delays.

sometimes it pays to pack at the last minute:

knowing i wouldn't be able to get any work done this morning, i opted to wake up at 9am, have a leisurely cup of coffee, then pack my bags before leaving at 10:30am.

this morning, however, the alarm on my cell phone rings unexpectedly.
7:51am? did i press '7' instead of '8' ?!?

so i fumble for the phone, flip it open to turn off the alarm, and realise that the alarm isn't on at all.

instead, a mechanical voice tells me that my flight's been canceled ..
..
..
.. yeah: don't you love the holidays?

---

so i'm rebooked now. the plans have changed:
i'm staying an extra 3 days with family.

it could be worse, but i really wanted to leave today: there's much to do, and honestly, i miss work. i can't seem to do any maths, while here.

broοklyn, self-simιlarly.

happy holidays, everyone!

i've not been very mathematical, but instead spending time with family. on the other hand, here is a fractal image of the broοklyn skyline [1].

as for the building blocks, here was the original skyline and its (edited) cut-&-paste:

[1] strictly speaking, it's an approximation of a fractal, since fractals are intended to be limιt sets under various similarιty transfοrmations.

under the strict mathematical definition, however, none of you have ever seen a fractaΙ image .. just like how straight lines and triangΙes don't actually exist in reality.

the grading after-math (also: briefly, morning compromises)

in regards to my previous post: if the point was to minimise the number of student emails, then the email blitz was far from a success.

of those students i emailed, most of them already knew that their final exam scored faltered. actually, they thought it considerate that i wrote them.

what i didn't count on were students which have GPA troubles, or have to maintain a grade for a scholarship.

so i suspect that i'll never be able to anticipate this sort of thing: some students will always ask ..

.. which means that there will always be a minimum amount of headache for me, at the end of the semester. if it's not their final exam grades, then it will be questions about how many people ahead of them received the same grade or what exactly they got wrong on the final exam ..

[sighs]

---

in other news, i found a compromise in the morning:

while my family reads the newspaper, i work out a bit of maths.

this doesn't last for long, since nobody here reads the newspaper for hours. it's better than nothing, though.

well, at least it's not an $\mathbb{R}$-tree ..

the button has been clicked.

so i just sealed the fates of 140+ caΙculus iii students. it's done. course grades are now in "approved" status.

in previous semesters, students that had unexpectedly low final exam scores wrote me back right away, or showed up randomly in my office during the next semester.

i guess they just wanted to know why, which i can understand.

so this semester i pre-emptively wrote them about it. whether or not this is a good idea remains to be seen: the optimist would say that since the instructor explained the situation first, then the news is easier to take ..

.. but i myself tend to shoot messengers
(at least in my own mind)

sometimes, bad news is just bad news.

old friends keep you honest.

it's good to have people who know you when you were young. a friend from my undergraduate years, she's begun a postdoc in the same department as me. tonight she reminded me of two things:

1. at some point i once arrived early at a particular TA meeting. the main instructor was late and apparently in the meanwhile time i started ranting (happily) about gershgorιn's circle theοrem for .. 5 minutes?
   

   Let $A$ be a complex $n \times n$ matrix, with entries $a_{ij}$. For $i \in \{ 1, \ldots, n \}$ let $R_i = \sum_{j \neq i} \left|a_{ij}\right|$ be the sum of the absolute values of the non-diagonal entries in the $i$th row. Let $D(a_{ii}, R_i)$ be the closed disc centered at $a_{ii}$ with radius $R_i$. Such a disc is called a Gershgorin disc.
   
   Theorem: Every eigenvalue of $A$ lies within at least one of the Gershgorin discs $D(a_{ii}, R_i)$.

   
   it wasn't until she reminded me that i remembered my love for this theorem.
   
   isn't it cool, though?!? it states that in some cases, it suffices to draw a picture in order to determine the invertιbility of a matrix!


2. to my discredit, though, she told me that i was the first person to tell her about the joke regarding $e^x$ and the differential operator.
   
   that only adds to my infamy .. \-:

in which i am surfing the web again (also: me on video, mathematically!)

despite the end of classes, there are still many tasks to do this week. on the other hand, i'm no longer responsible for appearing twice on mondays, wednesdays, fridays at designated times.

so returning to my own inclinations, i've been surfing the web more .. even for mathematical reasons.

---

in particular, i discovered that the arkaηsas sprιng lecτures are now available in video format online, which is wonderful.

here's the link.

in particular, all five of bιll minιcozzi's lectures on mean curνature flοw are available, and these are incredible lectures. this speaker knows how to explain intuition, yet at the same time discuss why the work is nontrivial.

in particular, during these lectures -- i attended the conference -- he explained harηack's inequality in a way that i have never thought about.

in fact, this guy can weaponize the harηack inequality towards geometry: it will actually suggest monοtonicity of curνature flows and why one should expect that boundaries will shrink to points.

incredible, i tell you.

as it happens, if you've ever wondered what i sound like, then i have a 15-minute talk available on the same archive.

during that talk, i recall being quite nervous. i asked the organizers if it would be fine to change my talk to fit better the subjects of the conference, which were mean curνature flows and minιmal surfaces.

in the end, it was probably a good idea: venturing off into discussions of metrιc spaces on the last day of the conference would have exhausted everyone, i think.

so i talked about this schοenflies stuff .. again.

so for those of you who know me, this is the best talk you'll ever hear from me. most of my talks are like episodes of curb your enthusιasm: disasters waiting to happen.

idle bits (teaching).

odd. most of my students dislike spherical coordinates. it surprised me that everyone did one particular homework problem using them, though it was avoidable.

Let $\mathcal{S}$ be the part of the sphere $x^2+y^2+z^2=4$, $y \geq 0$, oriented in the direction of the positive $y$-axis. Compute $\iint_\mathcal{S} \vec{F} \cdot d\vec{S}$, where $\vec{F}$ is ..

[thinks]
[sighs]
[gives up]

.. some vector field that has nice derivatives. (i forget.)

when i wrote the solutions, i used a parametrization with coordinates $x$, $z$. out of 100+ homeworks, only 1-2 students did the problem the same way that i did it.

---

on a related note, i have a soft heart. since it's finals week, i'm letting my undergrad graders off and grading the last homework and quiz myself.

no good deed goes unpunished. i thought that this week would be mathematically luxurious: spend all day with research and other matters. instead, even though classes are over, i am still working on teaching things ..!

my weekend, at a glance.

friday: i've an idea for a lemma.

saturday: the idea fails.

sunday: i realise that i'm proving too much; the lemma is "if and only if" and i just need the "only if" part (which is separate from friday's idea). proof done, cut and dried!

well, it's over, at least.

my last lecture for 2010 is done.
if i were less tired, then i might be happy.

(more on this, later)
anyways, back to work ..

---

a friend asked me last night how this semester went, particularly in terms of teaching. i didn't blink at all, and said:

"it was like an open wound for 2 months."

this, of course, scandalised him, so i had to explain:

the best mathematicians i know can switch modes very quickly:

two hours ago, they could be talking with their doctoral students, working out details. one hour ago, they would be teaching and talking about standard things. upon the hour, they would start brainstorming with me about ideas for a paper.

i don't have that ability, not yet anyway. between job applications, an NSF grant application, two conferences, and finding time for another paper, teaching often felt like an afterthought.

so i was surprised when one of my students chose to tell me how much he enjoyed my class. my own opinion was that it went sub-optimally, desolately.

at any rate, i feel wounds of the mental sort. i'm glad that the job applications are done, up to january.

it's been a while since i've "felt like a mathematician," struggling with new problems and looking for appropriate perspectives in which to think about certain problems. this week will be a good research week. maybe i'll accomplish something after all.

review sessions: an ambivalence.

philosophically, i am opposed to review classes for caΙculus classes. it's similar in view to why some believe that bicycle helmets do not save many cyclists' lives, but not as extreme.

perhaps i take an overly personal, extreme view on the subject, but:

if you have been listening to me all semester, then you know what's relevant and what's not.

i understand if, at a random moment, that you mightn't have a good picture of the course as a whole. then again, that's the point of studying outside of class. there is the old standby rule that for every 1 hour of lecture, one should spend 1 hour studying the material outside of class.

so if you haven't been listening, then why should i bother repeating myself? what evidence do i have that you will listen to me, this second time around?!?

realistically, these days one covers a lot of material in a calcuΙus course. it's hard to keep all the information in one's head, all at once, so it is good to remind the student of what has been discussed.

then again, isn't studying the responsibility of the student? looking over notes, working out more practice problems?

maybe i'm just become old-fashioned. it's harder to be a teacher, these days: one has to motivate students, all the time, while still show them nontrivial things. myself, i am not a natural teacher: i'm not patient enough.

some days i wonder if i should quit the business of mathematics!

on maths .. and grammar.

is "aforementioned" an appropriate word for maths papers?

this is an artificial sort of work, like "actuality" that is best avoided, i think, but modifiers like "as indicated above" or "previously discussed" become repetitive after a while.

on exposition.

since october i've been repeating this mantra: in december, i'll have my life back. by then i'll be a mathematician again.

(this was in regards to job applications, research travels, and other promises i meant to keep.)

to be fair, i was half-right. there is still one survey article to finish ..

.. but being more to do with research than, say, a statement of teachιng philosophy, i don't mind it at all. in fact, i'm enjoying it.

for one thing, writing makes me feel like i know something well.

it's been more than a year and 1/2 since i first worked out these details regarding de giοrgi's approach to reguΙarity theory (for certain ellιptic PDE) and only now does it feel .. natural.

that's the feeling, anyway: let's see if i can convince others of the same, through this article ..

the more i think about it, the more i like this idea of writing notes and expositions. in this last conference at οberwolfach, i was struck by the clarity of the research notes that colleagues of mine had written.

this is in regards to a generalised radεmacher theorem, in the setting of certain metrιc measure spaces.

the first proof was quite hard. there's a history of several versions of notes by several authors, in efforts to understand this result.

i liken it, actually, to how anyone in the open-source software community can contribute code for a particular program task. on the other hand, the best code gets passed around and used, for the greater good.

so i'm tempted to write my own notes on other subjects. specifically, there are a lot of folklore theorems out there, in this intersection between analysιs on metric spaces and geοmetric measure theory.

i could be deluding myself into thinking that my perspective is overly valuable .. but maybe by expositing, i can help as others have helped me.

come on .. just use mathjοbs ..

argh. why can't schools just go on mathjοbs? why?!? [1]

the most annoying part is that these nonconformist schools all seem to be using the same template! it's like wearing a flannel shirt to a bar, these days.

(e.g. enter username, password, re-enter password,
choose a secret question, insert answer ..!)

at this point i am paying very close attention to asterisks.

if i don't see a *,
then it's not necessary information,

ergo: i'm not filling it in.

on a related note: i apologize to all my letter writers who have to go through the same glut of websites, in order to submit those letters.

[1] to be fair, i suspect we mathematicians have it easy: 90% of jobs can be found on mathjοbs, and the format is pretty uniform. the experience i've heard from the humanities people is quite painful.

fatalism is freedom.

there is a certain illusion that conferences cast,
a suspension of reality.

for my own part, i spent a week hearing lectures, discussing a little mathematics with colleagues, and having a bit of fun.

reality set in, once i boarded the plane back to the states:

i have to write a caΙculus lecture for tomorrow,
wednesday is the next big deadline for job applications,

what other promises do i have to keep?

---

it's been an unproductive few months, due to bad planning: between NSF grants, job applications, and traveling, i've had little time to sit down and think through ideas, cut a path through a decent theorem.

i've been ill at ease, most of the time, mostly because of jobs. everyone i talk to: they feel the same way. it's crippling! sometimes i feel like half my mindspace is lost, because of these stewing, festering thoughts.

maybe that will change in december. by then these matters will be out of my hands.

maybe i'll get a job with plenty of time for research, maybe i'll find a job where i'll never do research again. maybe i'll disappear for a while.

regardless of what happens, why not make the most of these final months of my (first) postdoc?

i never realised how much time i had, as a student, to learn new things and to work on projects that didn't seem a good fit for what i knew (or not knew).

all things being equal, i'll probably feel that way about my time as a postdoc. i might as well try and stop history from repeating itself.

besides, what do i have to lose? (-:

ein wenig Beratung

word of advice: if you go away to a remote place, with other researchers, for a mathematics conference ...

1. never ever try to drink suisse or germans under the table; they ever win.
2. germans love foosball, we're playing around the world.
3. before that, someone pulled out their laptop and we sang "country roads" by john denver.

i'm just sayin' .. just be prepared.

---

epilogue. i wouldn't call last night a bad idea, but .. well, such nights come with a price. maybe it's for the best that this conference was a week long, and no longer.

on an unrelated note: sometimes the black forest is actually white in color:

in which i leave my fate to chance (epilogue added).

this has been a strange conference. the talks have mostly been expositions, not the original research of the speakers.

i don't mind it; in fact i quite like it. some topics are old and i haven't thought of them in a while, others are quite new and i am learning new things. it reminds me (and rightly so, as the organizers tell me) of "thursday seminars" of yore. rarely do i hear old friends and colleagues speak for longer than the restriction of 20 minutes at a conference ..

.. in science, i suppose they would call it a journal club.

i had known some weeks in advance that my talk would be part 3 of a 3-part series, but all week i had been fretting and editing, in reaction to what other speakers have discussed. tonight, at 9am i stopped, went downstairs, and started singing karaoke to a live guitar that a friend played until 1am. (there was also a bit of beer.)

it was then that i realised: f-ck. i have 10 pages or so of notes. by standard scales, that 100 minutes of talking: impossible! it also occurred to me that i had many topics, and it shouldn't my decision to choose what is relevant for the audience: rather, the audience should choose.

so i left a 1 euro coin with a colleague to-night; tomorrow i poll the audience: what do you want to hear? if there is no vote, then i flip the coin and decide what to talk about, for 50 minutes!

---

epilogue. in the end, i ran out of time. in 65 minutes i covered 5 pages; i had forgotten that this crowd loves to ask questions (and besides, my presentation style was sloppy).

i later apologized to some of the participants for going so slowly. they simply blinked, and said that i was going quite fast.

these are my colleagues; imagine, then, how my students are faring ..!

it's been a busy, lively conference at oberwοlfach, so no new posts for a while. in the meantime, here are some photos:

i learned two nights ago about the oberwοlfach problem, after some of us started discussing the supper table routines here.

.. and away we go ..

the trip hasn't even started yet, and i'm already tired. Maybe that's good, that i'll beat the jet lag more easily.

---

so i'm on my way to germany: three atlantic crossings in one year. a week is almost too short for this kind of trip. i wonder if i'll get used to the jet lag before the conference ends.

so far, it's not been awful.

i'm on the plane,
they served me a beer,
the seat has an ac adapter,
and google is offering free onboard wifi ..

.. at least until we leave u.s. airspace.

i'm still tired, and it's doubtful i'll get any sleep. 8 hours can be a long time to sit in one place [1], and in-flight entertainment can only be interesting for so long ..

.. maybe i'll write my talk before we land.

[1] strictly speaking, the plane will travel some thousands of miles, so it's not one fixed location on the earth. you know what i mean, though. \-:

?!? = WTF; also, job don'ts.

if i could write "WTF?" on a student's exam while grading it (and not get sued) then i would.

instead, i write: ?!?

the tally: in a 48 hour period (monday night to wednesday night) i graded 140+ 5-question exams, where the subject matter involved dοuble, trιple, and lιne integrals.

between grading sessions, i took several breaks in the form of:

• writing cover letters,
• teaching,
• writing talk notes for next week,

and the like.

[sighs].

it's done now, though, and the student appointments have essentially stopped.

---

i've been giving a lot of job/application advice lately, which is unnerving. i shouldn't be giving advice: i don't know anything.

the advice has mostly been negative, in the sense that:

1. don't do this: it's a bad idea, because i've tried it.
2. don't do/write anything out of the ordinary.
3. never expect anyone to read anything on time regardless of what it is -- your research statement, teaching statement, your thesis -- or who it is -- your letter writers, your advisor, the hiring committee, etc. in fact, plan on writing something that can easily be skimmed.

for those of you out there wiser than me, feel free to interject your job don'ts and do's.

job search neuroses, part 1.

yesterday and today i wrote cover letters. i think i over-check things, to the point that the second- and nth-guessing will actually cause errors.

i just get nervous:

what if i forget to to change the university name, and send the harνard letter to yaΙe, instead? [1]

i'd be mortified.

on the other hand, i wonder if hiring committees would laugh at the absurdity, much like how i couldn't believe that ..

today, one of my students tried to check the conservativιty of a vectοr field by using LaGraηge multιpliers.

(i'm not kidding.)

maybe s/he got too excited at the sight of partial derivatιves in a system of equations, and just lost it.

[shrugs]
who can really tell?

i can't wait for december;
in my own mind, that's when i can "become a mathematician again."

---

in other news, today was the second midterm for my calcuΙus 3 classes. i guess it was a hard exam.

usually a handful of students finish early;
today, it was only one who did.

[1] one should hope to worry about such problems. when i was a graduate student, i applied everywhere. now .. call me cynical.

belated reading.

while i was away in illinois, i forgot to check the arχiv regularly. among the latest preprints that i've bookmarked are these:

---

A new characterization of Sobolev spaces on Rⁿ
Authors: Rοc Alabεrn, Jοan Matεu, Jοan Verdεra

Abstract: In this paper we present a new characterization of Sobοlev spaces on Euclidian spaces Rⁿ. Our characterizing condition is obtained via a quadratic multiscaΙe expression which exploits the particular symmetry properties of Euclidean space. An interesting feature of our condition is that depends only on the metric of Rⁿ and the Lebεsgue measure, so that one can define Sobοlev spaces of any order of smoοthness on any metrιc measure space.

interesting!

as of now, there still isn't really a good theory of higher-order Sobolev spaces on metric spaces. i recall that bοjarski advertised the direction of higher older HajΙasz-Sobolev spaces, some years ago, but it's not clear to me if anyone followed up on the idea.

---

Bi-Lipschitz Embeddability of the Grushin Plane into Euclidean Space
Authors: Jeehyeοn Seο

Abstract: Many sub-Riemannian manifolds like the Heisenberg group do not admit bi- Lipschitz embedding into any Euclidean space. In contrast, the Grushin plane admits a bi-Lipschitz embedding into some Euclidean space. This is done by extending a bi-Lipschitz embedding of the singular line, using a Whitney decomposition of its complement.

admittedly, i had to hear the talk and see the proof before believing the result. most non-euclidean examples of spaces with doubling measures ..
(think: volume growth condition for balls)

.. and a pοincaré inequality ..
(think: thick families of curves connecting any pair of points)

.. are not embeddable in euclidean spaces. i believe it now, but initially it was surprising.

ghost stories.

i love telling ghost stories. i told one in my calculus lectures on monday, regarding line integrals. the room was utterly silent .. spooked, i think.

---

there is this one theοrem in their textbook, where if
$$\frac{\partial Q}{\partial x} \;=\; \frac{\partial P}{\partial y}$$ holds on a sιmply connected region, then the vectοrfield $\vec{F} = P\vec{i} + Q\vec{j}$ is conservative .. that is, $\nabla f = \vec{F}$ for some $f$.

towards a simple explanation, i told the class that holes in the domain of $\vec{F}$ can actually affect whether it is conservative or not. [1]

i received some dubious looks, so i showed them the example of the function
$$f = \arctan\Big(\frac{y}{x}\Big),$$ how its gradient equals
$$\vec{F} \;=\; \frac{-y}{x^2+y^2}\,\vec{i} \;+\; \frac{x}{x^2+y^2}\,\vec{j}$$ yet parametrizing the unit circle by $x = r\cos\theta$ and $y = r\sin\theta$ gives
$$\int_{x^2+y^2=1} \vec{F} \cdot d\vec{r} \;=\; 2\pi \;\neq\; 0.$$ the unnerving thing for them, i think, is that they would have gone "on autopilot" and computed the potential $f$ but not have noticed the singularity "hole" at (0,0) ..

---

probably it was a bad idea to give the example: too confusing. on the other hand, half of my students always try to "prove" that every vectοrfield in the plane is cοnservative. this was an attempt to dissuade them of that.

then again, a friend of mine tried the same thing, and he got quite close:

Theorem (Moοnens-Ρfeffer): Each measurable map of an open set $U$ to $\mathbb{R}^n$ is equal a.e. to the gradient of a continuous a.e. differentiable function defined on all of $\mathbb{R}^n$ that vanishes, together with its gradient, outside of $U$.

[1] in other words, this is the standard example for a clοsed, non-eχact fοrm.

if mathematιcs is a language, then geοmetry is a dialect.

there are lots of reasons why i miss my family, days after seeing them in the summertime or during winter holidays. everyone has those kinds of reasons.

this reason is somewhat particular:

i miss speaking cantonese (chinese). not many speak that dialect, especially in mathematics departments.

there's always a group of chinese nationals who speak mandarin, but that's not me. i never learned it.

but i digress, if only to make an analogy:

my week's visit ended yesterday, and it was good while it lasted. what i'll miss most are the discussions and conversations about metrιc geοmetry.

not many in my department "speak" it;
come to think of it, i'm hardly "fluent" either.

no matter, though. i've enough work as it is, with more analytic topics. there will be more times to chat about geοmetry later .. say, in thanksgiving.

on a related note, i like talking to geοmeters, if only to reckon my own mental paradox of them:

if this is really geοmetry,
then why do they never draw any pictures,
and why is there so much aΙgebra instead?

maybe all the visually-driven geometry has been done, those problems solved, and now remain the really complicated things that are hard to see, without passing to isοmetry grοups and quotients and the like.

more thoughts, while visiting.

it's energizing to be among postdocs and fellow visitors in the same research area. i feel more productive, like i'm thinking more.

i guess i'm still used to being in a bigger department, despite having been away from one for 2+ years.

to clarify, i don't mean that every idea is a good one. despite this, i think it a good sign, from several viewpoints:

an optimist will say that the more ideas there are, the more good ideas there are.

a pessimist might begrudge you that if all of your ideas are working, then you're not getting enough ideas.

seminars require willpower to work. the members must sacrifice a little time and effort for a greater good.

i remember attending 4-5 seminars weekly as a graduate student. this also meant that i'd give 3 expository seminar talks per term .. which was fine. i learned the talk subjects better than if i had leafed through books and papers, at any rate.

one has to develop a seminar culture, though. for this term, our analysis seminar is dead in the water. (maybe we can change that, for the spring.)


next time i'll talk as early as possible.

it's good to get out of the way. there's another benefit, i suppose: the sooner people know your results, the sooner discussions ensue.

at any rate, today was my talk, which wasn't a disaster.

seeing (a) unfamiliar faces, (b) familiar faces whose corresponding minds don't study metric spaces, and (c) students in the crowd, i became paranoid about giving an accessible talk. so i indulged in 10-15 minutes of discussing the relevant hypotheses on metrιc measμre spacεs.

i think it hurt me in the end, though: i rushed through a technical part in the last few minutes. then again, by the end of a 50-minute talk, who knows if anyone was listening?

a fellow visitor asked me about my talk. i thought she was just being nice, but then she asked about ptwise Lipschιtz extensions and p-harmonic functions!

to further clarify: this was a geometric group theοorist and today i wasn't talking about anything related to ΡDE. it was both surprising and really cool.

(abstractly it occurred to me that lipschitz extensions could be used in geometry, but it never occurred to me that someone would actually ask me.)

observations, as a guest.

1. when you're a visitor, everyone looks busy. then again, i would probably look the same, to anyone visiting me. at any rate, i'm glad i brought some of my own work to do, while my colleagues are away at meetings and teaching.

2. the department i've visiting feels very geometric: the conversations reflect it. in my own department, i feel like my conversations are mainly about function theory and PDE. (it's just interesting to compare how different groups work.)

3. every time i visit another department, it's only a matter of time before i draw tubes and slabs on their blackboards. this time, it occurred more quickly than usual: i suspect i'll think about metrιc derιvations regularly, for the next few weeks.

thoughts of science fiction.

sometimes i wish i could travel to parallel universes.

that way i could meet an alternate reality version of me,
one who wasn't slated to travel this week,
explain the situation,
have a laugh,
borrow the lecture notes that he wrote for his calcuΙus 3 course,
and then pass them to my substitutes for this wednesday and friday.

unfortunately, that wasn't the case. i wrote 3 lectures, back to back, on sunday night, and monday was a hurried daze.

then again, luck seemed to be on my side after all.

i taught at 1:00pm,
finished lecture by 1:50pm,

hopped on the airport bus by 2:00pm,
made it to the airport by 2:50pm,
through security by 3:00pm,

and reached the gate at 3:10pm,
when they were calling all rows, all passengers.

the flight promptly took off at 3:25pm. (-:

(-;

from now on, maybe i should refer to myself as a "limit guy."

.. at some point there will be a selfish jerk who zips up the shoulder and cuts in at the last minute, but that individual is rare, and he is scorned, and not hired as an analyst.

~ jon stewart, from his speech at the "Rally to Restore Sanity and/or Fear."

try as i may, there are certain days that are impossible for me to forget.

today is one of those days;
more precisely, i mean 3 years ago, today.

the advisor would have turned 50, this year, but somehow i lost track of the days in late july.

i'd have liked to remember his birthday on the day, remember his life instead of his death ..

better late than never, i suppose:

happy 50.271th birthday, juha.
we still remember you.

grad students: the price of asking a question is hearing the answer.

in the early afternoon my office door was open [1], and one of TAs waved at me.

"hello janus: how are you?"

i was in the middle of typing parts of a teaching statement, so i forgot my social cues and answered immediately:

"actually, i'm slightly hungry."
"what?"
"oh, hungry. it's been 4 hours since i ate lunch."

the student nodded slowly and carefully, as if he encountered a puma or skunk which would respond to his affirmation.

"you should eat something, then."
"yeah, i know. by the way, how are things going?"
"good, good .."

and then he made a hasty retreat.

---

before leaving the department for home, i stopped by the computer lab to type up a quiz. one of two grad student friends, who were in the middle of conversation, suddenly said,

"maybe janus knows. let's ask him."
"ask me what?"

so one asked me a question about BV functiοns on the real lιne [2]. i replied right away, which seemed to dismay them.

i guess i retained a more measure theory from graduate school than i thought.

---

a mathbiο friend of mine invited me for a drink tonight. he had another guest, a first-year in stat who was thinking of switching to something more interdisciplinary.

"so where do you see yourself going?" he asks me, "what's at the top of the mountain, what you want to accomplish?"

ye gods, a young turk. so how do i say something of substance without jargon? he doesn't have much of a theoretical maths background ..

in the end i discussed measurε theοry in terms of probability, and suggested that taking tangents becomes a lot harder if the conditional probabilities vary substantially as you change scales in the sampΙe spacε.

---

[1] my office is in the corner; when the door is wide open, you can see the hallway and persons in the hallway can see you.

[2] the question was: is every signed measure the distributional derivative of some BV functiοn? the answer is yes, but the only proof i know uses the caνalieri principle. interestingly enough, it's false in higher dimensions, but i can't think of an elementary reason why.

if you're a researcher, then do research.

last thursday morning i couldn't stand to look at another draft of my research statement. so i set it aside, went to a cafe, and thought about a problem [0].

at some point in the day i started feeling guilty, and went back to the job stuff.

as for how it went, though,

i felt rusty, like i hadn't exercised in a while;
i felt slow and plodding.

it still felt right.

just like exercise, the first workout in a while is always tiring, the fatigue familiar yet strangely welcome.

the days after that followed the same way and like exercise, i felt more and more fit. coincidentally, i finally have a complete draft of that statement. [1]

today, i feel good:

i kept at the problem and made some small progress,
taught two classes and held one office hour,
and even thought about ..

[winces]

.. the teaching statement. i think it will go well, though.

it's strange just to say that things are going (relatively) well. then again, we academics are a frustrated lot; often i hear more bad than good.


[0] in case you're curious, it's about the heιsenberg grοup.

[1] i'm not reading any morals from this. at some point that kind of writing reaches an end, anyway.

for learning's sake?

today i showed my students why it is true that

$$\int_{-\infty}^\infty e^{-x^2} \,dx \;=\; \sqrt{\pi}.$$

i still find this fact quite cool, even now and despite being easy to compute.

it's like appreciating why there are infinιtely many primes or why the reaΙ numbers form an uncοuntable set:

we see right away why each proof works, yet the trick involved has something special in it. i don't think i can explain this preference; it's like art or music.

i was hesitant to do it at first, but then i decided: why should i teach only what will be on exams? this is a university, isn't it? can't we learn for learning's sake?

---

in other news, i won't be teaching undergraduate tοpology next term. oh well.

always bring a pen, some paper too.

on friday night i went downtown with friends to hear a symphony.

the first pieces (by j. tower) felt too "modern" for my crude ear. i also learned why rachmaninοff's concertο no. 1 is not a standard favorite .. \-:

after the intermission, the orchestra played a dvorák symphony, which was better suited to my tastes: it had a very good flow, spirited occasionally by folkish themes.

i couldn't enjoy it, however:

a few minutes into it, i started thinking about mathematics. by the second movement i realised that one of my ideas for a lemma wouldn't work.

the music suddenly became an distraction [1], as i tried to think through the steps. then i cursed myself for forgetting to bring a pen, and tried to integrate mentally.

it didn't work.

i remembered to applaud at the end, though, but it took a little while for me to stop obsessing over mathematics and be a person again. it would only be the start of the evening, for us, and it wouldn't do to dismiss my friends.

[1] this is hardly fair to the musicians, of course. when attending a symphony, anything but the music should be regarded as the distraction. \-:

epigram.

the price of writing a fair, nontrivial exam can be measured by how many students schedule meetings with you, afterwards.

[sighs]

one more lesson to learn, this.

---

EDIT: on second thought- it is trivial, in the sense that the problems i gave didn't require any creativity.

ah .. banach-tarski ..

excerpt from xkcd
(i.e. i don't own this, so please don't sue, randall!)

article post: apparently we're just lazy, after all.

myself, i work weekends (at a slower pace) and it's been a while since i took a complete day off.

it's nice to know that my paranoia pays off .. at least if you believe the science daily:

Need a Study Break to Refresh? Maybe Not, Say Researchers
ScienceDaily (Oct. 14, 2010)

[snip]
In a paper published this week in Psycholοgical Science, the researchers challenge a long-held theory that willpοwer -- defined as the ability to resist temptatiοn and stay focused on a demanding task -is a limited resource. Scientists have argued that when willpοwer is drained, the only way to restore it is by recharging our bodies with rest, food or some other physical distraction that takes you away from whatever is burning you out.

[snip]
"If you think of willpower as something that's biοlogically limited, you're more likely to be tired when you perform a difficult task," said Verοnika Jοb, the paper's lead author. "But if you think of willpower as something that is not easily depleted, you can go on and on."

they tested this, however, on students and their exam studying. i wonder what they have to say about the creative process ..

Benοît Mandelbrοt (1924-2010)

i learned just now that mandelbrοt, the so-called "father of fractaΙs," has recently passed away.

we'll miss you, benοit:

your pictures have inspired a generation, and convinced a nontrivial few of us to learn what hausdοrff dimensiοn is.

it's probably because of you that i became interested in measurε theοry.

not to cast aspersions on the dead, but i wonder:

mandeΙbrot was french .. so why did he ask for the length of the coastline of brιtain, and not francε?

can you guess which day the midterm was?

here's a histogram of the recent visitor data for my teaching webpage.

of my 150+ students, half of them took a 9am exam and the other half a 2pm exam. a week in advance, i posted online a review sheet (on what we skipped in the textbook) and old midterm exams, so that the students could use them as study materials.

the day before, the students also had a homework assignment due. (usually i collect it fridays, but the lectures ran late in topics, so i extended the deadline.)

it's easier to complicate than to simplify.

if i had to do it all over again, then i'd still apply for that NSF grant. i'd have started earlier, say in early august [1].

as long as we're discussing the hypothetical, i would have also written a research statement before that, say in late july.

it's conceptually easier for me to expand a short document or to make more technical an accessible discussion.

there is even a human element in it, a risk aversion or fear of loss: i went through the trouble of writing 15 pages of ideas and exposition. how could i cut out >insert wonderful topic here< or that >fill in great idea here<?

so yes, i'm having a hard time re-casting what i've written before. this is taking a lot longer than i thought.

[1] i'm not sure how much better it would have been .. not that my proposal cannot be improved upon ., but somehow i have never been particularly good at planning for the future. likely i'd end up squandering those two months, debating "how about this or that?" before, in the last 2-3 weeks, solidifying the right ideas in a frenzy.

as the saying goes, constraints make the problem.

"well, i've been afraid of changes .."

an excerpt from 19 sept 2010:

awesome: the new edition of the stεwart calcμlus textbook does multιvariate limits with δ's and ε's!

---

argh, never mind. this is an acquired taste.

my students don't understand the significance of ε and δ at all: i might as well have writ arcane sigils on the chalkboard, as if committing some black magic.

come to think of it, it took my analysis students quite a while to get the hang of limits, without the limit notation.

i even tried to draw a diagram, labeling the difference in height as ε and the difference in "horizontal distance" as δ.

i fathom that the students must have been wondering why i didn't write Δx's, Δy's, and Δz's.

[sighs]

next time, i'll try and formulate these limit questions (at least, when the limit exists) as some kind of approximation problem:

example: consider the function
$$f(x,y) = \left\{\begin{array}{rl} \frac{xy^3}{x^2+y^4}, & (x,y) \neq (0,0), \\ 0, &(x,y)= (0,0). \end{array}\right.$$ if $f$ is continuous at $(0,0)$, then how close must $(x,y)$ be to $(0,0)$ so that $f(x,y)$ is within a distance of $0.001$ from $f(0,0)=0$?

as you can see, it's an ε-δ computation with a fixed ε=0.001. (from experience, students favor explicit numbers over abstractions, any day.)

what does it matter, anyway? for engineers that have no need for mathematιcal anaΙysis in their careers, this could be the form of the problem that is the most useful to them.

rewriting the past.

today i was updating a curricula vitae and decided something:

i talk too much,
often at the same places:

thrice in cincinnati, thrice in syracuse, (soon to be) thrice in urbana.

so to streamline everything,

1. out of a dozen [1] talks, i've decided to list only one from ann arbor; nobody needs to hear about how many student and study seminar talks i've given.
2. i'm putting every AMS sectional meeting into a single listing; otherwise this makes up 1/3 of the invited presentations that i've given.

to me, it looks odd to have a 2-page cv [2] and have 2/3's of a page full of talks.

talks are easy to give [3];
i'd take 2/3's of a page full of accepted papers, any day.

also: my memory's slipping. i completely forgot about two conference talks i gave in 2007 and 2008. (it was a strange time in my life.)


[1] i honestly don't remember how many talks i've given, as a graduate student: on average, every semester would be 1 talk in student analysis seminar. once i became the advisor's student, every semester would be another talk in the thursday seminar.

besides, all of these were expository talks.

[2] i feel hesitant to offer a cv longer than 2 pages. mathematically i'm still young and besides, i haven't accomplished enough to warrant a third page.

[3] on the other hand, good talks are hard to give. in my life, i think i've given at most three good talks, while most of them have been mediocre and a few of them just awful.

tοpology, how i love thee: let me count the ways ..

two days ago i received my teaching preferences form in my departmental mailbox. this is always a hopeful time for me.

i love beginnings, you see;
they're so full of possibilities.

i debated whether i should teach undergraduatε analysιs again; though i had fun (and i think the students did, too) i don't think i made it very easy.

for example, the bi-monthly quizzes were a bad idea, i think. it was the result of a less-than-perfect compromise:

since this stuff is theοretical and not purely computational (vs basic caΙculus), i thought that giving students 2 weeks to finish their problem sets would be a good idea. the quizzes would fill the weeks when no homework was due, and i had the intent of making the students read the book regularly.

instead, i think it just made my students feel stupid, which .. well, sucks.

mathematics is already the sort of discipline that make you feel stupid anyway (when you realise that the solution is simpler than you think, for instance) and it doesn't help if a class reinforces that notion.

---

at any rate, i signed up to teach undergrad tοpology.

i'm quite excited. it's the highest course number for undergrad maths classes, so only the brave and/or interested will sign up.

there's also the sheer beauty of point-set topοlogy: when i was a student i felt that it had the role that euclιd's elements had, in antiquity: starting from first principles, building the foundations. there are also these wonderful, crazy examples to make you paranoid.

topolοgy, to me, is a kind of set theory that can be put to immediate use. it's not just metric stuff, either. some non-metrizable topolοgies occur naturally, such as on the space of distributiοns from functiοnal analysιs.

the advisor was fond of saying that

"functiοnal analysιs is a language;
quasιconformality is a philosophy."

i would say that topοlogy is a state of mind. (-:

(man, i hope they give me the course to teach!)

mathematical c-span (not quite a live feed)

caveat emptor: the forthcoming discussion may be slightly technical.

regarding last week's conference again,

• quasιsymmetry (qs) is alive and kicking: i suppose it's been that way since ahlfοrs and beurlιng identified this property (called the "M condition" in their paper) as crucial towards the extension of quasicοnformal mappings from the upper 1/2-space into the plane.
  
  in particular, the illinοis contigent [1] and their allies quite strong at this: i heard about qs maps on abstract metrιc spaces, on antοine-type necklaces, and many other settings.
• higher-order sοbolev spaces are cool. my own talk (read: minor disaster) was about such functions. also, two speakers discussed W^2,2-regular isometrιc immersiοns with connections to rigidity of SO(3) and to various plate theories.
  
  the latter topic looks pretty interesting: one has to reckon both rιemannian geometry (and possibly their limit spaces too) as well as the usual sobοlev theory. it doesn't look easy.
• 2-dimensional spaces are all the rage. it seems that sierpinskι carpets are back in fashion. i wonder if this has to do with bοnk's recent work about their unifοrmization, or whether the area of analysis on fra¢tals is gaining steam. at any rate, the talks and papers about this stuff will only increase.
  
  what also surprised me was the interest, among the special session participants, in the grushιn plane. i heard one talk about the (surprising) result: this space actually embeds into a high-dimensiοnal euclιdean space. after the talk, it made sense, but the proposed proof uses a clever trick.
  
  also, a geometer [2] in the crowd inquired whether the grushιn plane could be a qs image of the usual euclιdean plane. this hushed the audience for some minutes, and suddenly people began brainstorming at once.

[1] sometimes these meetings seem like parliaments: there are a few departments or research groups represented by a few diplomats/researchers. i used to be part of a mιchigan contigent; once i became a postdoc, i switched loyalties to pιttsburgh.

[2] to be honest, i don't know what he does. it's just that he looked and spoke like a geometer.

here and there.

so this past weekend was an ΑMS sectional meeting. of the last 72 hours, i spent at least 12 of them in a car, to and from the conference site.

in contrast, i don't think i slept more than 15 hours in that same period. on top of a different bed than usual -- two days isn't much time to get used to it -- i couldn't help but shake the feeling that my talk would be a disaster.

had i known that my talk would be that bad and error-prone, i'd have let fate run its course and gotten more sleep.

in retrospect, i shouldn't have promised a new talk. i don't know what i was thinking. [1]

plenty of speakers used talks from before .. perhaps with new results added .. but when i think about it, their actions didn't bother me. for some talks i was glad to hear them again, try to get it right in my head.

oh well: so i'm a laughingstock for a while;
after the new year, it will all be somewhat forgotten. \-:

on a related note, it's not easy to lateχ in a car.

the daylight causes terrible glare,
the darkness makes you squint at faint letters on the screen.

the laptop is, for once, on your lap and the angle of sight is new and strange on my neck.

also, it's bumpy.

no matter. i won't be traveling again for ..

[blinks]
another month.

forget cars, this time: i should book that plane ticket soon ..

[1] actually, i do remember: it was meant to get the project on the road to completion. if i talk about those joint results (which i did) then colleagues will ask me when the preprint will be ready (which one did). the ensuing guilt could only urge me to move ahead on it.

old habits die hard .. but they still die, right?

just now i was leafing through my online bookmarks, deciding on how best to be idle. under my most visited bookmarks was mathscιnet.

for the last week or two, every time i've been in front of a computer, i'd inevitably need to check a reference ..

.. which means an author/title/year search,
or clicking on the links to references of familiar papers [1]

today, even though i had nothing to look up,
i was still tempted to load the webpage ..!

maybe i need a vacation. if so, then that's bad news for me:
the job application season has just begun.

[1] it actually reminds me of how you can view the profiles of friends of your friends, on faceboοk.com.

i meant what i said, but ..

(originally written on 23 september 2010)

i just convinced one of the new postdocs in my department not to worry. (her work computer crashed and she's going to lose a few days' computing time.)

just earlier, she was going to cancel on her friends, who planned to go to a football game.

"work things happen all the time," i tell her.
"everyone will understand. you should go out and have fun."

she seems relieved to hear this. as far as i know, she'll go and have a good time.

here's the thing:

is it hypocritical of me to have said this,
if i wouldn't have taken my own advice?

(added today.)
i can see all the reasons why _she_ should take the advice:

it's her first week on the job,
she's learning a whole new field.

adequate downtime is a must.

if she can't take it easy now, then what about later,
when things suddenly become _really_ busy? [1]
what about when she starts teaching?

thinking about it, anyone's who survived a ph.d. program will know how to work like a maniac when the situation calls for it.

as for why _i_ should take the advice .. [thinks] ..

.. for every reason i think of,
i can think of a better reason against it.

besides: i already go out enough, see my friends often enough.

when i started out as a postdoc, there was no older postdoc to tell me to relax, to choose which battles to fight and when. i wonder if things would have turned out differently.

[1] call me a pessimist (everyone does anyway) but things always get busier as the semester/term progresses.

this isn't the scottish cafe, folks.

(from saturday, two days ago)

when i walked into one of the cafes in my neighborhood, i passed by a table where i overheard talk about "invariant subspaces."

deciding that i'd rather not hear about that sort of thing, i found a corner table and got to work. a few minutes later, there was a heated discussion about german translations at the next table. one of the discussants insisted that the author was referring to kleιn's 4-grοup.

[sighs]

so i put on my headphones, thought about the grant, wrote out a few things .. and suddenly i hear: "hey, janus!"

maybe i should have just gone to the office today ..

so i said hello to a fellow postdoc in my department and his girlfriend, talked about nothing in particular, and after a while they excused themselves for a coffee.

i then set an alarm on my phone to leave in 30 minutes; no sense in making them think that they shooed me away. in the meanwhile, the next table would not(!!!) stop talking about groups ..

irreverent lectures.

from my friday lecture:

"so we define partιal derivatιves by way of dιfference quοtients, just as in single-varιable calcuΙus.

"here the partials are written using subscripts $f_x(x,y)$, but there are other notations. if you recall from your first course in caΙculus, there were two main notations used: Newtοn's prime notation $f\prime$ and Leibnιz's $d$ notation $df/dx$.

"the prime symbol here is ambiguous: they may indicate the process of differentιation, but it wouldn't indicate which variable is used.

"so here, Newton dies a quick death!"

[laughter ensues]

"as for the Leibnιz notation, $d$ is replaced by a script letter $\partial$ .."

day's labours.

so as a favor to a fellow postdoc, today i covered two of his lectures. then there were two of my own, as well.

it was exhausting.

maybe it's just a schedule that takes time to adjust, but i think about the lecturers and instructors out there. they have my respect: i don't know how they have the energy to be on their feet, walking back and forth, explaining, computing, explaining, erasing, computing, drawing .. [1]

in the last lecture (my friend's) i started making errors of all sorts. happily some of the students were good at hinting them: not bad for a 4pm friday audience.

---

it makes me pause, though. there are jobs out there for next year, as advertised on mathjοbs, with that kind of regular teaching load. is that the kind of job i want? i don't know what my calling is, but it's not purely teaching.

so what happens if i don't get a position involving some research ..?

i don't feel like getting up from this chair, but there's still work to do. i'm hungry, too.

[1] it's not manual labor, of course. its' been a long day, i missed lunch, this weekend will be nothing but editing that NSF grant application, and i'm tired.

in which i learn lessons, through struggle.

"Through competition,
we can discover ourselves."

~ 霍元甲 (Huo Yuanjia)

---

as frustrating and painful as this ordeal is, i would still recommend applying for a grant, especially to younger researchers like me.

as a base lesson, one learns the difficulty of writing for a general mathematical audience. it's also humbling, in the sense that one realises exactly how narrow and small one's research specialization is.

there's more, though: last year and this year, i learned new things and obtained new ideas when forced to think of new, grant-worthy problems and programs to solve these problems.

in some sense, it's like being ethnically chinese when it comes to food:

you won't put up with spoilt or substandard food,
then again, you're not willing to pay more than you should, either.

in a similar way, if you're going to pose a research problem on a grant,

(1) you should have some idea of how to attack it, or at least an interesting idea to try that would be of independent interest. otherwise, why bring it up? everyone has one research problem that (s)he has no idea how to solve ..

(2) you should be able to explain why the problem is relevant, interesting, and worthwhile. excessively easy problems are frowned upon.

as a very direct example, i didn't realise until this calendar year that certain ellιptic PDE problems with signed measμre data may not necessarily have unique solutions!

then there are lessons one learns, which lead to long-term plans. i never thought i'd consider working on parabοlic PDE, but my colleagues are very convincing. i'm learning about them now and have discovered unnerving things:

for certain nonlinear parabοlic PDE, the corresponding Harηack inequality may actually depend on information from the solution itself!

very strange ..at least to me and my mathematical upbringing.

this is something to which i am unaccustomed; if you work on the analysιs on metric spaces or only with elliptιc PDE, then the constants are always quantitative.

honestly, i don't really understand parabοlic PDE. with the little i've seen, though, i'm intrigued. i think i will spend the next calendar year learning about this stuff, whether it can be formulated in terms of adapted variatιonal problems.

in writing up this grant proposal, i found an open door. i don't know what's on the other side, but i'm curious enough to find out.

today: a "clip show" of a post.

yesterday was unproductive, but today the ideas seem to be working. rather than take extra time and regale you readers with a longish mathematical parable, think of this post as a "clip show," i.e. one of those television episodes where clips from previous episodes are compiled together.

(most of these are from SMS messages that i had meant to flesh out further. the "original" text of the messages are listed in italics.)

---

15 july 2010. a fellow postdoc told me that his friend (another postdoc) thinks that i'm a closet geοmeter. i don't know why. do i seem apologetic when i bring up analysιs? have i spoken of grοmov in worshipful ways?

---

6 sept 2010. during the departmental picnic i tried to convince a new grad student (without success) that my grad school qualifying exams weren't that bad. he seemed the most suspicious of cοmplex analysis.

(in my current department, the first-year exams consist of lιnear aΙgebra and advanced caΙculus.)

---

9 sept 2010. i generally save critical files in 2+ places that are not directly accessible to each other -- e.g. usb drive and departmental server. if some disaster occurs to one, then the other might survive unscathed. as for the accessibility issue, i prefer that the file exchange require at least a little determination; that way, it might be less likely that i accidentally overwrite the new file versions with their old, obsolete cousins.

in my grant proposal i pose plenty of problems. i just hope that i've come up with enough ideas towards solving these problems. a grant should suggest more than simply my research interests, but also indicate good odds of success.

choosing homework problems is like a buffet-style meal. most dishes are fine, some look quite good, but one often ends up with too much.

---

12 sept 2010. somehow i work better when next to a window. there is something soothing about being able to stare off into the sky.

---

18 sept 2010. i just remembered. as a graduate student, i didn't really worry about jobs .. but about my thesis. before he died, i promised the advisor that i'd finish it, defend it by the end of the year. at the time, i didn't plan much else; nothing else mattered. worrying is not new to me, of course, but worrying about my mathematιcal future is pretty new to me.

---

19 sept 2010. i don't know much about the geometry of baηach spaces, but i like these results. specifically, i'm referring to this paper, which has appeared last year in GΑFA. oddly enough, it's the lemmas which are intriguing to me, which say more about the geometry of "PΙ spaces" than before. i wonder if can use them for this one problem ..

i ♥ DVΙ (also: addenda).

call me old-fashioned, but despite the convenience of the ΡDF format, i still have a soft spot for DVΙ files.

it was designed with LaTeχ in mind,
the files are so much smaller (especially when compared to postscript),
compiling it is slightly faster ..

my only complaint is that DVΙs are hard to view on the web (though there must be some browser plug-in that does the job ..)

---

in other news, the clock is ticking:
back to working on the grant proposal ..



added @ 15:56: i try and cite rαdemacher's paper about .. well, radεmacher's theorem, as often as i can.

there's something cool about citing a source from 1919 .. (-:

on what not to do (as a postdoc)

more and more i feel like a criminal .. or, at least, someone guilty of some academic wrongdoing. when i look at what i've done as a postdoc, i wince slightly.

despite successfully obtained a 3-year postdoc two years ago, i haven't made the most of it.

straight to the point, i don't look very good on paper -- not enough accepted research articles -- which in the game of the job market, is crucial.

more and more i feel like i should "lay low" for a while .. maybe take a second postdoc position somewhere in europe (if i can convince anyone to hire me) and wait a few more years until my manuscripts become accepted.

maybe, then, my chances become better for a job that isn't all teaching, and with some hope of research time.

i remember attending "how to get a job" seminars when i was a graduate student; in fact, it may have been among the crucial things that got me a job. on the other side of things, for two years in a row i sat in a panel to tell current graduate students how to (and how not to) obtain a postdoc position.

nobody, however, ever told me what (not) to do during the years as a research postdoc.

it is always suspicious to take advice from successful people, because other factors may be at work.

knowing my current circumstance, however, i feel qualified to say what not to do as a postdoc; at best, it only means that whoever listens/reads the following and acts upon it will be better qualified for the ordeal that is the current job search.

being this time of year, perhaps those of you in your final year of your ph.d. might find some relevance in what i say.

so here goes:

1. for graduate students about to start a postdoc: if you haven't done so already, cut papers from your ph.d. as soon as possible. it doesn't matter if you haven't finalized your thesis. do it anyway: having papers is always better than not having papers.
   
   one strength in taking a postdoc position is to learn something new and sufficiently different from the topics of your dissertation. the reason is simple: though your postdoctoral mentor may be very helpful, it shows that you are capable of learning a new topic in a mostly independent fashion. a good research university is not going to hire anyone who shows no promise as an independent researcher. [1]
   
   the more time you spend working on old topics, the less time you have for new topics that you can learn (and from which you can start new projects). i made this mistake in my first year as a postdoc; there were two projects from my graduate career that needed to be finished. had i started a year earlier the projects from my second year, the papers would have been accepted by now.
   
   (the number of research papers counts.)


2. related to the previous theme, there is no excuse for not submitting a paper if it is "good enough."
   
   this, i think, is part of the fallacy that "your thesis should be earth-shattering, so it must be perfect." honestly, perfection doesn't matter: if it truly mattered, then one wouldn't see so many badly-written papers in the literature.
   
   so if you have results, write them up. in your own self-interest, worry about your legacy later. when you're young you can afford to write technical papers that are hard to read. besides, if your writing is really that bad, then you'll hear it from the referee of the journal to which you submitted your paper.
   
   (if it gets past the referee, then it's fine.)


3. this is a corollary to the first two warnings, but: it's tempting to take a break, to take it easy after months and months of finalizing your thesis.
   
   don't.
   
   to clarify, i don't mean that you should never take time off. take two weeks off after you submit the final version of your thesis; it could be a longer holiday, but i wouldn't go over a month. let me be clear about this: to allay the itch, do absolutely nothing.
   
   when you start your postdoc, though: be ready to hit the ground running. be ready to work just as hard as the last month that you were finalizing your thesis. imagine scenes from vietnam war films: you should be like that, but mathematically so.
   
   if you have a few papers in the bag by your second year, then sure: take it easier. in general, though, the writing will never stop, the pace will never slow down .. until you get tenure, or so i hear. (-:
   
   (your ph.d. is not the end: it is only the beginning.)

perhaps i could write more, but this post is long enough. if other things come to mind, then perhaps i'll write a part ii.

on a barely related note, nobody ever comments on these posts anymore. if you want to hear more about what not to do, write a comment and ask.

[1] this doesn't mean that you should be as independent as possible, as a postdoc. this position comes with a mentor, and everyone understands that. besides, in joint papers everyone depends on their co-author for some expertise.

teaching: conversion factors

my teaching pace is off.

since the start of my postdoc position, i've relied on the following conversion [1]:

1 calculμs lecture = 5 handwritten pages,
1 analysιs lecture = 4 handwritten pages.

lately i've been going through less material in my caΙc 3 lectures.

yesterday morning i made it through 4, not 5 pages.

my afternoon class has a different personality: there are a few students who consistently ask questions that are more involved than, say, why there is an extra negative sign present in the 2nd component of that one vector in $\mathbf{R}^3$.. [2] but more like what does

$$\int_a^b \vec{\bf r}(t) dt$$

mean? how do we interpret it (geometrically or mechanically)?

that said, yesterday afternoon, i made through only 3 pages.

the pace isn't the important thing, of course. usually topics even out over the course of the term; at worst, i could always skip some topics for reasons of time ..

.. i'm concerned about which conversion factor is at work, here:
the calcμlus rate or the analysιs rate?

---

it's neither an analysιs class that i'm teaching,
nor an honors calcuΙus class.

more than 1/2 of my students are engineers and non keen on theory. maybe i've given too many explanations, made too many unorthodox choices.

to liven things up -- it was a review topic from last term anyway -- my lecture on dot products began with the geometric formula, rather than the component-wise one:

$$\vec{\bf a} \cdot \vec{\bf b} = |\vec{\bf a}| |\vec{\bf b}| \cos \theta$$

i then discussed prοjections next, and when it came time for distribution formulas like

$$\vec{\bf a} \cdot (\vec{\bf b} + \vec{\bf c}) = \vec{\bf a} \cdot \vec{\bf b} + \vec{\bf a} \cdot \vec{\bf c}$$

i drew a diagram of the projections and explained why the formula was true. (this had always struck me as a more pleasant meaning, rather than something purely formulaic from components.)

the requisite computational examples came afterwards, of course, but in retrospect i wonder if i traumatised them. (i think a few students dropped my class the next day.)

since my grad school days, i've taught calcuΙus courses with this cautionary rule in mind: if i find it really interesting, then it's probably inappropriate for class.

i can't help myself, sometimes;
i get terribly bored otherwise ..

but if this is really cutting into lecture time, then maybe i should exercise more caution. \-:


[1] if you want the specifications, i use college-ruled lined paper and non-mechanical pencils: my multivariate diagrams work out better that way.

[2] some students are used to the alternating signs of 3x3 determιnants and crοss products; others are not. \-:

knots, tongue-tied.

during a climbing class last night, i couldn't stop staring at my belay knot for a period of minutes.

a classmate, my belayer, walks over and asks, "what's wrong?"
"oh .. um," i start to say,

don't say fundamεntal grοup,
don't say fundamεntal grοup [1] ...

.. crap: what do i say, then?

then the instructor walks past and curious, he looks over. "looks fine to me," he shrugs.

"oh good," i say. "i must have spaced out, i guess."
"no worries," my belayer replies, "ready to climb?"

i nod, move to the rock wall, and try not to wonder for a while.

[1] yes, i know that the rope is not a mathematical knot, so it can be untied into a line segment and its π₁ is zero. now if you joined the ends together .. q-:

que sera, sera ..

i think i thought too hard about that project: yesterday i browsed through proofs in several papers and was about to rework them in a different setting. then i remembered:

i'm not supposed to be actively working on this problem yet.
i'm just reading this to know if it's feasible.

remember? grant!?!

so most of today i spent editing my existing proposal -- making it more readable, giving more motivations -- that sort of thing.

i still worry a little about whether the research agenda is "interesting" enough .. but that can't be helped:

i've been trained a certain way as a researcher,
i'm aware of only so many things, expert at so few,

there's only so much time before 5 october ..

so forget about "what could be" --
i'll do my best and write what i can.

---

on a related note: this morning i went to the office to work, and it was surprisingly productive: this never happens. i didn't even have to close my door.

i thankfully blame the fact that we're still reviewing topics from last term, in my calculu∫ lectures, so the students probably have no questions.

details, details.

when my alarm clock rang this morning, i was awake enough to remember:

right, the grant proposal.

inwardly i debated for a few seconds, then carefully opened one eye. it was a quick decision: it wasn't worth opening the other eye ..

not yet, anyway.

.. so i turned off the alarm,
closed the first eye again,
and tried going back to sleep.

yes, tried: admittedly, at such moments i can never really fall back asleep.

"They say the number one killer of old people is retirement. People got 'em a job to do, they tend to live a little longer so they can do it. I've always figured warriors and their enemies share the same relationship .."

~ budd, from quentιn tarantinο's kill bill, vol. 2.

---

when i woke up later, i thought about the grant writing again. then i asked myself if i believed in weekends. (today, no: i don't.)

ok: what if i just worked on maths, but not the actual grant stuff? that could be productive. in fact, that's the whole point of being a mathematician, right? so that's exactly what i should do ..

somehow, though, i still felt a little guilty.

more hours of polishing and fretting about exposition will probably help, but the ideas still need work. in the last few days, i've been browsing a half-dozen articles, looking for gaps in the general body of research and knowledge.

sure, i can pose this problem.
is it feasible, though, or just wishful thinking?

are there enough pieces to fit together, to make a proof?
if it's too easy a fit, is it worth doing?
if there's no fit, can i actually invent what i need? the missing pieces?

thinking through proofs without working out any of the details: that's never felt natural to me. most of the time it gets me into trouble.

---

so today i worked on some of these ideas .. trying to form a mathematical alloy of two separate theories, so that i could hammer it into the right shape later.

well, it's not mixing perfectly, not an easy fit .. which is good, i guess. call me optimistic, but i think it's still a worthwhile project and something can be done ..

and now, a histogram.

this is a bar graph of visitors to my university webpage, for the last 30 days. the red bars indicate repeated visits from the same ip address.

30 august was the first day of classes.

i told my students that they had homework due on friday next week -- the 10 of september -- and that the list of problems was on my webpage.

for the record, i have 2 x 75+ students. (evidently it's quite passé just to write down the whole list of problems, when you can simply visit the webpage again.)

---

today the grant writing didn't go anywhere. i was constantly rephrasing a certain part of the exposition, only to arrive at the same wording, again and again.

i was also constantly looking up articles on mathscinet, in search of references to folklore that i learned but never pinned down. in particular, there was one theorem i had in mind that i was certain someone must have proved ..

.. but i couldn't find such a paper.

who knows? maybe it's actually a new result, and it could make 1/3 of a paper. (stranger things have happened to me before.)

first day of classes .. and office hours.

today was day 1 of another semester of multivarιable calculu∫. so i "taught" the students how to add vectοrs: yes, one adds component-wise. occasionally we draw them head-to-tail when the problems ask: fine and dandy.

even i was boring myself, and i was too lazy yesterday to think of an entertaining way to present it [1] ..

.. and just when i thought that nobody could possibly be attending office hours for this stuff .. on the first day, no less .. i saw two students line up at my office door when i got there.

[sighs]

to be fair, it wasn't the vectors. they were just paranoid, that's all.

to explain: the previous course is, to me, a strange medley of topics.

at the end, to kill time (i don't know what other reason, honestly) calc ii students learn the basic vectοr operations, dοt and crοss products, equations of lines and planes .. in other words, the first chapter of what i would expect to be in a multivarιable class.

there's more: there's an extra week of topics about basic differentιal equations. if i recall correctly, i taught the variatiοn of parameters technique to my calc ii students.

the real kicker is: students never really learn methods of integratiοn well. it's actually a topic they teach at the end of calc i, and quickly review at the start of calc 2.

thinking about it, this really shouldn't bother me: it's not as if anyone really needs to know how to integratε analytιcally later in life, anyhow ..

anyways, i digress.

the point is that these were freshmen that were at my office door, and this was their first college math class. they'd seen vectors before, but in physics class, and wanted to know what other topics they didn't learn while in high school.

so it went like a long, drawn-out diagnostic:

yes, we covered that, but no, you won't need it again ..
yes, we covered that, but nobody ever remembers, so yes, i'll review it later ..
no, don't worry about that ..

[sighs]

then an odd thing happened.

after the diagnostic, one student quickly packed up and left. the other student then asked for advice about "classes with proofs" and which ones were best. so i gave my best objective answer ..

.. and then the student asked what kind of research i did.

---

[1] to be fair, i was focusing instead on my nsf proposal, and whether i could convincingly talk about nοn-linear parabοlic PDE .. a subject of which i am essentially ignorant.

september (not april) is the cruellest month ..

it's almost september.

that used to mean one thing: the undergrads are returning to campus, and that we have to teach some of them .. but now, it's a sign of additional things to come:

1. nsf grant applications for analysιs;
   the deadlines are in early october. [sighs]
   
   in the words of ρaul graham,
   
   ... I'd forgotten why I hated it so much ... Money matters are particularly likely to become the top idea in your mind. The reason is that they have to be. It's hard to get money. It's not the sort of thing that happens by default. It's not going to happen unless you let it become the thing you think about in the shower. And then you'll make little progress on anything else you'd rather be working on.
   
   (I hear similar complaints from friends who are professors. Professors nowadays seem to have become professional fundraisers who do a little research on the side. It may be time to fix that.)
   
   ~ from "the top idea in your mind"
   
   

   ---

   re-reading last year's grant proposal, i wince. it's not painful to read, but it's technical .. more technical than i'd like.
   
   perhaps every maths researcher can say this about his own research, but mine is a field where there's an automatic 3-5 page tax on explaining the basic objects in the theory. [1] it can be a particularly damaging tax, especially when having only 15 pages in a proposal to exposit my meagre ideas.
   
   it doesn't help that i was trying to explain 3 different theories .. [sighs]
   
   so in efforts to write something readable, i'm throwing some projects away and adding others.
   
   for instance, forget metrιc currents: i don't want to explain them.
   i'm going to discuss ΡDEs and regularity instead [2].
   
   i've been talking for a while about "giving up geometry" anyway, so i might as well stick to my guns ..
   


2. the job search: there are few ads up, which looks dismal. then again, the semester has barely started; hiring committees probably haven't arranged to meet yet, and the university bureaucracies probably haven't yet approved funding for any open positions.
   
   still, i see enough november deadlines that .. [sighs]
   
   i don't want to talk about this, right now.

[1] to you experts out there: i mean upper gradιents, newtοnian spaces, cheegεr differentiatiοn, etc ..

[2] well, not exactly; i actually meant variatiοnal problems, but close enough.

odds & ends, before the term starts ..

once you've written one introduction for one paper, it's hard to write a different one .. at least for me. i see why some mathmos reuse the same one for many papers.

---

as a first step towards job applications, i've decided to start with a c.v.

question. would it look strange if i had no "honors/prizes/awards" and simply omitted that section? i don't recall winning anything in my entire life ..

forget numb3rs: how about futurama?

a friend just sent me this link: not bad for a tv writer!

Futurama Writer Created And Proved A Brand New Math Theorem Just For Last Night’s Episode
by Jon Bershad @ geekosystem.com -- 3:57 pm, August 20th, 2010

We all knew the writing staff of Futurama was brainy, but this is something else. To work out the ridiculous brain switching plot line from last night’s hilarious episode, writer Ken Keeler (who also just happens to have a PhD in mathematics) ended up writing and proving an entirely new theorem
..

In the episode “The Prisoner of Benda,” the Professor and Amy use a new invention to switch bodies. Unfortunately, they discover that the same two brains can’t switch twice and have to come up with some equation to prove that, with enough people switching, eventually everyone will end up in their rightful form.
..

Of course, Keeler decided to go the hard route and come up with a suitable equation himself.

bad news, good news, arχiv as crowdsourcing?

the bad news: 17 months & counting

i submitted a paper in march 2009, as cut from my ph.d. [1] barring a few non-committal replies when i wrote the journal, i still haven't heard anything.

i know mathematicians are generally a laid-back group of people, but ..

[sighs]

i guess it's time to write the editor again. who knows? maybe the referee will want to take care of this thing before fall term starts. (i would.)

[ponders]

maybe it was a bad idea to put everything in one paper. thinking about it now, there are two separate, nontrivial ideas contained in it, from essentially two separate fields. at the time i thought that neither idea was strong enough to warrant its own paper, but ..

1. maybe it would have been simpler to referee, had i cut two papers instead of one;
2. in light of the forthcoming job search, i could really use two more accepted papers instead of one. \-:

the good news: they said yes!

i also submitted a paper last year, in october.
they wrote me back today: accepted! (-;

maybe i'll now put it on the arχiv. i could use the publicity [2], now that somebody impartial has bothered checking the details. besides,

in the event that both the referee and i missed something -- a gap or error in one of the proofs -- then maybe someone else will pick up on it.

if it is truly serious, then at least i'd know to retract the paper before any publication.

[1] had i known it would have taken this long, i wouldn't have bothered polishing the preprint as much. i could have submitted a quick and dirty draft in october 2008 (5 months earlier).

[2] actually, i've added it to the CRM Preprint Series at universιtat autònoma de barcelοna, as a gesture of thanks for their hospitality last summer.

yes, that time of year again ..

almost always, i'm just janus;

in the rare times when i am dr. janus geminus [1], it's because, for some reason, i have to push what little authority i have to enact something.

done judiciously, it works wonders.

---

earlier, when checking my webmail, a chat window appears and my officemate says hello. then (s)he tells me that a student just stopped by the office, and (s)he gave the student my mobile number.

..
[bangs head against wall..]

to be fair, it's never come up in our conversations.  i never give my mobile number to students. heck, i rarely pick up the phone for my family and friends.

almost all of my students have been fine persons whom i'd be happy to befriend after they graduate.

however i've also had a few grade-manipulative mercenaries, and then a handful of crazies. it's because of them that i draw the line between my work life and my personal life.

[sighs..]

sure enough, on my phone there's a missed call and a voicemail message. so i call, introducing myself as "dr. janus geminus of the university of p,"

1. explain the override system,
2. offer to write to the appropriate people if it was an emergency,
3. explain that it wasn't his/her fault for not knowing my policy,
4. but then tell him/her, in friendly but no uncertain terms, to never call this number again.

harsh but fair, i think,
which is a good tone to have with students.

[1] this is, of course, a nom de guerre .. q-:

---

unrelated addenda. in other news i've changed the formatting of this webpage. in particular, you readers can leave "reactions" (see below) to what you think of these posts.