat you alive. that's what it does to me.

how does anyone stay sane in this business? how does anyone cope with setbacks and failures and uncertainty, on a daily or weekly or long-term basis?

why does it bother me so much? why should i be a mathematician if it bothers me so much? this cannot be healthy. maybe my mind is diseased.

so i tried to let it go, and go to sleep. i don't know how i could have believed i could, but part of the point of being an adult is to try anyway ..

.. because sometimes, you don't have a choice.

---

at 1:30am, i had filled a glass of water, drank it, and went back to bed.

at 2:05am, i retreated to the toilet for a while, and went back to bed.

some time later, i had no reason to get up, but i did. i sat in my chair, at my desk, and i stared out the darkened window for a while. then, at some point, i went back to bed.

a few minutes later, it occurred to me.
well, what if the curves are C¹-smooth instead?

so i got up again, and wrote for a while:
5 pages of meandering, scratches, but i managed a rough idea.

huh. that might actually work.
the hell with it: i'm tired.
i'll work it out tomorrow.

so i fell asleep, and woke up late this morning.

---

it might work, but it probably won't. i see some possible trouble spots already, but at least i know where the argument might break down.

i think i can live with that. i wish i could tell you that it all worked out, that i proved a conjecture that isn't mine or the adviser's, but that wouldn't be true.

there are no such things as endings, happy or not,
for life goes on, and on, unsettled and unpredictable.

i wish i could tell you that i learned why i keep at it, keep at mathematics, but i still don't know why, really. i just do.

a festering idea, and silly neurotic setbacks.

so i'm supposed to be writing, and that's what i did today: staring at a LaTeX screen, off and on [1], from late morning to late afternoon ..

..then a basketball break,
in which a friend of mine broke his arm on the pavement,
and we stopped playing..

but then dinner, and more LaTeX.


"so what's the point?" you ask, "things are going. what's the big deal?"

"ah," i'd reply, "but i didn't tell you what i was writing, did i?"

---

there's an idea, which has been bugging me. it's related to the error i recently discovered in my work.

[see this previous post for details]

since that error, i haven't really trusted my own math or any of the few proofs that i've written. in fact, when working on new ideas, often i'm relieved to find several errors, because

if i made such a grievous error,
and if it cost me that many theorems and corollaries,
then i have to become better at finding those errors.

you see, i wasn't the one who found the error in my proof. someone else did, but it's not embarrassment that bothers me.

what bothers me is that by missing that error, i demonstrated that after all these years, i might not know what is a proof and what isn't, what is good math and what is bad.

if you can't tell what is good math, then how are you supposed to do good math?

---

"so, you're a fraidy-cat neurotic," you'd say, "so what?"

that's the thing, and that's where the aforementioned idea comes in. it doesn't fix the error, but if it's correct, then it proves some of my previous conclusions and corollaries.

those conclusions were based on an error-prone idea, but what i'm saying is that i found another way around, another method of proof. it's simpler and the method is more concrete than my last idea, but ..

.. i don't trust it.

the adviser is out of town ..essentially indefinitely.. so i'm on my own on this one. fool that i've been: i should have given a study seminar talk on the background of my research, last semester or before. i could then bug someone to troubleshoot my proof, and finally find that elusive error.


i've been wrong before and it could well be wrong: that's life, and that's math. not everything works out, and as the saying goes,

if all your ideas are working,
then you're not getting enough ideas.



i feel silly for saying it, but i just don't want to lose my conclusions for a second time. the first time was hard enough.

[1] i meant that i was off & on, not the screen. trust me: my laptop isn't that dependable!

technical difficulties and detours.

occasionally someone asks me if i can read and write in chinese, and then i would respond with, "no, sorry: i happen to be illiterate."

this makes for very strange timing, to anyone who hears no more than that one sentence. it startles them to no end, and i think it's because they thought i meant english.

today, however, i felt computer-illiterate.

---

despite google-searching various keywords and combinations, i tried unsuccessfully to install several LaTeX packages onto my old laptop arielle, who is running Ubuntu Linux these days.

no luck.

so much for finishing the second section [1] of the first draft of that research article, today. i was planning to do my typing in the office; because it's not as comfortable as the apartment, there was the slim chance that i could be more productive.

well, that's a lesson for another day.

---

on the other hand, i spent the day reminding myself of the mathematical arguments that i would have needed to write in LaTeX.

fortune of fate, that: i'd have typed and saved and deleted and re-typed, maybe even opened up a new document in frustration and start from scratch. that is one fault of mine: it's too easy for me to "re-invent the wheel" and not make use of things that i or smarter people have thought about and polished already.

it's also painfully comforting: i think my younger self was smarter than me, my current self.


[1] don't let that impress you. i haven't written much of the first section yet, because i don't quite know how to introduce the rest of the paper (which isn't written yet, either). likely i'll write it last.

undisciplined and unproductive.

august soon arrives.

nothing seems to be working .. research, i mean. i'm too stubborn to let it go, let the other questions go, too, and return to the writing that i've promised and promised to do but have never fully embraced and finished.

[sighs]

i wish i could say that i'm tired, but i'm not. i've had a week or two of holiday already, so it can't be fatigue.

the ideas just .. aren't working.

back to full-speed: the aftermath.

in regards to what i wrote before,

i went back to it, today. the new idea hasn't failed yet, and maybe it will. but maybe it won't, and i can fix a proof and have a few theorems again. here's to hope, anyways.

well, the new idea has mostly failed .. but in an interesting way. then again, it's not interesting enough to write about.

maybe "failed" isn't the right word, either; "remains inconclusive" (read: useless) is more appropriate.

ah, well.

at least i learned something. maybe something else will come to mind, and maybe that will work ..

in the meanwhile, this has been enough of a diversion. it's time to write up that other draft, again. \:

half-speed.

the night before last was mostly sleepless, and yesterday i felt like i was half-zombie, half-human. it's a sad state of affairs when caffeine now harms more than helps: sure, it might keep me awake, but there is an extremely unstable equilibrium for the amount of caffeine which precisely compensates for insomnia.

you see, it's too easy to slip over, to get a slight shake in your wrists. they get worse if the thoughts start racing and you want to write them all down but your hand is too damned shaky and slow. then you get frustrated and try to slow down, but slowing down when you're up on caffeine is about as hard as speeding up when you're caffeine deprived.

then you have to concentrate.

use less words, because words take time and one has to remember to spell correctly.

draw more, but draw simply and meaningfully. make sure the symbols (those arrows!) will make sense later, because everything's going to be a jumble and you'll have to sort through it again and write it well.

i don't have the willpower for that, anymore. caffeine is supposed to be the good sort of crutch: to get you started and going and on the road to thinking and writing about maths. it shouldn't be a broken crutch that you have to fix with careful rounds of glue and tape.

forget the up, which is not quite a high; at this stage of the game, what good are flurries of thoughts, if you can't write them down and check them later? after all, the human mind is imperfect and rigorous thinking is unnatural, even to civilised brutes wretches such as ourselves.

---

so i spent most of yesterday with my brain processing at half the speed that it usually runs. i felt slow, and i was cold.

the idea i had, it didn't work.

i remember opening up the pages of a paper and without ambition, began to read. this is novel for me, because too often i have no patience to read papers very carefully; i get stuck, then frustrated, then convinced that i don't know anything and that i have no future in mathematics ..

.. which may still be true, but the frustration is quite pronounced, at such moments!

but i read a little and it wasn't so bad: section one didn't make sense, but it's an introduction and a microcosm of the whole paper. (i probably won't understand the enormity of the paper anyway.) section two was basic definitions, which always relieves me: no funny, fancy business yet, just definitions.

then section three was a little harder, but then something caught my attention. wait. i could use that, i think. i mean, if this is true and if the computation balks because of that ..


i carefully stuck a post-it note on the page, under the theorem, realising humbly that my mind was too slow to be creative, that day.

i went back to it, today. the new idea hasn't failed yet, and maybe it will. but maybe it won't, and i can fix a proof and have a few theorems again. here's to hope, anyways.

i never thought insomnia could be useful, if only for zombiism!

writing & pre-writing.

it was a week or so ago when i thought to start a draft of a research paper. for those loyal readers out there, yes: it's that one -- the one that i've been delaying and feeling uneasy about. however, now is the time to write, if not sooner.

so i'm following two main principles: the first is some advice i remembered -- but i forget who told it to me; it might have even been a comment on an old blog post -- and i've decided to stick to the format of the talk i gave on that topic, one year ago. if anything, it does give me an outline to begin my writing.

i've also decided to "start from scratch," so to speak: in the first section, titled "Introduction" there is a statement of a theorem from 1966, which motivates the work. in the second section, there is a statement of a new version of that theorem, and then come the preparations and definitions, the details and techniques.

there's much to flesh out, and even when i was preparing the talk, it had been a year since i did that research. now it's coming back with a vengeful, vengeful bite.

i made it to 2 pages of AMS-style LaTeX before i realised that i couldn't remember it all, and then remembered how intricate the argument was.

(to clarify, i'm not sounding my own horn; many of the ideas can be attributed to f. gehring, a rather clever and able man. only a few results are new.)

so now i'm pre-writing: trying to make sense of my notes, remembering why something is true, detecting if what i thought, two years ago, was really a proof. you see, when you've made one error which cost you a few theorems, you begin to fear errors .. everywhere.

at some point .. maybe tomorrow, i'll return to LaTeX again. but now i have to redo the maths that i should have done well when i was first doing them!

back in the saddle.

so it's been essentially two weeks of vacation time, with sporadic, fitful math thoughts in between .. which i later thought through again, with more care.

they don't quite work.

a little something might be there, but like my usual work, it's only mildly interesting. if the metric analysts had their own bar-&-restaurant, it would be the sort of thing that i'd bring up on wednesday nights after nobody said anything for a while.

then someone would say, "oh, okay," and nobody would say anything else for a while.

---

more important, troubling news is afoot, but it is not my place (or right) to tell it here.

summer: time is cheap, and i am unproductive.

it's been unproductive lately.

i think i'm getting too comfortable working at home. during the semester my work sessions were more productive because, strangely enough, i knew that on weekdays, i'd have to leave at some point for the department. that day there would be a class i'd attend, and then a seminar, or there would be my weekly meeting with the advisor.

the days were structured, and time was more precious. i did more with a small window of time then than i do now, in a day. it's not a comforting thought.

i can't help but think that the last few days could have been done in one concentrated morning and afternoon. the results aren't even that good or conclusive. i still don't quite understand the world of metric currents .. especially after that error in proof which has thrown awry all my sense of the topic.



in the meanwhile, i should be writing. i think i owe the world an article i promised to write, and yes, i will get to it ..

.. but it's always hard to leave something alone, when you feel you can do something about it .. even if it takes an unproductively long time.

"fortunately, mathematics is useless."

if i've ever said that i'm good at explaining to others the mathematics i do, then i retract those statements.

i'm terrible at explaining what i do to people. while on a shuttle from the airport, a biology postdoc asked me what sort of math do i do. it was then that i froze.

---

from now on, i will not call them applied mathematicians but mathematical scientists.

i'm convinced that theoretical mathematicians cannot explain themselves, even to scientists. we are like the O-negative blood type: our work can be used by non-mathematicians, but we cannot use the work of non-mathematicians. hence applied folks, if capable of speaking to scientists, must be scientists.

that doesn't mean that they aren't mathematicians. far from it: they may be excellent mathematicians, capable of great things and fine work. they just happen to be able to present themselves also as scientists. we theorists have no such flexibility.


also, i will hereby use the default answer of:

"oh, what i study is absolutely useless"

whenever someone asks me about the applications of my work. it's true, from my viewpoint. there are no immediate, non-mathematical applications which come to mind.

if they want to know how it is related to science, say how the field began, then sure, i'll answer that. but they have to ask explicitly that, because it's not worth wasting time and proffering illusions of mathematics as some science, or something very real.

---

mathematics is not real, or at least, not fully so. it is abstraction. for some inexplicable reason, when polished to something computational, it just happens to be useful .. in frighteningly many settings, and it will probably continue being useful.

but i still cannot say why, why it is useful. a mathematicial scientist would know better; i leave such things to such people. i'm happy to be useless, contribute nothing to today's society except sate its sense of progress. we are as useless as artists and poets and philosophers ..

.. but isn't it funny, how it is art and literature and philosophy that we remember from older civilizations, and not their technology, their science, and their engineering? we don't fully know how the egyptians built their pyramids, and except for reasons of historical posterity, i don't think we much care.

after all, with bulldozers and cargo planes and laser cutters and explosives, we could build better pyramids, and more quickly and efficiently.

we have abandoned steam engines and vacuum tubes. if we retain our computers, then still, their form will change. we already smirk at 386's, right?


but we remember culture; it persists in memory, despite not contributing directly to the progress of its society, in its time. so how is that, for "useless?"

lies that a mathematician friend might tell you, because i do.

so today i told several lies .. i think,
but these are very common lies for mathematicians.

twice today, i was asked about the things that i study: once by my sister and once by a friend from high school, and both of them are neither mathematicians nor know much mathematics.

oh well; nobody's perfect. q:
then again, they dared ask and that is something!

---

perhaps "lie" is the wrong word, and "simplification" is more correct. as a general rule, i never explain actual mathematics to non-mathematicians. the best i can hope for is to suggest an idea, or give a rough explanation of one key idea which is key in the work i do.

have i written about this before?
i must be getting redundant.

so to err on the side of caution, let me not rant excessively. spot the 'lies' if you find any, but

1. to my sister i described rectifiable sets as surfaces which look like terribly crumpled aluminium foil, because i couldn't think of a way to describe to a general audience the currents of federer and fleming, much less the notion of a current on a metric space.
   
   happily, she remembered what a tangent 2-plane was .. intuitively, at least.
   
   i then suggested the minimal surface (plateau) problem as an application, and to motivate the study of general metric spaces, i speculated that there are plenty of strange geometries which are interesting, even in real-world terms, like the parameter space for a bicycle or connectivity graphs from a computer network.
   
   
2. to my friend i tried to explain why limits aren't as intuitive as the terminology suggests. he fancies the example of achilles and the tortoise, and i suggested the pathology of porosity:
   
   what if the racetrack was full of holes, at all scales? so say that you want to sample various times to determine how achilles will finish; how do you avoid choosing bad sample times over the race?
   
   i then suggested to him that as a sequence, an orthonormal basis on an infinite-dimensional hilbert space [1] has no limit, even though no point is that much farther from another. as a concrete example of why infinitely many dimensions is reasonable, i suggested possible states of an electron ..
   
   .. even though i don't know any quantum mechanics. i wonder if that remains true.
   

[1] more true to words, "suppose you had two coordinate axes, and now three. now imagine four, which could be, so why not five .. and so on: imagine that there is no end, and there are an infinite number of independent directions .."

thus far (update from new york)

my visit to new york is half over, and as a vacation (from mathematics) it's been mostly successful.

there was one morning where i was the first one awake (my brother and a sister are also visiting this weekend) and i wrote as i tried to remember how things stand with my research.

on another day, i was killing time for some reason and tried to make rigorous an example that the advisor and i sketched out. the key word is "tried" and it's still not quite rigorous.

i'm sure it's true, or at the very least, contains something true and purposeful in the computation. maybe i'll find time for it on monday -- during a train ride into manhattan -- or on wednesday -- while waiting at airports.

in the meanwhile, i'm trying to be a person again, and relearn how to do normal things.

pondering, on whether or not to ponder ..

tomorrow i visit my parents and the house i called "home" in my adolescent years. except for a new pair of running shoes i bought last new years, i can't remember what i left in that dusty chamber that used to be my bedroom, and that makes it hard to pack.

no matter: it's summertime.
packing clothes is easy.

packing mathematics is much harder. i'm still thinking of taking the week off, but that feels wrong. as a friend of mine suggested, i can always split the difference and just read: read the papers that i've been meaning to read ..

.. because when will i ever have the time to sit and read them, anyway? it will definitely be more relaxing than .. say, research. after all, thinking can be quite tiring.

maybe i should take it as a vacation.

besides, what's the worst that could happen? if indeed i get so obsessed with a good idea that i break stride, and prove something great, then it was meant to be. in the meanwhile, why not enjoy myself?

then again, why does this sound like excusing myself, and being lazy?

bad news.

yesterday a fellow grad student stopped by my office and asked me about one of my proofs. he thought there was a flaw in it and i asked him to explain.

sure enough, there was. he apologized for not telling me sooner (he knew on monday) but i don't think it really mattered.

so i lost a handful of theorems .. mostly corollaries from that one "theorem" .. and there's not much of my collected work left.

it's been a bad 24 hours.

i met with the advisor today, and we came up with a few ideas. none of mine worked, so i'm trying one of his.

---

the thing which stings the most is that i didn't catch my own error, and it's not for any reason of vanity or embarrassment. the fact that i didn't spot the error means that i'm not as good of a mathematician that i thought i was.

just when things were going right, that i felt like i was making progress .. and now? now i feel like i'm starting over, pillaging the salvage.

not surrender, but a retreat.

i think i've lost another few days. last night and this morning i was tempted to throw away the notes and scratch work i've jotted since last saturday.

to clarify that: i'm a packrat. i never throw anything away, least of all written ideas .. that is, if they contain any value at all.

blame it on paranoia.

the last thing i'd want to happen is realising that an old idea of mine will work, only i can't remember the argument and i threw away the paper. i don't think i'd forgive myself.

for similar reasons, when i think about a problem, i start with the most obvious, 'stupid' ideas first [1]. it's the safest course of action, i think. if the 'stupid' ideas don't work, then fine: usually there is some lesson in why they don't work.

now what if the 'stupid' idea works, and you never tried it?

anyway, i've been thinking about a problem lastly and nothing's working. i feel like i've been arguing myself in circles to the same impasse.

i can never reach one particular step, and coincidentally, i can't seem to apply one particular hypothesis in that step of the argument.

i've concluded that i don't know enough to solve the problem. there's something i need to know, or discover, before i can tell whether this argument will succeed or fail.

but as of now, i'm not ready for this problem,
and that is a VERY frustrating thing to say.

it's not quite a surrender, but it's an order of retreat. it still feels like giving up .. and i HATE giving up.

more than that, i hate being unproductive. so now it's a matter of switching gears, and starting work on something else: picking battles, so to speak.

it still feels .. wrong, but i can't afford to lose any more days without learning anything. picking battles, and living to fight another day.

[1] i must be getting thin-skinned or something, but these days, words like 'obvious' or 'trivial' or 'stupid' bother me, when used mathematically.

a mathematician's disclaimer (cross-posted)

i originally meant to write about vacations on the other blog [0] but as do many things in my life, it turned mathematical.

so i've copied-and-pasted it below.

---

how often do people take vacations, in this day and age?

i can't tell and even if i could, my opinion cannot be trusted: i'm an academic and apart from one afternoon per week [1], i set my own work hours. there aren't so many hours, either, but there are enough to make it feel like something close to work and definitely not play ..

.. well, not for most days of the week, anyway.

* * * * begin: waxing mathematical * * * *

not all academics can be so casual, i suppose. we mathematicians are a strange class of academic. our field is rife with technical jargon and the terminology must be both intuitive and precise in order to serve any use. but we are not quite scientists: we have no laboratories and we have no loyalty to any workbenches. a chalkboard helps, but paper and pen will do, and if there is neither then we can always think it out carefully.

according to legend, archimedes drew figures in the sand, and eventually he died amidst his figures.

nobody seems to understand us, because simply put, mathematics is a language and there are few fluent speakers. i tell people that practitioners of theoretical mathematics are more akin to philosophers than to scientists. we may refer to 'thought experiments,' but those are more thoughts than experiments.

scientists must bow to reality. despite how abstruse and abstract the theories of the physicists, there is the motivation to explain observable processes.

instead, mathematics begets more mathematics, and mathematics has nothing to do with reality [2]. if we are helpful to science, it's because science poses problems that are mathematically interesting; otherwise mathematicians would have stopped listening to scientists, long ago.

science entertains mathematics. if we produce something of value, then let the scientists have it, make their progress, and reinvent the world in their own image. they have offered us stimuli and now we can ponder new curiosities, and now the engineers can invoke this progress and harness it to whatever end will serve or sate our society.

for example, a mathematician doesn't care whether string theory is correct or not. to an algebraic topologist, string theory is another motivation for problems in cohomology, and to a complex analyst or hyperbolic geometer, string theory is another reason to revive the notions of conformal structures and teichmuller spaces.

but i digress: we will always be misunderstood. we cannot share our work to anyone but other mathematicians. we are like a type of nerd, who will bander on and on about something that is pointless to everyone else in the world, but to us, it's SO DAMNED COOL. think of WoW gamers or foodie cooks or linux hackers; we are like them.

i know some mathematicians who love to talk about their work, but amongst non-mathematical company they clam up and are nervous, in the way that people become nervous when you tell them not to imagine their best friend naked.

"it's so damned cool, but it won't make any sense if i tell you and you will be annoyed if i do tell you, so i should shut up .. but it's so .. damned .. cool .."

as non-mathematicians, you will never know us in the same way that you might never understand an artist and his art. artists must really, really care and think their art is wonderful and important; otherwise, why would they devote so much time to it, when they could be doing something more fun? why would they exile themselves in isolation, when they could spend more time with their friends and loved ones, or at least, work a less lonely occupation?

think of them, and think of us.

if you really want to understand what we do, we'll try to explain. but be warned: we mathematicians don't mean to offend if we say, "it's complicated," because it is complicated and hard and often frustrating. if it were easy, we'd have sorted it all out and be at the beach ..

.. and probably, we'd be drawing an occasional figure in the sand .. at least, when nobody's looking.

* * * * waxing ceases * * * *


on an unrelated but funny note: i'll bristle at being referred to as an 'intellectual' but i have no problem with being called an 'academic.'


[0] yes, i have another blog: two others, actually. one you can find on my profile (see sidebar) and it's about vegetarian cooking. the other blog is .. well, the other blog.

[1] the weekly research meeting with the advisor. today we ended up talking mathematics for 3 hours or so; it would have been shorter, were i better at math.

[2] except in the platonic sense, i suppose, but i'll leave that to the philosophers.

something wrong, something right, and neither very promising.

it's been a frustrating 6 days.

i spent the first 4 days trying to prove something, aware of what shouldn't work in the proof .. due to known facts and non-examples .. and i couldn't prove it.

worse yet, it's because the argument is incomplete. as of 2 days ago, i had no more ideas or inspiration to fill the gap. so i don't know if the argument is wrong, either.

after these many [1] years in graduate school, i've learned not to feel that bad about being wrong or entertaining a stupid idea ..

more accurately, i would call it an emotional callous

.. but to be wrong and not to have learned anything .. ye gods, that's frustrating. it's like having a mild conspiracy theory stuck in your thoughts, and not knowing if you are indeed a crackpot or not.

i wonder, if only on a practical level, whether that explains why some conspiracy theorists are crazy .. that it's not the crackpot theory that drives them mad, but the uncertainty which does.

as for the other 2 days, i may have actually proved something, but the argument of proof doesn't look right.

you see, it's not my theorem. a few months ago i read a particular theorem in a paper from 1999 or 2000, and i couldn't understand the proof.

so i tried to cut up the author's argument into claims, and prove those. some i could prove and some i couldn't, so i thought of other claims which would make sense ..

.. but having let this take a life of its own, my current proof has noticeable difference from the author's proof.

so yes, i've had better weeks. \:

[1] and yes, i'm griping. i know that plenty of you have taken longer to finish your own ph.d.'s and that i have no basis to gripe about time.

for the record, i am griping about something else.

scattered and unproductive.

i don't think i'm being very productive. certain things i do by habit, like waking up and then thinking about mathematics for a few hours, over coffee .. taking notes and pondering problems.

lately i've been confused.

i think i've been able to coax all the 'easy' results from the fusion of these two topics that i've learned .. though i don't think i've learned them very well .. and now the work will become more difficult.

put one way, i would say that the theorems i've proven over the last two months have been pointed observations and casual conclusions (and not all of them mine).

i wouldn't call them all of them corollaries, but i've needed to use a nontrivial fact or two from the works of others. without them, these theorems would still be claims, dead in the water, without proof.

now it won't be so easy.

i have a claim in mind, but i can't prove it, and those facts from before are no longer applicable and not obviously relevant. i think i'll have to invent or discover a small something, but a new something, in order to solve this part of the problem ..

.. but i don't quite know what something it should be. i suppose that if i did, i'd probably settle the problem and have a laugh about it. for now, i'm not laughing.

most of the time, i'm frowning at those pieces of paper.

---

my mind feels scattered.

i can't really encapsulate my research as a thesis problem anymore. then again, we've always kept this second "problem" open-ended, like an exploratory mission of sorts. but before, it was easier [1] to say what i was doing ..

.. but now this idea runs this way, and another idea runs the other way. often i don't know which to follow, which is more productive or fruitful, and often i just pick one, hope for the best, and get to work ..

.. and every so often, wondering if "the grass is greener."

i also find myself switching around these sub-projects. for example:

yesterday & today, i thought about a problem which has more to do with the currents of Federer & Fleming on euclidean spaces, rather than the new-fangled theory in the context of metric spaces.

the flat forms of whitney and wolfe also appeared briefly, as those in the know .. well, could probably have known.


two days ago, i was pondering the (first) heisenberg group and what could be said there, in my line of work. i conclude what i always conclude:

does anyone actually understand this space? it's a beautiful geometry, but do we know how it works, analytically?


the days before that i was working on metric spaces which are general enough to do what i'd like them to do. the advisor then asked me about something, which was a good idea.

i'll probably try and work it out tomorrow .. unless i keep with this Federer-Fleming thing until i reach an intractable dead-end, which might take a while, and then there's the rare chance that it could work out ..

but you get the idea: "scattered." (:

---

[1] well, "easier" if you narrow down the world to a few dozen people, maybe even a hundred.

i'd like to think that i could explain the rough idea to a mathematician, but then come the "prerequisites," such as

* measure theory & functional analysis, for starters,
* a little differential geometry, for motivation,
* some awareness of the analysis on metric spaces, for relevance,
* some geometric measure theory, for intuition and analogy,

i might be able to gloss over certain things, but it would be hard-going.

a trip down math memory lane.

my motivation for research is very low today. since i woke up belatedly, at 11am [1]) i don't think i've had any thoughts towards anything mathematically new or substantial.

part of me wonders ..

1. if this is unconscious disappointment in myself: i wanted to wake up early and get a good start,
   
   (despite a late night of friends, beer, revelry, and a propane bbq grill)
   
   but it's already a dismal start and from this, what follows will only be salvage.
   
   
2. if my mood reflects the weather outside. the last two days were warm and sunny, one of them humid. today is cool and cloudy grey: fine weather, despite a little rain drizzle.
   
   it reminds me of pittsburgh weather and those happy times.
   
   but it is hard to hold to absolutes, and not compare relatively: yesterday made such a fine evening, and today there is a comfortable yet uneasy chill in the air.
   

it's hard to say.

---

over the past few weeks i've written notes on many pages, and having no inclination or mind for mathematics today, it was a conveniently productive task to do.

it's startling. it's like reading my old journals from younger years, but these are instead of the mathematical sort.

there are all sorts of digressions into separate questions related or unrelated to my work, at hand. some of these read like ambitions and others like delusions of grandeur.

i didn't realise how much i filter, when talking research with the advisor. with him i stick to the successes and the instructive failures: the notions that make worthwhile meeting time.

some notes from a day, two weeks ago, contained a flaw in reasoning. it took a few days of further, confusing notes and scratch-work before i deduced there was an error.

revising my ideas, this destroyed the fleeting, hopeful fancies of what this flaw would have implied. so i forgot them, and with them i forgot a few good ideas which became 'guilty by association.'



[1] yes, i missed the defense that i mentioned, yesterday. so much for honor. \:

"we are men of honor. lies do not become us."

i never thought i'd encounter this sort of dilemma:

    i've been invited at least twice [1] to the same party, which will celebrate the thesis defense of a friendly acquaintance of mine.

    so i suppose that, as a personal matter of honor, i must attend the defense. usually, this wouldn't be so tricky of a decision, but there are two difficulties:

1. the defense begins at 9 am.
   
   yes: am, meaning ante meridiem or morning .. or as i prefer to call it: pre-noon. informally, it means either sleep-time or breakfast-time, or if i wake early, math-time.
   
   
2. the thesis concerns algebraic number theory and uses 'scary' words like p-adic cohomology.
   
   me, i'm just a poor analyst [2] who only knows a definition or two from basic (co)homology and can barely remember how to do any diagram-chasing!
   

[sighs]

it is a difficult thing, to struggle with matters of honor.


[1] once was through email, but that was a mass email to invite .. well, each and every math grad student. the two other times were the ph.d. candidate (who will defend) and one of the hosts/tenants of the house/party site.

[2] to this day, i don't quite know what to call myself, other than the broad umbrella term that is 'analyst.' i've met quasi-conformal geometers, geometric analysts, and geometric function theorists, but i don't know if i would call myself one of them.

let me prove a few more theorems and write a few more papers; then i will stake a claim! (:

on a lighter note .. monkeyBALL.

i may be weird when i called the boundary of an n-cube an n-square ..

.. but from the way one author describes the Monkeysphere, i think he means a Monkeyball. q:

beware your soul ..

i have been LaTeXing lately, and perhaps it will become a paper ..

.. after a long, arduous process, of course,
with many edits and rewrites,

but perhaps .. \:

---

i might have mentioned this before, but when left unchecked, LaTeX will consume your soul. computer programming of any sort, even excessive typing, will do this.

the problem, i think, occurs when one forgets that the code is not the important thing. for instance, it's easy to obsess over making LaTeX look even prettier than it already is ..

.. or when writing a program in C,
where you may write a function procedure
to handle the exception caused by the previous function,
which you had written to replace the earlier function,
which didn't have an exception,
but didn't handle cases in the object type that you wanted,
because your previous class construct didn't compile,
and you never figured out why,
and at the time you thought it easier to start from new.

one thing leads to another, then another, and at some point you realise that you're obsessing over something that has nothing to do with good mathematics or good programming or good anything.

i've heard that graphics in LaTeX are such quagmires.

so suggests the advice i've been told, when once i studied computer science:

unless you are really good at coding, NEVER improvise in front of the computer screen. always code with a plan in mind, and if the plan changes, sort it out first and then go back to coding.

---

then again, meandering can have its benefits. while updating a preliminary bibliography, i remembered a theorem from one reference.

i think i can generalise the result, now.
so the LaTeX ceases, and the mathematics continues.

concerning some controversy of late ..

on a more serious note, i learned today that there is a banff protocol concerning increases in mathematical journal prices and (nonviolent) protest:

THE BANFF PROTOCOL

We agree neither to submit to, referee for, nor participate in the operation of any journal that charges an excessively high per page subscription fee, as compared to the average of the 25 highest impact journals in pure mathematics**

as for my stand .. let's say that i'll wait until tenure before i make a stand. \:

in which i acquit myself of murder, and debug a proof.

this may be understatement, but mathematics can be confusing. i'd even say that it can be quite hard.

today over coffee i worked for a few hours, until my observations gradually suggested statements of an absurd inclination. i don't know how much logic everyone knows ..

(though i know that many of you do know enough logic, say, to be dangerous)

.. but i am referring to proofs by contradiction; colloquially speaking, they refer to verifying the validity of logical propositions to the absurdity of their negations.

---

to give a silly example, one might ask whether it is true or not that this morning, i killed a man in beijing with my bare hands.

well, suppose that i did; that would have meant that i was physically in beijing this morning, but that can't be true because i have an alibi: the barista at caribou coffee knows that i (or someone like me) had ordered a coffee and a cookie, and whittled away a morning, staring at pages of strange symbols while looking confused and frustrated.

so i couldn't have been in beijing, and so i couldn't have committed that murder. it's not rigorous proof, but then again, real life doesn't make a very good logical framework ..

.. so you'll just have to trust me: it wasn't me. (:

---

i still can't determine what, exactly, is wrong with a recent argument of mine. most of it seems right, if only because i've attempted to write a transparent, lazy argument which borrows most of its strength from the good results of others ..

.. but something remains amiss. i thought of an example which is inconveniently close to a counter-example to the conclusion of that argument.

in the last hour i worked, i resorted to troubleshooting and what computer programmers might recognise as 'debugging.' in other words, i narrowed down the errors to a few places in my argument-code, and now it's a matter of testing those places.

i'll input test data and look for valid output, but the data isn't numerical. instead i'll input a known metric measure space and the output will be a collection of geometric objects.

(the objects are metric currents, for those who know.)

so it's down to this: getting my hands dirty with the details of an example .. but isn't that what mathematical analysis is supposed to be? (: