the minor dilemmas and decisions which arise from writing.

i intended to work for most of my free hours today, but all i seem to do is blog. so in the interest of doing work-related things, let me express a few thoughts:

ONE. i seem incapable of writing preambles to theorems anymore. often i sit and struggle and think of the right words (or the right point to make) to add before writing a precise mathematical statement in the form of a lemma, theorem, corollary, or proposition.

most of the time i give up, flip open my thesis, see how i wrote it there, and i think either:

* good enough; glad i thought of it before
    and then i paraphrase myself;

* god, that's crap. why didn't i just write
    <add mildly insightful comment here>?
    and so i do.

in other words, i cheat and i plagiarise myself. it makes me wonder how i ever managed to write this tome of a thesis.


TWO. i've often heard people say that "i'll probably write two or three papers out of my thesis," which leads me to think: wow. it must be nice to have that many good ideas.

my writeup is at 27 pages and counting. there's no introduction yet, and one application hasn't been added yet. if i never add it, then i might write a separate little note about it -- say, 10 pages -- but it would be a really weak note, as far as notes go. that theorem relies on two things:

* a main theorem, already written up and added to this first draft;

* another theorem which has essentially been proven,
    and if you know the subject well enough,
    you could probably reproduce it rather easily.

so as you can see, it wouldn't stand very well on its own. besides, once you combine two papers from the literature, then the same theorem would already be known. then again, one of those two papers doesn't exist yet. you see my point, though, right?

on the other hand, if i add the application into this paper, then it's headed into the unwieldy realm of 50+ pages. who in their right mind is going to read that, from a recently minted ph.d.?!?


THREE. there is also another theorem missing from the writeup and i'm not sure if it's worth the extra pages; i suspect it will need at least 7. the argument is long and technical and the method of proof is one that never sat well with me philosophically. [1]

also, i don't actually need the theorem to prove anything else, but it does round out the whole picture nicely. admittedly, in light of what i proved and wrote up thus far, i would ask a "natural" question that this theorem would answer.

again, the theorem wouldn't be able to stand on its own, not even in a note.

so what do you do when you do have ideas, but they cannot make papers on their own?


[1] the proof is correct, and i learned a lot by thinking it through, but the ideas don't feel like ones that i would have thought up. i had a lot of help with my second advisor and really, i think of it as his theorem, not mine.

he'd deny it, of course.

an old friend, an unexpected encounter.

after teaching my afternoon class yesterday, i sat at my office desk and tried to convince myself to work. i wasn't very successful. then i heard a knock and i opened the door.

if i tell you that it was my friend Rμpert, then at first it wouldn't be surprising.

what would make it slightly surprising is that we were math majors together at pittsburgh, and he left when i did, five years ago. we both went to math grad school: i to ann arbor and he to pasadena. of course you can't account for someone after s/he defends a thesis, but i wouldn't have expected him in pittsburgh.

what would make it surprising is that i actually understand what he does, as a mathematician. he's related to the hyperb01ic ge0metry mafia: the pseudo-anosov flows, the dynamics, the foliations, all of that. i've heard of his advisor, and i've taken classes and heard talks by people whose work he studies.

he might even know what a qua$ic0nf0rmal mappin9 is;
i forgot to ask him, earlier.


odd. he once told me that he studied these things called braid groups, and i assumed the worst, that is, he became a closet algebraist. q;


as for good news, he will also be employed soon, as a postdoc in the chicago area.

2 (or 3) quick observations.

every time i see a preprint on the arXiv that is written by someone i know, i get nervous.

shouldn't i get to finishing that one draft ...?

mathematics isn't a competition, of course. then again, the job market is.

---

my morning calc 3 class is fast becoming my rehearsal or opening-night sort of lecture: it's where i get used to my "lines" (i.e. my lecture notes), see what works and what does not. the afternoon class goes more smoothly ..

.. though it feels less organic, more routine.


i'm also starting to wonder if i'm lecturing well. i try to make the topics meaningful and how things fit together, but here's the rub: in a calculus class, there is (almost) always a textbook, and too often students only care about solving the assigned homework problems. so i wonder if my lectures are pretty but not very useful towards applying to problems.

i do examples, but as i said: i wonder.

---

i was right. i'm probably slated to talk soon at our departmental analysis seminar.

it's also interesting how the U of M model for analysis seminars has been exported: i've heard from a colleague at KSU that they have a "Thursday seminar" and i think ours will be of the same model. then again, the original format is a finnish one .. i think. [shrugs] on a slightly related note, i also heard word of a postdoc visiting our department from finland, for the year.

so again: it's pleasantly odd how certain characteristics here are similar to ones i've gotten used to, in ann arbor.

i wonder, though, if i have enough clout to invite my cronies for talks? (;

---

odd: today, more people visited my mathematical homepage than this blog. as far as i've checked, that's never happened before.

then again, most of the hits have been from servers in this university. i did give out a url to my students which links to the main page, so perhaps it's not so surprising it's happened. some people will click on anything.

---

anyways, so begins the weekend: i don't teach again until wednesday. that means, of course, trying to catch up on research and writing. wish me luck!

another idea down, and i'm on the clock.

well, my idea doesn't work .. that is, this one.
more accurately, i can't make it work.

even more accurately, i should say that the idea is also related to one that my mathematical brother m@rsha11 has entertained. more importantly, he thought of it first. in his case, it worked and it proved something nice:

fact. for k > 1, there are no nonzero metri¢ k-¢urrent$ on the hei$en6er9 9r0up (in the sense of am6r0si0 and kir¢hheim).

oh well. no sibling rivalry, really: he's a smart fellow and has earned his results. myself, i throw things together and see what sticks. at least i got a thesis out of it.

---

for the sake of optimism, if i back up a few steps in reasoning, there might be a way out of this. maybe i can prove something. then again, what i envision requires re-inventing many wheels of fun¢tiona1 ana1y$is, essentially.

i'm loathe to attempt that, now: there are things to write up, classes to teach, talks to prepare for october, and sooner or later the research group here will "invite" me to talk in our ana1y$is $emin@r.

also, there's something in my work contract about collaborating on new research with members of that same group, so better to wrap up these projects and toe the line. it will probably be fun, too.

anyways, it's time to be a diligent mathematician again and to stop blogging for now. there is a lecture to write, for tomorrow, and there is always LaTeX to do.

teaching, working.

i guess i always dismissed them as operations restricted to euclidean 3-space, but cross products of vectors are pretty cool ..

.. in the sense that determinants are also rather neat. (;

hopefully teaching will become a little more routine and a little less neurotic, that my TAs won't kill me if i panic the students, and that i'll have time for some research between writing lectures, giving lectures, and holding office hours.

say, is it a sign of research withdrawal if:

i go to bed,
fully intending to get a good night's sleep and a good start in the morning,

then toss and turn for an hour,
then get up and have a glass of water,
then eye my dinner table, with paper and pen on its surface
.. they're right there, ready to be used ..

and then sit down and work for an hour, fleshing out ideas?

lately i've been getting the fear that if i don't get research done now and if i don't write up my work now, then i will never have any (more) papers or research life.

briefly: a first 1ecture in ca1¢ulus.

this semester i'm teaching two sections of multivariab1e ca1¢ulus, and today i couldn't find my second lecture room. i asked one room twice and i think they thought i was a dolt.

another room initially had clean writing on the board and an instructor-ish person was standing near the lectern. when i checked later, the board was erased and the instructor-ish person was gone.

i asked: "is this a calcu1us iii course?"
many said yes.

i looked at my cell phone, as my watch died on me, a week or two ago: it was one minute into the next class period.

close enough. so i set my bag down, took out the textbook and my lecture notes, and started writing on the board. admittedly, i wasn't sure that it really was the right room, but to my credit, no other instructor ever showed up ..

.. then again, would you enter a classroom,
if someone else was teaching in it,
and there was a full class also there, taking notes?


more about the first day of teaching tomorrow. i feel dead on my feet, after today .. and curses: i didn't get any research done!

teaching unease.

the last time i taught a class was in fall of 2007, which was a calc ii course.

yes: i guess i'm a little spoiled from my grad school experience. i suppose the gravy train ends this year. from now on i'll be teaching several courses, every semester.

yesterday and today, i've been a little neurotic from my teaching preparations. i don't know why, either. i've taught before and i know my calculus: i do analysis, after all. heck, i even know what students there expect out of a calculus course.

so why does it still bother me?

---

on a lighter note, one of the TAs (for the attached recitation to my lectures) emailed me about teaching details. the email was addressed to a "Dr. Janus." [1]

i nearly started giggling. oh dear. except for colleagues and friends, i suspect that i'll be Dr. Janus, from now on.

[1] of course it didn't actually say "janus," but my surname instead. some of you know exactly who i am, in physical reality, so bear with me and my mild attempts at internet anomymity.

old ideas don't die; they don't go away, either.

thinking about it now, it's been months since i've discussed research with a fellow mathematician. so i suppose this means that i've been working in isolation. maybe that explains how i feel ..

.. because i feel slow and stupid.

my ideas aren't quite right -- i'm almost sure of that -- but i cannot think of better ones which will do. so i alter and adjust them and try those variations.

i feel that, mathematically, i spent the last few months making small permutations of the same basic, probably flawed notions and seeing the slight variations in their common failure. [1] this borders on obsession, of course, and i hope that when the postdoc begins, someone in the department can knock some sense into me and suggest more fruitful areas of math for me to pursue.

---

today i couldn't remember the definition of a curr3nt, in the language of f3derer and f1emin9. [1.5]

the space of k-curr3nts is meant to be dual to the space of differential k-f0rms, and a 0-curr3nt is equivalently a distri6uti0n; i remember the motivations. so there's probably a LCTVS structure [2] on the space of k-f0rms, just as there is such a structure on the space of test functi0ns, by way of partia1 deriv@tives and sup-n0rms.

i may be wrong; but i can't remember exactly and my copy of f3derer's gmt is some hundred miles away, so let's go with that. at the very least, there should be some continuity property for an arbitrary curr3nt, otherwise a 0-curr3nt couldn't possibly match up with a distributi0n, right?

sometimes i really hate myself, especially my poor memory.

i can deal with ambiguity. however, i can't deal with confusion. like anyone else, i looked up "curr3nt" on wikipedia: here's the entry, but i can't reckon what the continuity axiom should be .. if i'm even remembering it correctly.

argggggggggggh.

anyways, i'm bothering with the "classical" definition because i don't want to throw away one of my ideas.

to patch up one of my currently flawed approaches of proof, i need to use some property of metric curr3nts that an arbitrary (FF) curr3nt should not have. otherwise, i would "prove" something for all (FF) curr3nts (of finite m@ss that is conjecturally untrue.

often i wish i were smarter, but i'd settle for better ideas. \:


[1] non-mathematically, i spent the summer unemployed, recovering from thesis writing and thesis corrections, wall-climbing, and as they say in good will hunting, "seeing about a girl." q;

[1.5] this means, of course, that ethically i should renounce myself as someone who studies ge0metric mea$ure the0ry. on the other hand, i was trained to be a metric ana1yst, and until recently, all curr3nts to me were curr3nts in the sense of ambr0si0-kirchh3im. so: put one way, it's not my fault. (;

[2] short for "l0cally c0nvex t0po1ogica1 vect0r sp@ce," as you might learn in functi0na1 ana1y$i$.

quick post: la p1us de chan9e ..

my bags are packed and i'm leaving ann arbor today. i don't know when i'll come back, but likely not until i'm invited for something.

recently i was notified by the U of M computing staff that my account will expire in 2 months. as a step towards becoming a Pitt person (again) [1], my U of M mathematical homepage is now gone. it automatically forwards to my Pitt website ..

.. which is the same HTML code and still looks the same. so as the saying goes,

"la p1us de chan9e,
la p1us de meme ch0se." [2]

speaking of which, yesterday i found a gap in one of my recent arguments. admittedly, i'm not surprised -- it shouldn't have been so easy to prove such a thing -- but nonetheless i'm a little depressed about it.

a failed proof is like a pet goldfish dying;
i feel the same way about it.

well, i guess i can learn from the fallacy that i made, and try to determine what the truth really is. there's time to kill on the bus ride, anyway.


[1] short for `university of pittsburgh.'
also, i was once an undergraduate student there.

[2] i'm too lazy to look up the source, but i think the quote is due to m. pr0ust.

a day of interim

yesterday i edited and added to the draft of a paper which will (eventually) be complete. as for today, i feel rather useless and lazy. it's my last weekend in ann arbor and my usual work routine is shot to pieces by the subsequent changes and demands of leaving.

for example, i have a few things to pack, a small box of belongings to ship, and an apartment to clean (in order to recover some part of my security deposit).

today is not a day to add to long-term goals; it has too much of a transitional period. i can't seem to concentrate on details or to plan anything else.


so i've resorted to LaTeXing the most recent research ideas i had, some days ago. specifically, i'm trying to see if this proof idea is valid. this sort of writing is easier, as the ideas are still fresh in my mind, but it is still slow-going and my mind drifts away so quickly to other things ..

.. like, for example, blogging. \:

perhaps it's time to do something more useful, like packing or getting some exercise.

conjectures as obsessions

i think i proved something new today. it's a little lemma which is not useful for anything, but it came up because i've been conspiring about vector fields again [1]. well, perhaps it will one day be useful, but i haven't thought that far or deeply about it, yet.

it's also not too surprising. a special case has been known (but without explicit proof) and the statement is the same [2]. i suspect that i rediscovered their argument, but that will have to wait.

---

i think i've reneged on a vow again.

some months ago i said, half-publicly, that i was going to leave the field of analysis on metric spaces -- this specific stuff, at least -- and start working in a new but related field. at the time, i had just finished the final draft of my thesis and i was desperate to do something new, something different.

even now, i'd like to diversify my interests and see if i can become some kind of analytic "jack of all trades." but between then and now, i discovered something; rather, i remembered something about myself.

you see, i tend to obsess,
and i have a weakness for conjectures.

1. call it the habit of youth, but given a half-decent idea, i'd still go after the isoperimetric problem in the (first) hei$enber9 9r0up in a heartbeat.
   
   
2. there are few days when i think about maths and when i don't think about one particular case of the''f1@t ch@in c0njecture" of ambr0si0 & kirchh3im. i don't know why. i just do.
   
   
3. on those rare days when i am brave enough for the abstraction, i might think about the various chee9er conjectures on so-called PI sp@ces and their we@k tan9ents. then again, i'm often not brave enough; abstraction is not one of my strengths ..
   

.. as opposed to obsession, but is that really a "strength?"

call them obsessions or loyalties or motives: as a mathematician, i have them and probably i am ruled by them. i like to think that i am not alone, and i suspect that it is in the nature of how we teach mathematics:

for so many years in school, we are told to solve problems ..however algorithmically.. and as undergraduates, we learn proofs and work on problem sets. we become inclined to think in terms of problems.

i've told others before that i do not build theories. it's not in my nature; i am not my late advisor.

as for whether i am a problem solver ..solve problems?..
well, let's say that i think about my obsessions.



[1] then again, by definition it is impossible for me to conspire .. that is, to conspire alone. according to dictionary.com, "to conspire" means:

1. to agree together, esp. secretly, to do something wrong, evil, or illegal: They conspired to kill the king.
2. to act or work together toward the same result or goal.
–verb (used with object)
3. to plot (something wrong, evil, or illegal).

so unless vector fields are inherently evil, i don't think i was necessarily conspiring. oh well.

[2] see the latter sections of "$tructure of nu11 $ets in the p1@ne" (ecm proceedings) by A-C-P

a conspiracy theοry of vectοr fιelds (or: out of practice)

i wouldn't call myself a resident in ann arbor anymore. all of my belongings lie in an apartment in pittsburgh, but i nonetheless remain in michigan. so i suppose i am something like a visitor who is still clinging to his east hall office and his U of M computer account.
anyway.
these last few days have been a return to mathematics. i haven't set a routine yet, and the paper writeup has been slow. among other things, i'm dealing with a new laptop and new $\LaTeX$ user interfaces and trying to remember how i wanted this paper to go.
but today i didn't add to or edit my paper writeup; i didn't work on collaborations, either. i didn't even pursue this idea to prove a conjecture that i admitted to my mathematical sibs. [1]
instead, i tried to start a new foray into research. it has the flavor of what conspiracy theorists and other crackpots do: i observe several events unfold, and i suspect some underlying, overarching cause. i can't say why i have these suspicions, except that they are all topics which i've read about from geοmetric measurε theοry.
in particular, i see vector fields all over the place.

1. there's a paper by fragalα and maηtegazza from 1997, in which they formulate a tangeηt sρace for a measurε. the definition uses vector fields and schωartz's theοry of distributiοns, and they have proven a theorem which relates these objects first to nοrmal 1-curreηts and then to tangeηt measurεs as studied by marstraηd, preιss, and others.
   
2. there's still no manuscript available, but in some recent work of albertι, csörnyei, and preιss, they discuss the notion of a "weak tangeηt field" associated to certain leβesgue ηull sets in euclιdean sρaces. this is apparently intimately related to sets of differentιability of lιpschitz functions.
   
3. i've formed some (measurable) vector fields of my own, with some help from the earlier albertι-csοrnyei-preιss machinery. they are not spelled out in my doctoral thesis, but the data can be found in an important technical lemma in chapter 5 (about derivations on the plane).

i haven't proven anything. these thoughts of mine are merely "what ifs." if i had to be honest with myself, this isn't the most efficient use of my time.
then again, i'm out of practice in being a mathematician. sometimes it feels like i've forgotten how to do research, how to make steps in reasoning, how to prove things and deal with technical details. i think of this as a little rite of passage.
i've said that i wanted to be a mathematician again, but i didn't say that it would be easy. \:




[1] as it turns out, there was no gap in the argument. then again, there remain many details to check before i can say the idea has become a proof.

after a hiatus of moving (& parents) ..

my state of things:

i'm done with moving my material possessions. my parents helped, which means that all maths and blogging have stopped in the last week. [0]

i've still no university ID,
still no printing access,
but my login works.

i can blog but i still have to pay for the city bus,
in order to arrive here and write in this blog.

my new laptop (read: toy) ships tomorrow, i think.

i feel like a half-person, and i don't feel like a mathematician.

it's been weeks since i last LaTeXed and more additional weeks since i thought about new research. there's this idea i remember from a month ago, but that's just memory.

i don't know if the idea will work.

there's no reason why it would, other than it addresses a concept from ge0metric mea$ure the0ry and a concept slightly afield from analysis on metric spaces. it's a "throw spaghetti and sauce at the wall and see if any of it sticks (and tastes good)" sort of idea, a gamble.

it's the type of idea that you'd try if you've stopped being creative and need to try something, because you otherwise feel like a worthless human being.

---

it seems that i've been getting more hits on this blog lately, due to that one post about the "m@thj0bs rum0r w!ki" of 2OO7. it's either that, or someone's been having a "case of the mondays" yesterday.

so: if you are one of those waylaid souls seeking job advice though bl0gs or g00gle search, then sorry: i was just as confused as (if not more than) you are about what makes a good maths postdoc job application.

i still am.

however, if you want to learn vicariously what NOT to do, then read some of my past entries. i've made enough mistakes that someone should learn from them.

EDIT: apparently, similar things happen over at Reasonable Deviations. here is a list of searches which give them hits: [link]

---

on an unrelated note: today i had approximately chinese food [1] and received this fortune cookie:

a smile is a curve that can
get a lot of things straight."

cute, that. then again not all smiles are non-straight, so the meaning isn't terribly rigorous in its wordplay.

i'm splitting hairs, though: it's what happens when i haven't done any math in a while and lack mental stimulation.

[0] so in case you were wondering, that was why this blog has so recently been silent.

[1] it was at a "pan-asian" eatery. the food was mainstream enough for the uninitiated american palate, but authentic enough for ethnic asians.