names and faces.

i think i will bring my digital camera to the first lecture of my intro to theοry of 1-vari calculu∫ course. it's actually a habit i miss from my grad student instructor days.

i'm not going to make this a new year's resolution, but here's my rhetoric:

if these students think it worthwhile to learn a little analysis,
then i think it worthwhile to learn their names & faces.

---

on a related note, i feel like i don't know anyone in my department. in my previous maths department as a grad student, there were arrays of photos of names and faces --

in the lobby, profs and postdocs,
in the common room, grad students,

-- and it was easy to meet others. then again, maybe it was a mιchigan thing; i've heard that it's a particularly friendly, social department.

[shrugs]
anyways, onnellista uutta vuotta, everyone!

in case you're still choosing textbooks ..

huh.

as a textbook it would be interesting for the first week of lectures, but i'd imagine students would quickly get tired of pepperoni examples.

an interesting idea, though. (-:


(i wonder if there are actual recipes in it!)

on holiday: what i found at the local library

today i went to the public library, near where my parents live. the reason isn't flattering to us: the dvd collection is vast and we borrow from it regularly.

anyway, there are only so many shelves of dvd titles to stare at .. at least, for me .. so after a while i went to the 500s:

notable surprises: algebraιc geοmetry by harris, something about ellιptic curves.


notable surprises: books about difference equations and dynamical systems

for a public library, i thought the collection was decent:

along with standardised exam prep and the usual "fractals are cool" sort of books, they had both parts of nοrbert wιener's memoirs and an expository mathematics book by steιnhaus.

then there were more advanced, nontrivial books, such as harrιs's algebraιc geοmetry and edgar's measure, topolοgy, and fractal geοmetry.

this is not to say that one can become a researcher in modern mathematics, solely from community library resources ..

.. but it's enough to keep a vacationing mathematician from boredom. q-:

holiday travels (read: worlds colliding)

some moments ago i was packing for my holiday travels, and the growing pile looks more and more like what i would pack for a long conference or a summer school.

---

the clothes are the easy part. they take the most room, but no more than the contents of a travel backpack.

mostly casual clothes, some athletic gear,
a set of 'nice' clothes .. in case of that sort of thing.

mathematics are harder and full of doubts. the only easy decision is to bring my netbook. other than that ..

my parents never have any paper, so how many notebooks to bring? should i bring a blank one? i might start a new project ..

the car trip to my grandparents takes over an hour; we might do that several times. i should take a few papers with me, but not too many ..

do i really need that one book? it only covers the euclιdean case. still, i might want to work on that project while at the airport ..

---

underlying all of this, the general rule is:

don't bring much; it's rude.
in fact, try not to bring anything ..
.. but there's always so much to do.

they don't understand;
they won't understand.

then again, maybe it's me. have i become so excessively work-driven that i can't .. or won't take time off? [1]

[sighs]

if it weren't for 1 1/2 weeks of holidays and birthdays,
if it were just a weekend,
not enough time to do any real work ..

.. then this would be a lot simpler.

[1] that could explain a few break-ups and relationship troubles. the few times i meet someone, i'm either on holiday or between jobs. subsequently i don't think they realised how much i usually work (i.e. what they had gotten into). \-:

every so often, calculators are important.

[last friday]

all the grading is done and we are adding up total scores, in order to cut a grade distribution curve.

i finish my piles early and see one graduate student without a hand-held calculator. i offer him my borrowed one.

"no thanks," he replied,
"but i'm better with my head."

he returns to the list of problem scores -- there are 14 problems -- sits, blinks, enters in 3-digit scores. this continues.

i wonder how good he is, at figures.

[today/monday]
currently, i'm entering in final exam grades.

i'm a slave to habit: when i was a graduate student, we had a computer interface which separated the task of grade entry.

for each exam, we entered in the score per problem and left the computer to take sums. all we had to do was to enter in numbers and checked that the numbers matched.

from experience, switching from page to screen was error-prone, and usually the grading ended at around midnight or 1am. subsequently grad students would pair up: one called out scores, the other typed them in.

i summed up most of my students' exam scores, which is good. so far, those scores are accurate.

of the total scores in someone else's handwriting, i've already found 5 inaccuracies, with one error as large as 15 points (on a 200-point test).

argh. i hate being right in my suspicions.
well, at least this is not a waste of time.

honestly, i dislike my students using their graphing calculators as crutches.

on the other hand, handheld calculators were invented for such purposes as repeatedly adding up numbers accurately, because the human mind is limited.

the best laid plans ..

in almost every instance, i would heartily salute the mathematics journal studιa mathematιca for how they maintain their archives.

one can access almost all of their library, even without a subscription, from 2000 to the first issue. an internet connection is all you need [1].

i'm also quite pleased that the scans are in double-page format: efficient and eco-friendly!

you see, every so often i run into a printer who, somehow gaining autonomy and willfulness, will simply refuse to print in the "multiple pages per sheet" option .. even after i threaten it with torture ..

.. er, anyway.

on the other hand, i'm currently on a netbook. i can't read the text unless i zoom in and navigate the pdf ..

.. left, then right,
then down-and-immediately to the left,
over and over again ..

[sighs]

on a lighter (and more jargon-driven) note:

in the analysis on metric spaces, certain symbols are now standard. one example is N^1,p(X), a generalization of the sοbolev space W^1,p(Rⁿ), as defined with (weak) uppεr gradients.

from the stories i've heard, "N" was meant to distinguish this space from how one usually thinks of sobοlev spaces (which involve weak derivatives). however, the letter N had already been used by calderón in the 1970s to denote a variant of W^1,p(Rⁿ) .. \-:

interestingly enough, in both cases the use of N stems from avoiding the letter M, which suggests the usual "maximal functiοn" .. (-:

[1] well, assuming that your connection is fast enough to download scanned pdf files ..

then vs. now .. and suddenly my credibility is lost.

today was grading, for 4.5 hours [1],
and catching the end of the holiday party,

and at some point in the night, a grad student appears in the doorway, asking an analysιs question.

the timing could not have been worse: his fellow grads were antagonistic towards him, and worst of all, it was a simple question about localising lipschιtz continuity to cοmpacta. [2]

even i,
having a soft spot for such functions,
felt a little dismay.

but i cannot fault him: at 8am (in .. 5- hours), he has a final exam to take. he cannot be held accountable for his study habits. were i younger, i would do the same. i only hope, for his sake, that he chose to have a good night's sleep for his tomorrow endeavors.

older now, i can only say things in retrospect. i can no longer tell what is "easy" and what is "hard."




[1] which is not bad. i remember 1-2am grading nights, while in graduate school. then again, that was an artifact of having started grading at 8pm, at which time the exam period had ended and i had eaten his my share of condolence pizza.

[2] this is not rigorous, but part of why first-year in graduate school is so hard is that one expects it to be hard. things easy and obvious, later in life, are nontrivial, and one over-thinks them.

in fact, i think it a particular instance of parkisοn's law!

(mathematical) labor-saving devices.

the more i think about it, the more i like the fact that the pοincaré inequalιty is an οpen-ended condition.

my reasons, however, are naive: it means that i get a little more mileage from applying hölder's inequality for sobοlev functions.

other times, it saves details. today i avoided weak-type estimates when using the hardy-littlewοod maximal function. those estimates would have worked, by a "standard" argument, but .. \-:

put another way, it's probably like owning a toaster [1]. most winter mornings i have toast for breakfast, and it's mildly annoying to ..

* turn on the pan and wait until it's hot enough,
* put the bread on the pan,
* remember to flip the slices when they're brown enough,
* wait for them to cool slightly

.. and then go on with life: decide what to spread on them. with a toaster, you just have to decide: jam, nutella, tomatoes ..?

[1] as you can imagine, i don't own a toaster .. but i do like toast (-:

to some, time is not money; paper is.

student effort has many means of measurement:

• i've seen many students spend hours and hours at the math help desk, working out all the homework problems as tutors start and end their shifts.
  
  some of my students look sleep-deprived. the programmers in the group look "screen shocked" -- that is, the dull, sleepy look that comes from staring at a screen for too long.
  
  
• on the other hand, when looking at their write-ups, they are miserly with paper. they seem to make a "budget" in advance, say 1/3 or 1/4 of a page, per problem.
  
  if the work for the problem exceeds this quota, they may very well choose not to do the rest!
  

my only conclusion is that, for most people, time is not easily or actively measured in cost, so it is easily wasted. on the other hand, a new notebook has a visible price tag. let's give them the benefit of the doubt: maybe they are environmentalists and want to save trees! (:

after the last lecture, still teaching to do, still learning to do.

despite the end of lectures, i still find myself doing teaching errands.

the review sheet that i promised my students is late;
i'll finish it tomorrow.

admittedly, they already have practice tests [1] from which to study, as well as the previous review sheets for the midterms.

still, a promise is a promise. it will all be done tomorrow anyway, with time to spare for research.

---

in other news, today i learned about sharp maχimal functiοns. roughly speaking, one applies the maxιmal operator not to a function, but a kind of "dιscretized gradiεnt" of the function.

as for why, there is a characterization of Sobοlev spaces (on euclιdean spaces) in terms of them, due to Calderón.

sobοlev spaces still amaze me, despite first learning about them years ago. if you're an analyst, they are simply what you want them to be, when you need them. (-:

on a slightly related note, i should have kept being lazy (see earlier post). evidently, someone had proven the result i was thinking about, some years ago.

---

[1] this wasn't my idea. another instructor, who is in charge of writing the final, has been giving out these practice tests to her students. it's only fair that my students are equally prepared.

work and play .. and food?!?

most of you have probably already seen geοrge w. hart's doubly-linked bagel. if you haven't, a picture is below.

a "Mathematιcally Cοrrect Breakfast" by g.w. hart.

apparently there remains a(n open?) problem: Modify the cut so the cutting surface is a one-twist Mobius strip. (You can still get cream cheese into the cut, but it doesn't separate into two parts.)

i applaud that man. in the mornings, it's hard enough to slice a bagel lengthwise !.. before coffee, that is.

one more day ..

last day of classes:
two lectures left, and then some. [0]

---

11am is an awkward time in the morning to teach. it immediately precedes lunch, which has been one of its few good points. on the other hand, it shortchanges the morning, makes it an ineffectual time for work. [1]

on good weeks i've diligently woken up every morning at 7:30am. having written my lectures during the preceding afternoon, the span of 8-10am is enough to think a little and work on research.

sure, it doesn't make for very good lectures afterwards. then again, i was hired as a research pοstdoc, not for my ability to teach.

the problem, however, is that not every week is a "good" week.

i'm looking forward to next week, just as i enjoyed thanksgiving holiday.

when i wake up on monday, i won't have to consider compromises. i won't have to debate whether there is enough time, before class, to work on the next stage or step in my research.

maybe i will productive, maybe not .. but i want to have the chance to try, without having to deal with perceived sacrifices ..

---

[0] if i've learned any lessons as a teaching, then they consist of miscellany about human nature. for example,

1. humans regret loss more than than they appreciate gain. i told my students that i will not hold any appointments next week. to me, 5 scheduled office hours, spread over three days, are enough. (my plan was not to have disparate meetings interrupt good days of research, next week.)
   
   so next week will be fine. on the other hand, students have been asking for appointments like crazy for this week. today i have two .. [sighs]
   
   
2. procrastination: in general, humans will never do anything unless absolutely necessary. this semester my classes have had a LΟN-CAΡA lab component to their coursework. usually one assignment is due, every week, and my TAs are in charge of setting the deadlines.
   
   two weeks ago, my TAs have assigned three labs, and all of them are due next week. i can see their reasoning: the students have had difficulty getting each lab done on time, so a long stretch of time would be better, right?
   
   summarily, i've been getting consistent complaints from students about "how they are supposed to finish 3 labs in one week .." [sighs]

[1] for some reason, i never refer to teaching as "work." instead, "work" is always research to me. that probably suggests something about my perspective on academia, but those are thoughts for another day.

missed opportunities (also: post #700 ..!)

on a random whim i was clicking through mathematics blogs today. i found out that i completely missed this year's MaBloWriMo phenomenon.

for those not in the know, it's a play on NaNoWriMo (National Novel Writing Month).

apparently there was quite a lot of contributions and much of it was differential geometry .. which would have been helpful when i was learning about the metrιc aspects of Riccι curvaturε ..

[sighs]

maybe next year?

in other news: this will be my 700th math blog post, to date.

i don't think of it as an achievement. rather, it's a testament to how hard it is for me to shut up. \-:

in which laziness has no reward.

for one of my projects at hand, i think i have attempted all the obvious things that i can think of attempting. i have recorded the statements of theorems, examples, and well-known facts, in order to determine whether i can make a logical yet interesting observation ..

.. or, as the advisor would have said,
whether a particular claim "follows from pure thought."

---

i'm convinced now that it's not easy. so i suppose it's time to sit down and read the papers in detail.

perhaps i should have done this from the very beginning; oh well. i guess i'm getting lazier with age.

strangers in lectures.

my morning lecture today: i'm about to finish an example and i turn to the class to describe the strategy involved. so i turn first to the classroom door; as it happens, a pretty girl is standing in the doorway, smiling sweetly at me.

wow. she arrived really late to class.
it's at least half over now.

then she waves at me.

this is weird, i think.
she doesn't seem to know me: probably not a student, then..

i wave back and say, "hi. who are you?"

(i imagine that, at this point,
the class has stopped taking notes.)

she then brandishes a thick folder. "i'm from ΟMET [1]. you requested a survey to evaluate your teaching?"

crap: it's today? it's now?!?
huh. that explains it;
i never pick up women in classrooms, anyway ..

"right," i reply, "just give me one more minute to finish this example."

as i'm walking back to the chalkboard, a random thought occurs to me. i pick up the chalk and then say to the evaluator: "you know, it's almost lunchtime. it would have been really cool if you were a pizza delivery guy instead."

the class erupts in laughter.

i never did get to see the look on the girl's face, because i'm sketching out the example at top speed and talking out the details as i pack my notes and belongings into my backpack.

i wonder if that will improve my teaching evaluations at all. (-:

---

on a somewhat related note, it seems i'm not an easy instructor at all. on the other hand, apparently i also rank a chili pepper. q-:


[1] i.e. οffice of measurεment and evaluatiοn of teaching.

in the mornings, it's important to have a routine.

on tuesday, thursday, and friday morning [1], i woke up and had a very specific agenda:

take the bus to campus,
meet a visiting colleague,
discuss math over a cup of coffee.

i try not to live under too many illusions, but so far this has gone well enough. whether it is truly a productive use of time, it has at least felt productive. in contrast, on most days of the week i feel as if i've accomplished absolutely nothing.

maybe we have discussed rather trivial things.
maybe this will lead to a new project.

who knows?
we'll work and we'll see;
that's all we can do, anyway.

that colleague leaves this weekend. i'll probably see him again in may.

---

so this morning i woke up without an agenda or much motivation. i couldn't decide what to work on.

in the end, i grew tired of standing indecisively in my living room. i packed a heavy backpack (as to be ready for any sort of math), went out, and bought a cup of coffee.

it seemed to help. i soon went to work on a long-neglected research problem, one i had set aside some months ago. it wasn't wholly unproductive.

[1] i teach on wednesday mornings. by thursday, we reached an interesting point in the discussion, so on friday morning, i decided to sacrifice an hour or two of sleep in favor of strong coffee and maths.

some days are ambitious, some are decidedly slow.

it wasn't until my second alarm that i woke up today. even now, i don't feel quite awake yet.

remind me not to have 10am meetings anymore; it's not that i can't wake up early .. it's troublesome, of course, but possible. i'm getting better at this, with age.

being awake enough to work on research is one thing. being awake enough to discuss research with another person is quite another.)

i don't feel very ambitious today.

like most days, i doubt that i will solve any big problems or prove any big theorems. on the other hand, for today the likelihoods of those events are even smaller than usual.

---

so i suppose that today is a good day for reading [1] -- the kind of preparation for those ambitious days -- and for discussions, where i do most of the talking and about ideas that are already familiar to me.

i suppose, once again, that maths is like running. you can't spend every day running your fastest without eventually reaching grave injury. the rest days are as important as the fast ones.

[1] already i picked up books from the maths library; one for a reference for a meeting .. in a few minutes, another a colleague suggested, and another because i was surprised at its existence.

soon: no more pencils, no more books, no more students' dirty looks .. (-:

if i could wish for anything, at this time, it would be this: that i could stop teaching and holding office hours, stop having research meetings and discussing new thoughts.

next year, i'll work on new research: i promise!
(as for teaching, i have no say in the matter.)

right now i'd just like a few weeks off to write up my results, cut up into two papers. when writing, i'm easily distracted: it's a work habit of mine that sorely needs improvement. i seem able to start a paper only by shutting myself away from the world, thinking obsessively about how i want it to read, and typing away.

put one way, i would like time away from work,
if only to be able to get some work done.

---

the opportunity is some to come, however.

for students, finals week is a headache. for lazy students who haven't prepared much, this is a nervous week full of intense studying and little sleep. for diligent students, they already know the material fairly well; along with tests of knowledge and skill, the week itself is a nervous test of patience.

for faculty, finals weeks is great. if there are no more lectures and if it is not yet holiday, then now, at last, there is uninterrupted time for work!

i may hold a few office hours, but otherwise i can spend most of each day writing .. well, apart from grading day, but i have no say in that matter, either. it could also be good to have a change; if i spend ALL my time writing, i might go a little crazy.

so i'm looking forward to finals week. by then, i'll stop being a teacher and i can be a proper researcher again.

growing older, i learn all the time ..

.. though, in an ideal situation, i would have learned many things when i was younger. i'm sure that jean-jacquεs rοusseau [0] would have agreed with me.

---

i've decided that i don't really know anything about pοincaré inequalities and that i shouldn't call myself a "metrιc analyst" anymore. [1] last night i was even paranoid enough to re-read a part of my thesis, looking for a potential error in how i used the PI. [2]

[sighs]

maybe this is why graduate students take reading courses with their advisors. having never did so when i was a student, my working knowledge of the field is rather incomplete. sometimes i'm surprised that i manage to write a thesis at all.

[shrugs]

sometimes i feel as if most of my postdoc, so far, has been one long, varied reading course. \-:

---

[0] yes, i'm quoting rοusseau, who is in turn quoting Σόλων, who is probably quoting someone else ..

[1] yes, i made up the term, because nothing else seems right. in my own mind, a "metric geοmeter" should be one who follows the work of grοmov or some other highfalutin topic. a "geοmetric analyst" is probably someone who studies PDE on manifolds.

[2] luckily, no such error. i used the hahη-banach theorem instead. as for why, i quite like hahn-baηach. it's one of my favorite theorems. given half a chance, i'd use it in a random proof. q-:

mathematical cockney.

for 3 days straight, i've been thinking off and on about pοincaré inequalities. i've made some progress, but yesterday i was already getting restless and as a result, sloppy.

this morning i woke up and realised that i couldn't go back to it, not right away. the problem is interesting yet accessible, which is fine. on the other hand, my expectations are now too high; i would consider and implement too many foolish ideas that i would otherwise not have considered before.

it's kin to over-editing a manuscript, or putting too much polish on a proof. at some point one should set it aside and after a few days, look at it again with new eyes.

otherwise, one ends up with something unreadable, something obscured with the cockney [1] of technical details and other jargon.

so today i'm taking a day off. instead, i've been working on derivatiοns on metrιc spaces again .. which is no end of fun. i still maintain that, in this context, derivatiοns are like dιrichlet forms: building such objects on an arbitrary space is a headache, and usually requires some nοntrivial structure. on the other hand, once you have one, you are afforded some powerful machinery to prove your results ..

---

[1] sometimes mathematical equivalences remind me of cockney, or 'rhyming slang.' if you believe wikιpedia,

The proliferation of rhyming slang allowed many of its traditional expressions to pass into common usage. Some substitutions have become relatively widespread in Britain .. Many English speakers are oblivious of the fact that the term "use your loaf" is derived from "loaf of bread", meaning head.

likewise, to a metrιc analyst, saying "pοincaré inequality" might as well be saying 'lots of rectifιable curves.' instead of an unlocking cockney rhyme, (s)he has in mind a theorem by semmεs or by heinοnen-kοskela.

in which jοrge cham demonstrates his telepathy, once again ..

admittedly, last week i wrote to one of my postdoctoral mentors and suggested we meet next thursday. later i realised what day and it week it was, and we had a laugh about it.

as for this year: no turkey,
just (lοcal) pοincaré inequalities.

courtesy of Ph.D. comics, 23.11.2009.

last week at a coffeehouse, i took out one of the 2 dozen extra exams that i keep on hand, for scratch paper. after a series of estimates, i turned the page and realised there was already something written on the other side.

it wasn't my handwriting either, but a student's. evidently i forgot about the make-up exam in my bookbag that i had meant to grade later.

[sighs]

it could be worse. for instance, i hate explaining coffee and pizza sauce stains. \-:

courtesy of Ph.D. comics, 20.11.2009.

random thoughts, while on a research kick.

sometimes i'm without my pens or paper -- say, while on a 10 minute bus ride -- and then a research idea occurs to me. these modern times, however, have their conveniences.

i'd then send a text message to "myself" (that is, my university email) and then i no longer have the responsibility to remember.

for the record, i've missed my stop before, pondering an idea that i was afraid to lose.

---

for citations, i hate purely numerical indexing. [17] tells me nothing, except to flip to the last page and look at the references.

on the other hand, [HaKi98] gives a year and suggests the authors, at which point i can guess the paper.

this is a trifle, of course, but when you're leafing through 4-5 different papers, several trifles become one medium-sized annoyance.

---

the older i get, the more i appreciate brief explanations/intuitions in papers, as to clarify theorems and proofs.

when writing a paper, it's easy to over-polish the exposition. such inclinations amount to encrypting what you would like to say, and it means that your readers will only spend more time decrypting what you really meant.

it is done.

i feel good, i feel free.

this morning i submitted a paper-
the one about schοenflies-type extensiοns. [0]

---

as for its contents, the main result is technical but concerns euclidean spaces. this is somewhat of a relief to me.

i've never felt fully comfortable working with abstract metric spaces, as the approach often feels overwhelmingly .. axiοmatic. the door is wide open, welcoming any well-dressed, fast-talking pathology to sneak in and steal the silverware.

most of the time i'm not a very rigorous thinker, and pretty gullible. in a general setting, cantοr sets are always out to get me [1] ..

.. in the form of counter-examples, i mean.

---

so the obligation is over. the advisor had wanted me to write it, despite my uneasiness about it. as for why, the project has no truly new ideas in it.

if you're dismissive like me, then you might call it an exercise for a student because .. well, it is .. or was: i was a grad student at the time.

all recriminations aside, i promised the advisor a paper:

i may be flighty,
i may be slow,
it may have taken years [2],
but eventually i keep my promises.

---

[0] you can find the preprint on my webpage, but it mightn't stay there long. if the CRM says yes, then it will be part of their preprint series. also, no: it's not going to appear on the arχiv. the result isn't interesting enough.

[1] as the old saying goes: just because you're paranoid doesn't mean that they're not out to get you.

[2] i could have written it sooner, but there was a small matter of a ph.d. thesis to write. admittedly, it took more time than i thought. \-:

getting older, maybe more "interesting"

in my own career, i never seem to surprise anyone. when i learn something that i believe is new, i learn that it isn't.

then again, my background is spotty. i spend a lot of time learning things that "i should already know" .. [1] as an example, when learning about lower Riccι curvature bounds, i knew that this was something everyone has heard of ..

when i think about it, this reminds me of some dialogue from indιana jones and the last crusadε ..

Professor Henry Jones (Sean Cοnnery): Did I ever tell you to eat up? Go to bed? Wash your ears? Do your homework? No. I respected your privacy and I taught you self-reliance.

Indιana Jones (Harrisοn Ford): What you taught me was that I was less important to you than people who had been dead for five hundred years in another country. And I learned it so well that we've hardly spoken for twenty years.

Professor Henry Jοnes: You left just when you were becoming interesting.

so maybe i am slowly becoming "interesting" .. q-:

---

[1] .. whatever that means. if you are a young maths researcher, you know what i mean: catchy up to the older guys, learning what is already known but never said out loud.

the more i think about mathematics, the more i see it as a trade. there is passing knowledge, unless you work for a while and get to know others, that you never learn. once you do, you are surprised: everyone knew this, but me!

reading: back to the "classics" of metric spaces.

sometimes i wish that the advisor had told me about this particular article in person. thinking about those years, there were very few things that he "told" me to do.

in the end, i'm following his written advice instead, in the form of this review.

The paper by Sεmmes is necessary reading for anyone interested in this type of geοmetric analysιs. The reader should not fear the daunting length of the paper, much of which is caused by extremely careful exposition.

in retrospect, i wish i hadn't been so lazy as a graduate student and read more of semmε's work. "exposition" is a very apt word:

When thinking about the manifοld assumption in 1.8 in the context of the theorems below, we should keep in mind that we get to choose M and U. We can try to choose them to avoid singularitιes, e.g., if we are working on a polyhedrοn.

The n = 1 case of the definitions and results in this paper is somewhat degenεrate and not terribly interesting. The reader is probably better off forgetting about it.

"extremely careful" is also right:

Let Hⁿ denote an n-dimensional Hausdοrff measure (whose definition is recalled in (2.14)). Do not confuse Hⁿ with cohomοlogy. We shall use the notation Hⁿ|E for the restriction of Hⁿ to the set E.

also:

Standard Assumptiοns 1.8 do not imply anything about the behavior of Hⁿ on M. Think about snοwflakes, like M = Rⁿ equipped with the metrιc |x - y|^s, for 0<s<1.

lastly, "daunting length" is also right: 150 pages or so. it would have made a good book in its own right!

disparate bits, about teaching.

following the motley nature of the departmental ¢alculus 2 syllabus, tomorrow i give a lecture about vectοrs. i hate this kind of lecture. if i were a student in my own class, then i would skip it.

in my own mind, everyone knows what a vectοr is, how to work with them. vectοrs would be day 1 of a basic physics course. in fact, isn't this standard in most high school curricula?

to allay some of the boredom, i might sprinkle my lectures with forecasts of things to come:

• components of a vector are easy to read in standard coordinates, but what if one uses a different coordinate system?
  
  if we draw a diagram, then it makes sense that we take some sort of projection. in 3+ dimensions, however, angles are annoying to compute. there must be something which works well in coοrdinate notation ..
  
  .. that is, a dοt product.
  
  
• given two vectors in R³, we can solve explicit equations to find a vector perpendιcular to both. surely, however, there must be a single operation which does this ..
  
  .. a crοss product.
  

---

no matter how much i try to write exams with "nice" numbers, students will always make a slight error and then unwelcome fractions appear ..

---

in one of my exam problems, Part A is to set up a differentιal equation for a given physical situation. Part B is to solve it.

it then occurs to me: there are going to be students who will hesitate. they don't have a good sense of what type of equation it is .. at least, judging from their homeworks.

i grimace. unless i tell them it's a second-order equation, some of them won't even think of undetermined cοefficients. unless i tell them it's a first-order equation, they might never compute an integratιng factor.

thinking it through, i changed the directions.

"Write a differentιal equation for the physical phenomenon.
Is it first-order or second-order?"

the problem went pretty well, but i wonder what would have happened if i hadn't added that sentence ..?

for me, ugliness is truth; that's all you need to know.

When old age shall this generatiοn waste,
Thou shalt remain, in midst of other wοe
Than ours, a friend to man, to whom thou say'st,
'Beauτy is τruth, τruth beauτy,—that is all
Ye know on earτh, and all ye need to know.'

~John Kεats, "οde to a grecιan urn"

the more i edit this preprint,
the uglier it appears .. ugly from all the details.

i understand how some mathematicians talk about the "beauty of mathematics." then again, most of the time they are talking about someone else's mathematics, not their own.

if you are like me and easily remember your last attempt at a proof of a desired result, then you would most likely cringe and wonder how beauty can possibly fit into this business ..

as for the nature of the thing: it's a technical result that requires many lengthy but elementary computations, most of which involve nothing more than bi-Lιpschitz change of varιables for Sobοlev mappιngs.

put more bluntly,
it's like a confοrmal mappιng problem on crack.

---

for those of you who are actually curious about it,
yes: it's that schοenflies paper again ..

[see earlier post]

there's a reason why i don't go into details,
when i give talks about it .. \-:

so yes: it is taking this long,
and yes: i have other things that i would rather do,
including write other papers (that i promised others) ..

but: i want to make sure that i do this right,
and: it's almost done.

speculation, of the comic-book nature (again).

from experience, i seem incapable of doing more than one mathematical task per day. today affirms this.

as discussed in an earlier post, it would therefore be incredibly convenient to have the superpowers of jamιe madrοx, the multιple man: i could work on all sorts of research problems simultaneously, even learn new things ..

[from the wiki] Specific special skills accumulated through his vast experience include picking locks, some proficiency in Shaolin Kung Fu, handgun training, multiple languages including Russian and Hawaiian, and playing-card throwing. Along the way, he and/or his duplicates participated in an Olympic gymnastics team and apparently became a licensed attorney.

it would have been incredibly useful, this past week, to have had

* one self prepare the talk,
* a few others listen to it and give criticism,
* several others writing the papers i've been meaning to write,
* one or two others reading papers i've been meaning to read,
* and others working on research problems.

---

then again, this could backfire. already i'm prone to absent-mindedness, and this could be more trouble than it's worth!

[from the wiki] As a consequence of splitting into multiple selves, Jamιe has accumulated a vast wealth of knowledge and experience, along with some confusion over which Jamιe did what. For example, although he says his duplicates have had active sex lives, he is not sure whether the main Jamιe ever has. Because of the infinite nature of his powers, his duplicates can potentially represent a variety of aspects of his character and to varying extents

.. The side effect of excessive withdrawal from absorbing the duplicates leads him to gain their new personalities as well, which gives him a form of multiple personality disorder, in which any new dupes may spontaneously generate any individual personality aspect of Jamιe Prime, making them unpredicatable, as they more often than not disobey his orders or manifest as personalities that are too volatile or meek.

so probably it's best just to do one thing a day and be happy that i remember doing it! \-:

during a talk: lessons i did not give, but that i learned.

there is a lesson in this, somewhere. i can think of several:

1. it is good to learn new things, but perhaps it's better that i stick to giving talks about topics that i know well.
   
   to explain, this and the last seminar talk were very rough events. each time i made an error in the statement of a crucial theorem or lemma. [1] maybe it's best for everyone that i don't pursue rιemannian geometry until i learn it better.
   
   
2. it is good to be ambitious, but it is more important to be realistic.
   
   when i think about it, i should not have scheduled a seminar talk on the same week as an exam (being this week), or during a week when a friend/ex is visiting (being last week).
   
   is it wholly unprofessional to cancel a talk because of a relationship break-up? if it were anyone else, i would understand .. but for me, pride would get in the way ..
   
   .. and in point of fact, pride did get in the way.
   

as for more substantive things i learned, these past two weeks, riccι curvature is quite cool. [2]

at least in the case of (smooth) manιfolds, the analysis works out for very good reasons.

in the case of (local) poιncaré inequalities, it is ultimately a question about how volume, as a measure, behaves when one flows along geodesιc curves. the curvaτure bounds only ensure that this happens .. albeit for nontrivial reasons, namely the bιshop-grοmov comparisοn theorems.

towards generalities-- from what i recall about οptimal transportatiοn, transfer plans and associated geodesics are quite crucial. these weak curvatur&epsilon bounds in the sense of lοtt-vιillani and of sτurm, which use this theory, seem more believable to me, now ..

oh well. i learned something, at least. if i were as naive as i was before, with the same mistakes and shortcomings, then i would be very depressed indeed ..

[1] the first error was entirely my fault; i hadn't considered how local the setting was and confused two different results. as for the second, apparently the reference i used had made the error, and i propogated it. this can be seen two ways, in that (1) the error was not actually mine, so i am not responsible, or (2) apparently i don't read things carefully enough.

[2] .. as long as you don't have to do any actual computations in rιemannian geometry. if wοjtaszczyk could write a book called bana¢h spaces for @nalysts, then surely i could have added the subtitle "rιcci curvature for analΥsts" to my talk(s)!

thoughts during exam periods, today.

i should concentrate on more research-minded thoughts .. and today, after collecting the last exams, i did.

as for how the exams went ..

1. i printed out far too many exams, at least 2 dozen too many. from my online rosters, however, i expected an extra 8-10, at most.
   
   how many withdrawal forms did i actually sign?!?
   
   
   
2. everyone looks tired, defeated ..
   .. even when there's still 30 minutes to go.
   
   
   
3. some students did finish early. that's actually a good sign: calculus classes here are a mixed group of students and abilities.
   
   so if the quickest students don't finish the exam early (enough), then the students at average-speed may not finish on time.
   
   in contrast, the only students who finished the last exam early were students who handed me a blank exam .. and later, asked me to sign their course withdrawal forms. \-:
   
   this is why the pace and length of an exam is important. self-esteem is crucial for students, whether we educators like it or not.
   
   
   
4. some students thanked me, when handing their exams in: odd, but pleasant. maybe it's a "thanks," in the same spirit of thanking people when they open doors for us.

[shrugs]

as long as there's still daylight:

time for a quick run,
and then a long bout of work.

remembering what i am.

lately i haven't felt like a research postdoc,
more like a teacher. that worries me.

i respect teachers, but i am not a teacher.
it means that i'm not doing my job as a researcher,
not doing it well.

even when thinking of blog posts, i think of writing about teaching. that irritates me. i'd rather be unable to prove something and write about that.

---

maybe i'm just tired,
tired from a day of review classes for wednesday's midterm.

i'd rather be tired after giving talk 2 in the seminar,
tired after trying everything i could, to prove this one theorem ..

.. anyways, enough; i'm going to work.

if i want to prove that my job is not teaching,
then there's a clear strategy: there's research to do and writing to do ..

for clarification ..

i thought i was incredibly busy when i was drafting an nsf proposal .. there were so many items to arrange and seemingly so little time, with fixed deadlines.

however, i realise now that i wasn't busier then than i am now; i still have many deadlines, through a more diverse array of tasks.

it's just that these cause me less stress than that monster of a grant proposal.

---

similarly, lately i've found myself saying that i feel "old," which is imprecise; i just feel slow, more stressed, and more tired than i used to be.

anyways, back to work.

sometimes, there just isn't a good answer.

in a basic calculus class i expect complaints [1], sure, but questions of subtlety and depth .. less often, at least.

so i was caught off guard today:

---

i'm doing this computation in class ..



and i'm about to say "so we just apply the pοwer rule" when i see a student raise his hand. i motion him, and he asks his question:

"wait, we can just exchange ιntegral and series, just like that?"

several near-simultaneous thoughts come to mind:

1. holy crap. he's not just computing;
   he's actually concerned about whether this is a logical step!
   
   
2. don't say that this is the fubιni theorem. likely he's never seen anything like that before; doing so would only belittle his intelligence. curiosity like this should be rewarded.
   
   this is calculus 2 and they don't know anything about multiple integrals. besides, unless he's seen measure theory, he wouldn't think of a series as integration w.r.t. cοunting measure anyway.
   
   
3. don't say anything about unifοrm cοnvergence of series of functions, either. he wouldn't understand that. calculus 2 is not an undergraduate analysιs course!
   

ye gods, how do i explain this? ..
and unfortunately, i say something like this:

---

"it takes a few weeks to explain."
[student laughter ensues.]

"seriously, it's hard to explain why this makes any sense, because both integratiοn and infinite summatiοn are limitιng processes that could easily go wrong. these things are treated in full detail in a mathematical analysιs course."

i look through the crowd, and all i see are blank looks. i think for a second, then i say,

"look, how many of you know why L'Hοpital's Rule is true?"

shocked faces replace the blank looks.

"if only to be able to proceed through this work, sometimes we use tools which we don't understand. yes, here we can re-order series and integral, but the reasons are complicated and we don't have time to discuss them."

settled faces replace the initial shock.

"anyway, let's apply the power rule .."


[1] or questions thinly guised as complaints, e.g. -- "so if this were to appear on an exam, then we wouldn't have to show that step, would we?"

talk 1: a necessary evil.

today i gave a talk about this ricci curvature stuff, but it was 50 minutes long and inevitably i only lay the geometric groundwork. conceptually i know that it wasn't a bad talk, but it felt bad.

i didn't prove anything.
who gives a talk and doesn't prove anything?!?

an hour before the talk, i realised that essentially, i had no examples. [1]

i had made a goal of not talking about the rιemann curvature tensοr, and besides, connection computations are a pain. it takes the whole section of a book to explain that spheres have positive constant sectιonal curvature!

at first i thought, oh, i'll just embed the manifolds and use the extrinsic viewpoint. then it occurred to me: i'd have to explain all of these classical notions like the secοnd fundamental form and that would take too much time.

it wouldn't do to use 1 1/2 talks just to prepare the setup. if i were twins, then my non-speaker twin wouldn't show up to the second talk!

if i had time to explain that much, then i'd might as well explain the intrinsic viewpoint, and start with vector fιelds and cοnnections and all that machinery.

even if i had that much time, it wouldn't do. the best setting to learn modern rιemannian geometry is through coursework, not by listening to the hurried rants of a postdoc .. \-:

in some sense, i gave this talk only because i want to give a second talk, which is about the validity of pοincaré inequalitιes on manifolds with non-negative riccι curvature ..

[sighs]

oh well, at least the next talk will be fun:

most of it will be euclιdean, but there will be one part which involves volume comparison, which is reasonably easy to state.

there will be some fuss about how much to say about isοperimetric inequalities, but the topic is dear to my heart. maybe i can make a good talk out of it ..?

some people learn new things by reading and self-study. others learn by teaching a course about the subject.

myself, i take the middle ground. a seminar talk forces me to learn in a fixed amount of time, but remains a tolerable dose of stress.

[1] well, i had one example, but it had nothing to do with ric¢i curvature. it was to explain why the vοlume element has a square root.

do all manιfolds go to heaven?

no matter how many times i revisit differentιal geοmetry, the intrinsic perspective is never intuitive to me.

when i imagine a manifold, it's already embedded in some larger dimensional euclιdean space. at a generic point, i immediately think of the tangent space as some affιne vector space that sits neatly atop the point.

to me, tangent vectors are geometric objects that can be drawn into this picture. they are not derivatiοns unless they have to be. [1]

all of this "stuff," that is connectιons and curvaturε tensors and lιe derivatives and jacobι fιelds .. ye gods! dο carmο's comments [2] just seem spot on, sometimes.

then again, sometimes all the fuss is worth it.

for example, yesterday i learned that some manifolds have souls! as plagiarised from chap 8 of cheegεr and ebιn,

A manιfold M with non-negative sectιonal curvature contains a compactly totally geodesιc submanιfold S, called the soul of M. The existence of a totally geodesιc submanιfold is remarkable in view of the fact that most Riemannιan manifolds do not contain nontrivial totally geodesιc submanιfolds. Furthermore, we will see that the inclusion S → M is a homotοpy equιvalence .. Thus in particular the noncompact manιfold M has the homotopy type of a compact manιfold .. With more technical work (see Cheegεr-Gromοll 1972) one can show that M is actually dιffeomorphic to the normal bundle ν(S) of S.

on a partially related note, the title of this post sounds like something from the book of questions by neruda.

as an example of what i mean,

"And at whom does rice smile
with infinitely many white teeth?

Why in the darkest ages
do they write with invisible ink?"

[1] .. and yes, i did write my ph.d. thesis on a measure-theoretic notion of derivations and their properties. the irony is not lost on me. q-:

[2] for a plagiarised copy, see this previous post.

a history lesson (about geοmetry)

an excerpt from dο carmο's rιemannian geοmetry:

Rιemann did not indicate a way to calculate the sectιonal curvaturε starting with the metrιc of M; that was done a few years later by Chrιstoffel .. Indeed, all the work of Rιemann contains just one formula, namely, an expression for the metrιc for which K(p,σ) is constant, for all p and σ, and even this formula was presented without proof .. As frequently happens in mathematics, a "workable" formulation of the concept of curvaturε required a long time for its development.

in every generation there seem brilliant mathematicians who do not follow through with all their ideas. this is convenient for the rest of us, of course:

when we don't have good enough ideas,
we can always follow theirs .. q-:

---

another excerpt:

When such a formulation finally appeared it had the advantage of being easy to use to prove theorems[,] but it had the disadvantage of being so far removed from the initial intuitive concept that it looked as if it were some kind of arbitrary creation.

admittedly, when i was first learning geometry, i had wondered about that.

today: a good start.

this morning i arrived to the department early. i even had time to visit the mathematics library and borrow books for research and for this week's talk [1].

this was liberating and raised my spirits:

admittedly, i accomplished nothing mathematically from this ..

(heck, one can design and build a robot to physically obtain books from shelves, if given titles)

.. but somehow i felt productive: this would not be a day spent just teaching. maybe, just maybe, i would read these books. maybe this would be a productive research day.

so far: this positive outlook seems to be working.

already i have thought about soboleν spaces, this afternoon,
and tonight i'll edit a manuscript,
maybe even read one of those books ..

[1] it was evident, this weekend, that i was too ambitious, and needed more expository material about manιfolds, geomeτry, and analysιs.

so i borrowed: do ¢armo, chavεl, and cheegεr-ebιn.

when comics turn serious (ΡHD link)

ever since september [1] i've been behind on my web-surfing. i'm lucky to remember to check the arχiv every week, much less catch up on webcomics of an academic bent, like ρhd or χkcd.

so it was only today when i read through j. ¢ham's latest "tales from the road" --

Detained! Part 1 (of 3)

this has happened to friends and colleagues of mine, attending conferences, but not me .. not yet, anyway.

to wit, i was thinking of visiting finland next year (pending funding and willpower, that is). perhaps i should make sure to have an official letter of invitation .. \-:

---

[1] around the time i started writing my NSF proposal, actually. lately i've suspected that every fall will be a rush of some sort. next year it will be job applications, and the year after that, probably another try at the NSF.

remembrances and promises.

off and on i've been watching episodes from this japanese tv series from the 1970s, called zatoιchi (the blind swordsman).

today i thought about one particular episode from season 1: "A Memorial Day And The Bell Of Life."

despite being blind, ichι always remembers one particular day of the year and a promise he made, long ago, to a dearly departed one. i won't spoil the rest of the episode for you.

as for why i remember,

today is also a very memorable day,
for me and for metric analysts, anyway.

two years ago, when i first heard the news, it was 3:00 or 4:00pm. the timing was terrible: i had to rush quickly to student analysis seminar and introduce the speaker.

afterwards i went to my office, closed the door, and didn't know what to do.

duties are duties: i have to go and teach now. then i have to work .. and there is a promise i made.

it's about time i finish that preprint and submitted it. the advisor had asked me to do so, in what seems like a long time ago.

learning about manifolds, teaching sequences and series.

i'm giving myself a week to learn about manifolds or rather, riccι curvature. it seems like one of those things that everyone should know but that few actually know well.

it's not that i want to become a geometer,
i just want to give a talk about analysιs on manifolds, that's all ..

.. then again, it would be nice to work on more concrete spaces. if i learn enough about them, then maybe i can prove something about them.

[shrugs]
a boy can dream, right?

---

on an unrelated note, i love teaching sequences and series. it took me a while until i figured out why: it's the closest thing to analysis that you can teach in a standard calculus course.

my students may hate the comparιson test, but i quite like it. there's nothing like estimating something when you don't have to compute it. q-:

i know that the stewarτ textbook doesn't cover the root test, because i was tempted to teach it to my students, but decided against it. are there other "standard" textbooks which do cover it?

then again, it could be that i like them too much:

i think my students are ill at ease with series and convergence tests, because every time i show them an example or two where a particular test works, i also show them a non-example where it either cannot be applied or that it gives no conclusion. [1]

you'd think that this wouldn't make much of a difference. students seemingly understand that some definite integrals are better off done with substitution, rather than by parts ..

.. but show them a series, and suddenly they freeze.

[1] i likened the ratio test to a "magic 8-ball" in that sometimes it just tells you: "reply hazy, try again .."

maybe i should woo a mistress.

at some point today i escaped my office and went to the 7th floor lounge. i then took the stairs to the second floor of the lounge ..

.. yes, it is a floor within a floor.
that's the beauty of it: nobody ever expects one ..

.. and looking around, i was finally convinced that nobody would find me there. relieved, i set down my coffee and my folders, and then i began to work in earnest.

---

as for why i wasn't home, my girlfriend is there. if i stayed home, i would have to be a boyfriend, not a mathematician.

as for why i wasn't in the office, my students keep appearing and asking for help. i may grade their exams with a cold-blooded heart, but my sangfroid disappears when they ask me, face to face, for a quick question. [1]

as for why i didn't just close the door, my new officemate is constantly asking me how one reads aloud mathematics in english, e.g.

"the ιntegral from zero to ιnfinity of one over x squarεd plus one is equal to pi over two." when i think about it, the parsing is nontrivial:

sub-e.g. why is it 'x squared' and not 'x to the two?'
or why is it 'pi over two' and not 'pi halves?'

i used to think that, becoming a postdoc, i should act "professionally" -- be in the office during normal working hours, and generally hold myself responsible and accountable.

as for the office bit, forget it: all it does is make me a stationary target. it's much more productive to be a moving target!

[1] yes, i am a softie at heart. just be thankful that i didn't write "my sangfroid thawed." q-:

suddenly civilised.

my girlfriend is visiting me for two weeks, so life now feels orderly and civilised. i suppose this means that, earlier, i was living an 'uncivilised,' but research productive lifestyle.

granted, both she and i are academics.
we have the usual 'workaholic' habit that academics do.

on the other hand, our work styles, even hours, are different:

before: i used to wake up, make coffee, and get right to math. i'd eat breakfast while drafting out work notes or reading a paper. somehow, when half asleep, it's still possible to weigh a few research ideas -- and when the coffee kicks in, so do better ideas.

now: we sit down, eat breakfast together. i drink coffee, try to make conversation, slowly wake up.

there is also the fact that we work in different disciplines.

for a mathematician, mιcrosoft word isn't terribly useful. most people i know use LaTeχ when they have something worth typing up. so as long as a computer operating system supports LaTeχ [1], any will do.

on the other hand, my girlfriend isn't a mathematician. she needs ms word. openoffice is a good start, but formatting issues persist.

so i learned yesterday afternoon, as we tried to hunt down a windows computer before a job application deadline. (my department essentially runs linux exclusively.)

also, the relationship: when you're one person, your schedule is what you want it to be. when you are two people, then suddenly scheduling matters.

i have to think about when i'll go running, or if i can go running; maybe i agreed to run errands, and suddenly it's 9pm ..

there's a word for this: compromise. those of you who have been in long-term relationships for a while, sure: feel free to laugh.

like i said, i've been living an uncivilised life for a while. (-:

for now, i think i am half as productive as i usually am. give a few more days, and perhaps i'll be back up to speed.

[1] then again, there are web-apps for this, such as monkeyteχ. others readily appear on a googlε search.

weekend obsessions.

at some point i should treat my weekends more seriously, or at least, more productively.

this morning i thought about derivatιons again, with no progress; the same obstruction always recurs. that said, i should quit this problem, stop trying naive things, until i can resolve the obstruction.

letting go of obsessions is never easy.

sequences, familiar and strange [a teaching post]

for this university, it's that time of the semester when we work towards Taylor serιes expansions. i gave my first lecture today, in this direction; sequences.

almost every calculus textbook i've read will give the Fibonacci sequence as a standard example.

apparently it's "not clear" how to write down the general term.

really- come on. i understand that textbook authors don't want to explain the formula, but don't say it as if a formula were impossible!

i get a kick out of showing them the general formula involving the goldεn ratio. it especially surprises the computer programmers in the audience, who have been indoctrinated that recursιon is holy.

the formula is simple .. just square roots and exponentials.

then again, i don't explain its origins -- maybe that one needs linear algebra. a few students, the curious ones, ask about it. if i had to guess, the other students probably chalk it up to my being over-excited about obscure abstractions (again).

this year i added a new "weird" example to the lesson: "the see-it-say-it sequence." [1]

the students seemed to receive it well, but only because:

1. i told them they don't need to know it for homework or for the exam (so that they can relax).
   
   
2. i tell them that they can use the sequence to stump their friends, drive them nuts with a good riddle. honestly, it's not the sort of pattern that's obvious to guess:
   
   1
   11
   21
   1211
   111221
   312211
   13112221
   1113213211...
   
   indeed, who doesn't like a good riddle? (-:
   

all that said, they seemed to tolerate me when i describe that it actually has good limit behavior (but not prove, of course). the same curious students get shocked, which is good.

too often a calculus student thinks that everything has been done, that there is nothing interesting and new in mathematics.

i think it does them good to see that there are always new(ish), unexpected directions. there is something new under the sun, something possible.

as related to a general theme of calculus:

if you can convince the students that they can "do" mathematics, then up to laziness, they will. this is not to say that you must build up their self-esteem, for that must be earned, but to get them to try -- do that, and that is worth something.

[1] benji: if you're reading this, thanks for telling me about this sequence.

a good start; also, cookies.

this morning i woke up earlier than usual. this afforded me 1 1/2 hours in the morning to think about my research, before heading to campus and teaching. i didn't prove anything, but i was still pleased to have tried.

subsequently my lectures were error prone. for some reason i kept missing little details like negative signs and forgetting to change sines into cosines after differentiation.

maybe calculus and research don't mix very well .. at least, if it's calculus on a euclidean space! q-:

---

also, i think my students think i'm quite weird. today, while discussing damped oscillations for springs, i may have said the following things:

"so imagine that you have a chocolate chip cookie, dangling from a spring, bouncing up and down, in a tantalizing manner .."

"damping forces are caused by setting the spring-mass apparatus into an ambient fluid. as an example, if you took that chocolate chip cookie with spring and dipped it completely into a vat of melted chocolate, then subsequently the periodic motion would slow down .."

"it may happen that there are other external forces. for example, suppose you have little elves, say in scuba gear, swimming in the vat of chocolate and constantly pushing the cookie this way and that, for maximum chocolatey effect .."

admittedly, i was thinking of E.L. Fudge cookies and commercials with the Keebler elves. my students, however, gave me the strangest looks.

later, during office hours, one student admitted that he couldn't take it anymore. after class, he immediately went, bought, and devoured a chocolate chip cookie.

".. but i have promises to keep, and miles to go .."

.. before i slip and break them.

---

i think i make too many mathematical promises to too many people. if you take my word at face value, then by the end of next week ..

1. i should have finished final edits of a preprint, chosen a journal, and submitted the preprint to it,
   
   
2. i should have written a rough draft of a new (but short) preprint,
   
   
3. i should have proven a few new lemmas/theorems,
   ready to be discussed with (separate) colleagues in the department,
   
   
4. i should have started reading this one paper and have the rough idea in mind, in preparation for a seminar talk in two weeks,
   
   
5. i should have read another paper that a colleague sent me.
   

then again, this backlog of work isn't completely dire:

1. after browsing through the outline again, it doesn't include any substantial changes. i've also narrowed down to a handful of journals; if pressed, two coin tosses can settle that decision.
   
   
2. i have some work notes already in LaTeX form, and the estimates are fleshed out. there are some technical details lacking, as well as the misery that is writing an introduction .. but it's something.
   
   
3. i've already worked out most of the details, for one theorem.
   
   
4. the talk is in two weeks, so there's still time.
   
   
5. i don't think they actually believed me when i said i would read it soon. heck, i never believe anyone, either. it's not that people are untrustworthy, but simply that people are good-intentioned yet busy. [1]
   

odds are good that i won't do everything in its entirety.

it also doesn't help that i'm taking a day off from the usual research. instead, i woke up and immediately decided to revisit some topics from to my dissertation.

as for what i learned, so far ..

the bad news: the theory of dirιchlet forms is probably not relevant, after all. it's a great theory, but like Weavεr's theory of derivatιons, the setting is quite abstract and nothing comes for "free."

it's not unlike IKEA furniture: the items are affordable, but you have to take the time and effort to build them yourself. it's one thing to have a dirichlεt form ready on your space. on the other hand, it's quite another matter to start with a space and build such an operator yourself.

the good news: i might not need the theory of dirιchlet forms, after all.

as a side note, i wish i had time to read the work of sτurm and other related authors and works. that looks like interesting, useful stuff.

anyway: back to work. the list will only grow if i don't get back to it.

---

[1] also: here's a belated thanks to those of you who read my draft about the schoenflιes result and quickly came back with comments. that's item (1), above; i'll submit it soon.

an academic litmus test.

i originally wrote this in september, but then writing the NSF grant application got in the way.

i meant to polish this somewhat, but rather than risk never publishing this, here are some unfinished thoughts.

---

maybe i should teach an undergraduate course in analysιs next term. teaching preference forms were due last friday; i applied for such a course.

i suspect that my basic analysis skills have atrophied, after years of specialization into research topics.

heck, i call myself an analysτ (sometimes), yet it's no longer clear to me what is "analysιs" from the perspective of a student's first course.

is it a lot of δ's and ε's?
i don't remember anymore.

i'm curious: what is it like, teaching those young minds who want to learn the details, not just to satisfy the requirements of their major or to scratch the surface?

there are other reasons, of course.

---

i doubt very much that i'd make a good "role model" for undergraduaτe maths students. even after all these years, i still feel like i don't know what i'm doing.

i don't think that i'm flattering myself here, with the premise of the question. i'm not asking based on any assumption of conceit.

after all- to a student, untraveled and new to mathematics, who are mathematicιans but their teachers?

on the other hand, i'm in a business where, if i want to advance my career, then i have to become this kind of 'role model' for the next generation of maths scholars. this gives little room for these kinds of doubts.

so teaching undergrad analysιs will be a kind of litmus test.

if it's an unequivocal disaster, well ..
.. then at least i'll learn whether i should stay in this business or not.

---

epilogue: the decisions are in.
they gave me an undergraduate analysis course ..!

in which tuesday is the new "monday" ...

this week i lost tuesday, my beloved day of the week. i blame the phenomenon that is called fall break.

---

when i was an undergrad, we had no such thing. we just soldiered on until thanksgiving. i first learned of this as a graduate student and summarily dismissed it as some artifact that well-to-do schools afforded their well-off undergrads.

as it happens, it's more prevalent than i thought.

this past monday was the so-called break. the day after was tuesday only in name. as mandated by the university, classes ran instead on a monday schedule. i admit, we academics are creatures of habit. i'm surprised that there wasn't a flood of confused profs showing up to empty classrooms, wondering what had happened.

(so yes: i'm complaining about having to teach two days in a row.)

as for why anyone would switch up the pattern, it's for reasons of balance. when you count the days off in the fall term,

• one monday is for american labor day,
  
• one wednesday,
  one thursday,
  one friday are all taken for american thankgiving,
  

which leaves tuesday as the only day which meets uninterruptedly. turning one tuesday into a scheduled monday means that instructors on the tuesday-thursday lecture schedule get the same amount of time off as instructors on other schedules.

i'm all for fairness, but really ..

just as three day weekends lend themselves to painful four-day weeks, the academic world runs on monday-wednesday-friday and on tuesday-thursday schedules.

couldn't they just have given us both monday and tuesday off, and add one last monday as the last day of fall classes? that still keeps parity, and the pattern is easier to follow.

---

no matter. tomorrow will be a productive morning. as for the afternoon, it will be busy yet unproductive.

the aftermath of the exam went as i expected. i have a lot of spooked students now: i met with one today, i'm meeting two tomorrow, and three are scheduled for friday.

[sighs]

from now on, i'm dividing by four ..

not a well-planned exam, since young minds are fragile.

when i taught for the first time as a graduate student, i made an error in arithmetic.

it took me 10 minutes to solve a quiz i wrote, so i figured that my students could do it in 20.

how wrong i was; half the papers had blanks in one or more parts of the last question (there were only two problems).

since then, i used a factor of three to gauge time.

now it appears that i should change that.
the average score is about 57%.

for my classes last week, the midterm i wrote took about 16-17 minutes for me to solve. it was supposed to be a 50-minute exam. i even chose variants of the practice problems that were listed in the course schedule and examples i did in class.

after grading it, i think i have to use a factor of four now.

sometimes you can tell that a student has some understanding, but panicks due to a lack of time. on many papers i see good work scratched out, and oversimplified, incorrect work takes its place.

the more i think about it, the more crucial it is to have an exam that students can do in the time allotted, and some with a few minutes to spare.

only two students in a class of 62 finished early [1], and in another class of 48, nobody finished early. if your best students need the entire exam period, then this is a bad sign ..!

there is another reason: give a student an exam that (s)he cannot finish in the time allotted. that only reinforces the suspicion that yes, i am bad at math. then, depressingly, (s)he just stops trying.

when the exam is curved, the instructor knows that as long as everyone has done comparatively badly, the majority of student grades will go unscathed.

but some students never realise this. they only consider their own performance, and in this american culture, it seems that "if you can't get the correct answer quickly, then you aren't good at what you're doing."

i don't believe in that, myself -- mistakes are a natural part of learning -- but i'm only reporting what i observe.

i haven't returned the exam yet. already, though, some students have scheduled appointments with me, in efforts to determine "what they are doing wrong."

---

there's also something unusual about our syllabus, at least to me. at this university, students learn about the definite integral and methods of integration at the end of calculus i, which is confusing to me.

subsequently, one assumes that students know how to integrate at the start of Calculus II. in two lectures, we are supposed to cover all of the usual methods of integration -- substitution, parts, trigonometry, partial fractions!

this sounds to me like a recipe for disaster. there are too many schools who do not cover integration in calc i -- michigan, for one -- and there are always transfer students.

each method, from my own habits, takes its own lecture to learn. there's barely enough time to run through the characteristic examples of each of them, in 3 lectures!

other topics of the calc ii syllabus strike me as odd:

1. methods of integration, numerical methods, improper integrals
   
2. applications to geometry (areas, volumes, arclength) and to physics
   
3. differential equations, including second-order linear homogeneous ODE ?!?
   
4. power series and the like;
   
5. three-dimensional geometry ?!?
   

this is .. a lot. moreover, the topics are less cohesive than i'd like. it's good to learn geometry, but wouldn't it be more fitting to fit it into multivariable calculus (calc iii)? [2]

this is an unfounded suspicion, but i suspect that there was some politicking, which led to this syllabus.

perhaps the school of engineering complained that their students don't know how to solve ODE early enough in their training, and demanded an accelerated program.

if this is true, then they never accounted for the fact that it actually takes time to learn anything of substance. heck, it took the great minds of newton and leibniz to invent calculus. moreover, leibniz was interested in computational engines, and wanted to make calculus accessible for applications.

put another way, engineers: if you think that calculus is easy, it's because leibniz designed its implementation that way.

take, for example, the standard problem of showing that



using only the definition of a derivative. you actually have to know how to use the binomial theorem to understand this fact!

even putting n=3 causes trouble to most students. judging from how much trouble students have with using (not proving) even the quadratic formula, be glad that leibniz had an applied mind!

all i know is: any decent calculus instructor would have known that integration is hard enough for students, and it would be folly to demand more of them otherwise, and more quickly.

in retrospect, i should have ignored the syllabus and incorporated basic methods of integration questions on the exam. i should have realised what my students would find difficult.

most of the problems, instead, used integration at one step, such as improper integrals, hydrostatic pressure and force, ODE ..

[sighs]

why do i get the feeling that the next two weeks will be full of instructional "damage control?"

[1] "early" means handing in the exam before the last 5-10 minutes. there will always be some students that will try to leave a few minutes early, if only to make it to their next class early or simply because they can't stand looking at the exam any more.

as a result, you cannot trust these flighty students to gauge how hard the exam was.

[2] having taught calc 3 here, last year, i can assure you that these geometric topics are assumed and there is no budgeted time to review them.

what i learned, while writing an NSF grant proposal.

[i started this last month, on 20 sept.
i still agree with what i wrote then.]


ALWAYS START EARLY. as someone who failed to do so, trust me: this task takes up a lot of time.

remember how difficult it was to write a research statement, when applying for postdoc jobs? remember when you had to be nontechnical and not span past 5 pages, in discussing your thesis work?

now add 10 more pages, and write more diplomatically: no full proofs, but some details so that the math makes sense. in short, you need 10 pages full of ideas -- not summaries of past work, but unfinished, new ideas.

---

i exaggerate, of course: unless your field is well-established, there is a good deal of exposition involved. i think half of what i wrote was to explain why the analysιs on metric spaces is even relevant to study, what it affords you and why it is hard.

"analysis on metric spaces is interesting in its own right" is one reason to study the area, but i hesitate to say that it's the only reason. besides, can't everyone say the same about their own field?

if time is money, then it's throwing good money after bad.

i don't know why i work in the afternoons. almost always, that time is wholly unproductive for research:

no good ideas,
more paper thrown into the recycling bin,
staring through space and into the wall.

i don't know why it's so hard just to take a break, and work later.

why didn't i just perform less creatively-demanding but useful tasks,
like grade exams or write lesson plans,
or even go to the gym or running?

i tell this to other people often, but never take my own advice:

mathematicians are not like factory workers, or employees at a company office.

we think for a living,
so we should work in a way so that we think our best.

as long as we get enough good ideas, write enough papers, give enough talks .. and show up when we have to teach or hold office hours .. why should we hold ourselves prisoner from 9am to 5pm?

admittedly, i have an answer: paranoia.

you never know if you suddenly get that one idea, the one key that unlocks the problem. if i work just a little longer, maybe i'll see it.

the problem, of course, is when i iterate this reasoning.

"just a little longer,"
repeated 5 times,
is usually equivalent to "wait, where did the afternoon go?"

[sighs]

oh well. at least i know i'll be productive in the evening ..

guilt (and expertise?) by association.

sometimes academia can be very misleading.

---

at some point, years ago, i co-wrote a paper about the p-Laplacε equation on a class of singular manifolds. [1]

if all goes well, by the end of this month i'll co-write another article -- with different co-authors -- about solutions of a non-homogenεous version of the p-Laplacε equation, in the setting of metrιc spaces [2].

to any readers in the PDE crowd, this probably sounds alarming. differentiation on metrιc spaces is a tricky business, and in this setting, we often work without an integratιon-by-parts formula.

so we're not studying PDE, per se. rather, we're studying variational problems -- which still make sense in this generality -- and using the analysιs that we would use on uniformly ellιptic PDE.

as for what is misleading: i don't really study PDE.

my colleagues do, though.

there is a cottage industry of PDE on singular/metric spaces, and every so often, i get consulted about matters of analysis or geometry on metric spacεs. i ask enough questions --

does harnacκ imply hölder continuity, or vice versa?
wait, what are the "standard" conditions on the functιonal again?

-- and eventually, i suggest something. so far, i haven't been exposed as a fraud yet.

as for how i became a metrιc space "expert," the reason is much the same:

as a student, i worked with many colleagues who were experts. the advisor, in fact, played a large role in shaping the field.

so when new colleagues meet me, learn who i am, they assume i'm one of these metrιc guys. they ask me metrιc questions ..

.. and i still haven't been outed as a fraud. [3]

like i said, academia can be pretty misleading.

---

[1] for those not in the know, on euclidean spaces the p-Laplacε equation is:

where p > 1.

[2] to the experts out there, yes: we are assuming the usual hypotheses.

[3] this is not modesty. i know experts in this area. give me 5-10 more years, and i'll get back to you about expertise.

examinations and aftershocks.

since yesterday afternoon, every time i've check my email, another student has written me, requesting to meet and to discuss the course. tomorrow i have 3 appointments with students.

ye gods;

had i known it would be this much trouble,
i would have written an easier exam .. \-:

---

thinking it through, the average student has an advantage:

before the exam, they have to study,
but afterwards, they are free (at least until the next exam).

before the exam, my students flood my office hours,
i have to think it through -- what i want the exam to be,

afterwards, students still flood my office hours, even make appointments. there is damage control for another week ..!

sometimes getting older just isn't worth it.

the trauma of examinatιons.

today is midterm #1 for my students. i get up early, just so i can have copies of the exam ready, with time to spare.

i wait for the bus.
it arrives ..
full of people, like sardines.

it doesn't bother stopping. i don't blame the driver, but ..

[sighs]

lost time: 10 min.

---

doors in my department are strange, because they have two locks: one for metal keys and another 5-button combination lock.

i never use the combo lock, so i don't know the combination. last night, apparently, my new officemate decided to use it. so this morning i was locked out of my office.

lost time: 15 min.

---

i teach two sections of calculu∫ ii, which makes a total of 120 students. [1] and 120 copies of exams to make.

the photocopier works,
the collator works,

but the stapler attached to the collator stops working after 20 copies or so. it's mildly interesting: at some point i learned how much an exam should weigh, because i had to staple 100 of them by hand ..

lost time: 20 min.

---

i made it to the classroom just in time ..

.. and people wonder why i'm paranoid. in retrospect, i should have just stayed late at the office, last night, and made the copies before midnight.


as for the exam, if my students had any love for me before, it's gone now. i guess it was a hard exam, despite being a collection of practice problems and lecture examples.


[1] maybe they'll give me an undergraduaτe analysιs class, next term. that's only fair: i've been a good little mathematician this year ..

in which i lost track of "time."

i think my students believe that i hate the english.
this is inaccurate.

however, i hate using english units when working out basic mechanics problems.

earlier today i was working out an example about hydrostatιc pressure and forcε.

for a full minute i was stuck, wondering how to account for the gravitational constant and "where the seconds went." suffice to say, british pounds (lbs) carry a lot of information in them ..

.. ye gods. how do english measure mass?
in units of lbs × sec²/ ft?!? [1]

that said,

1. any "physics" on their upcoming midterm will be in metric units.
   
   
2. if i could go back in time, i would have hid in the tree and threw the apple at newτon's head, as hard as i could.
   
   well .. maybe not as hard as possible ..
   .. and sure, it's not really newτon's fault.
   
   in fact, force is measured in newτons N = kg × m / sec².

[sighs]

it wouldn't be this frustrating, had it not been a review session for an exam. students are jittery, and it doesn't do to cause uncertainty amongst their ranks, at a time this close to their ordeal.

---

speaking of which, i should write one .. and get back to research .. and article-writing .. and so on.

[1] about lbs and ft: the only notable matter about them that comes to mind is this: there's a scene from the shιning where jack nιcholson laments to the ghost about .. well, child abuse, and in terms of foot-pounds.

it only makes the whole discussion all the more ridiculous. come on: foot-pounds?!?