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 ..