"words, words, words .. what is the matter ..?"

well, i was right not to predict 20 pages because,
for some reason, the preprint has now reached 25.

[sighs]

there's nothing wrong with 25, but i thought that this would be a short and easy paper .. well, "easy" by my own standards ..

that is: if i proved it,
then it can't be that hard. q:

admittedly, i added a table of contents, which is 3/4's of a page, and the references fill up another entire page.

it's still incomplete, so: back to work ..

this picture better be worth a thousand words ..!

maybe it's good not to have too many goals or make many promises.

there are too many interesting ideas out there,
there's never not enough time,
and every co-author eventually becomes impatient.

in other news, i'm still working on this preprint about measurabΙe differentιable structurεs. i think i've created a monster.

to make sure that the theorem is non-trivially new, i've come up with a new example to which the extant theorems do not apply. explaining it is taking much more time and space than i thought, though ..

.. in fact, it's become a new, separate section of the paper!

conceptually, it's not hard; it's a self-similar construction. in fact, it looks like this:

i'm embarrassed to say how long it took me to make this simple image. at any rate, fractals are definitely the analyst's best friend ..! (-:

EDIT @13:19: i uploaded the wrong graphic; the actual set isn't supposed to be symmetric. argh ..

quietude

it's been a quiet week here. thinking back on it, i should either have taken a few days off completely or worked proper, full days.

it was only some days ago that i realised that teaching forces natural breaks in the week, even the day. that said, i'm not used to working 8 hours at a clip, solely on research. instead i worked for most of the daylight, idled after 6pm. even then there were plenty of breaks.

some people are fine taking complete vacations for a week. maybe grad school trauma is still fresh in my mind, but i still can't do it. the last proper vacation i had, going on holiday with my girlfriend at the time:

we climbed in the mornings, to avoid the heat of the midday sun;
while stretched and napped (for hours!) i took out my notebook, listened to the sound of the waves, and work on math.

admittedly, it was one of the best vacations i ever took. the food wasn't bad, either; only in thailand can you have a decent papaya salad.

[sighs] i guess we mathematicians have quite a reputation ..

.. even from colonial days:

"Thomas Godfrey, a self-taught mathematician, great in his way, and afterwards inventor of what is now called Hadley's Quadrant. But he knew little out of his way, and was not a pleasing companion; as, like most great mathematicians I have met with, he expected universal precision in everything said, or was for ever denying or distinguishing upon trifles, to the disturbances of all conversation .."

~ from the Autobiography of Benjamin Franklin.

"so you can teach: ok. can you do research?"

I guess i'm a good teacher: everyone says so, faculty and students alike. there are certainly worse things to be told.

every time i'm told that my teaching is good, though, right away i think about how pleasant it would be if someone would compliment my research (genuinely).

this is not to say that i disrespect teaching. i like teaching to students who are engaged in the material, and over the years i find that i like explaining concepts to newcomers [1].

make no mistake about it;
teaching may be easy, but teaching well is hard.

maybe it's that i spend much more time researching than teaching (including preparations), and i would rather the response be proportional to this time allotment.

maybe i suffer more in getting good ideas, making them rigorous, and polishing them, so that i value critical responses of that end more than another. so instead of a proportionality of time, i weigh it as a proportionality of pain. \-:

maybe i just want to be good at everything, that i just want it all! q-:

[1] part of it, i suppose, lies in the challenge. can i explain measμre theory to someone who took calc ii? can i convince someone who doesn't know any math why there are infinitely many primes? can i motivate why baηach-alaοglu [2] is so cool?

[2] admittedly, it's probably my favorite theorem of the moment. baηach-alaοglu is like a get-out-of-jail-free card .. if you have the right "lawyer" (in the form of a reflexive space!)

a proof of least resistance ..

for the final exam in my proofs class, there were five problems. i told my students what two of the questions were, but not exactly.

1. reproduce five definitions, which will come from certain sections of the textbook (and yes, this is the same kind of problem as before).
2. reconstruct one of two theorems: one of them was the Cantοr intersectiοn theorem. on the day i print out the exams, i'll flip a coin [1] which will decide which one appears.

as it happens, the proof of Cantοr's theorem that i gave in lecture was different than the one written in their textbooks, and i told the class that either one is valid for the exam .. and if you want, you can write your own proof.

[snickers ensued]

i thought my proof is intuitive, when i lectured it .. but i hadn't seen it in a while. for some reason, i thought it wouldn't be that different from the book proof ..

.. until i started grading the exams, and realised:

wtf? this thing is a monster!
why didn't i just give a nice, short proof ..?

.. ye gods, did i prove this on the fly?
i'm pretty sure that i wrote up notes for that lecture.

was i trying to be cute?

to explain, i was trying to motivate the theorem at the time, why anyone should believe that it's true [2]. so i indicated it in two steps:

1. nested intervals imply monotone sequences of the corresponding endpoints. if you take infιmum and suprεmum (respectively, of left- and right-endpoint sequences) then you get a possible interval;
   
   (in other words, do what the picture tells you to do, but do so rigorously.)
   
   
2. as long as we can prove that this "limiting" interval is nonempty, we're done. this only happens if inf < sup ..
   
   .. but that can't happen, if you treat inf and sup like limits. to do this rigorously, use an ε-closeness argument ..

it's a "geometric" proof, sure, but not the most efficient one. then again, it was the only way i could remember how to do it ..

.. so props to the students who actually proved it that way, on the exam!

on a related note, some students actually gave their own proofs .. distinctly different, too, which was quite cool. (-:


[1] as it happens, it landed heads.

[2] i spent an inordinate effort, all this term, trying to make some of the tricker theorems intuitive. as i told the class: "if you don't have an intuitive idea of why the statement is true, then odds are good that you won't be able to prove it." in other words,

(logic) + (intuition) = (maths).

the joy of writing (with no sarcasm meant, strangely enough)

i don't know what i was thinking, some months ago. for some reason i thought that this one particular project would take only 15 pages ..

.. call it the optimism of beginnings.

already i'm on page 18,
there's still (at least) one technical lemma to add,
and it's missing an introduction.

[sighs]

i would guess now that 20 pages should be about it. based on past experience, however .. let's just say that i'll focus on writing, and it will take as much space as it will need.

this is not to say that the writing is going poorly or painfully. it's quite the opposite, actually, even fun. by putting all of this stuff into LaTeX, i feel like i know one thing particularly well. there's also something enjoyable about phrasing an idea in the exact words that you want them.

i might even throw in a problem or two, if only to reward anyone who will actually read the final version ..

.. with the understood disclaimer that nobody else might find the problem interesting, of course. q-:

i've always liked those articles with explicit conjectures and open questions in them: somehow, it's like "giving back to the community .."

in which i encounter a catchy title.

i couldn't stop grinning when i saw this title/abstract from the cνgmt server:

(the) Hitchhiker's guide to the fractional Sobolev spaces
Eleοnora Di Nεzza - Giampierο Palatuccι - Enricο Valdinοci

These pages are for students and young researchers of all ages who may like to hitchhike their way from $1$ to $s \in (0,1)$. To wit, for anybody who, only endowed with some basic undergraduate analysis course (and knowing where his towel is), would like to pick up some quick, crash and essentially self-contained information on the fractional Sobolev spaces $W^{s,p}$.

maybe i'm just a nerd at heart. (-:

this is just as awesome, though, as when i learned that there was a "dark side" to the caΙculus of variatiοns.

american english vs. "mathematιcal english"

there are some bits of mathematical english that i find myself using more and more, in everyday conversation, like:

"to this end .."
"it follows that .."
"note that .."
"consider the following ____" [1]

sometimes i wonder if others .. say, my students .. understand that i'm using these words in a slightly different sense than usual:

"Now consider the following linear transformatiοn," said the professor, who, upon walking to the left corner of the chalkboard, began to write at twice the speed as before.

The student paused, and considered the possibility of keeping his notes thorough and complete.

After a moment, he set his pen down and closed his notebook. "Nah," he said to himself.

i wonder how often that happens .. (-:

as for the word "indeed" -- how i use it, anyway -- it appears right after stating a claim, thereby signaling the start of a sub-proof .. but only when the claim is not stated separately like a Lεmma or Sublεmma. as an example:

"without loss, $f_k$ also converges pointwise a.e. to $f$. indeed, the sequence {f_k} is norm-bounded in $L^p(X)$ with $p > 1$, so a standard functiοnal analysιs argument, using the Baηach-Alaοglu theοrem and Mazμr's Lemma, implies that .."

to be honest, i don't know how i developed that habit .. maybe from the advisor? i probably picked up a lot of habits from him.

---

on a barely related note, i quite like old-fashioned words.

at some point i had wanted to give a lecture in which i'd state a claim and start its proof with the words "verily, we note that .." but i never got around to it, though. (-:

---

[1] after a while, reading maths papers feels a little formulaic. there are only so many instances of "such that" which i can bear. on the other hand, perhaps the formulaic jargon we have is good ..

.. because it means that non-native english speakers can present their ideas just as well, for everyone to read. it makes my life easier, at any rate: my french is rudimentary, my german essentially non-existent. then again, i wonder if i can read de giοrgi in his original italian ..?

every semester's end, i get a few more grey hairs ..

the term papers are graded. that's done. i feel ..

.. much like how i felt when i came back from india, last year: tired, feverish, and jet-lagged from the ordeal of being stuck in a moving vehicle that circles the globe at unnatural speeds.

i was firmly without any motivation to accomplish anything .. yet fully aware that i had to return to work, right away: there were projects to write up, and an nsf grant application to prepare.

still .. somehow, i managed, after that.

the pace doesn't let up. tomorrow is a final for one class, so there is grading right away ..

[sighs]

i had these happy dreams of close-to-a-week's time, during finals' week, when i could hack through a complete draft of these recent results i have. (the process is more technical than i though, and there are many choices to make.) [1]

when i was a student, the end of the semester never seemed so bad. my own experiences were calm ones. sure, there was studying to do, but the days were mine to plan as i please ..

.. ah. to be young again. \-:

[1] no: it hasn't escaped me .. and yes: my dreams are about work. to clarify, i mean being able to work on the things that i want! q-: