free mathematica1 intuitions: the τricki

interesting: there's a new collaborative mathematics knowledge database called tri¢ki.

according to their FAQ, tri¢ki differentiates itself from the usual mathematica1 wikis by emphasizing problem-solving methods, rather than subject matter.

admittedly i didn't understand what this meant at first, so i clicked on a random article and found how to "use t0po1ogy to study your gr0up."

they're not just listing theorems, but also heuristics and principles exercised in action. four examples are peppered through the discussion, which is quite helpful ..

.. not that i'm going to study gr0up theory or anything. as for something more relevant to my own work, there is also "constructing ex0tic sets and fun¢tions using 1imits."

so "know-how" is a descriptive word, here. so is "intuition" or "folkways."

it's like the web-version of bugging your friendly neighborhood specialist into explaining why her/his field makes any sense and/or how anyone ever proves anything in her/his field.

the internet is a repository of lots of free stuff. the tri¢ki happens to make (some) mathematical intuition free, as well.

---

on the other hand, i can imagine it being a headache to troubleshoot and edit articles for correctness. it's one think to spot-check whether a theorem is correct as stated, but imagine a debate as to whether a particular method is useful or not ..

in which research can be like .. homework?!?

if i think about a single research problem for too long, then i get sloppy. i chase after details that don't matter, rather than stick to the single, ultimate objective.

i liken this to spending too much time in front of a computer screen, when programming.

after a while, one starts writing code to fix previous code, whereas the objective should be writing code to fit the parameters of how the program is supposed to work. i remember this well, from my undergrad days: after coding for a few hours, leaving to eat, i'd return to the screen only to see ..

.. well, cr@p. useless, cr@ppy code.


to quote a friend of mine,
"don't attack the solution; attack the problem."

---

this is why i hesitate to spend entire days thinking about one problem.

however, there is a way out of this.
note the keyword: one.

a day can be split into several problems or projects. i think it took me too long in life to have discovered this .. or rather, rediscover.

when you think about it, this is exactly how one does homework problem sets. it's too risky to attack one problem until it's done; at some point, you need to have something to show for all that time and effort.

i guess i haven't done any homework in a while. a ph.d. will do that to you, i guess. (;

so lately i've been juggling problems. there is a risk in this:

if you attack one problem and are unsuccessful, then you mope and curse the world, life, etc. for a little while. on the other hand, if you attack several problems and are unsuccessful at each, then that depression lasts a little longer.

then again, maybe i should be doing what i would have done as a student with problem sets: talk to someone else, see if they could solve it. at any rate, i could use a few more collaborations.

preliminary post: summer workdays [completed]

i'd almost forgotten what non-teaching life is like. faintly i remembered that one has the entire day to devote to research.

what i didn't remember is that when there's no progress, then that cloud of "no progress" follows you from morning to evening, casting uneasy shadows on everything.

more on this, later; i still have some time and willpower left, today, in hopes of possibly accomplishing something ..

---

now i feel guilty. you see, there's not much to add. unscheduled days feel empty to me, especially when they're over.

at 10am today i began working and thinking, then caught a bus to campus to the office and worked for a while. "work" is probably not the best term possible; i remember reading, looking for something, being confused for a while, caught up in details that don't matter.

i left the office, not having accomplished anything. walking home, it occurred to me that 0ptimal transp0rtation will not work for this idea i had. it didn't help that i lugged, from the math library, vi11ani's new book on my back: a 900-page tome that's twice as heavy as my laptop ..

.. which is an unfair comparison;
netbooks are quite light.

anyway.

i seem to lack an ability to pose reasonable research problems, which does not bode well. when i hear talks or read abstracts of papers, the problems the speakers pose make sense; i mentally kick myself for not having thought of something like that. some days i wonder if i'll have any good ideas, ever again.

mathematical amnesia/unpreparedness [from yesterday]

remind me never to choose a Lap1ace transf0rm problem to grade, if i have a choice in the matter.

oh well;
at least it's over now.

---

the weather is chilly today, not a distraction. i had the morning free for research and i read about s0bo1ev spa¢es once again.

after this many years of having such function spaces around, too often i still feel like i don't know anything about them.

that's been a recurring feeling:

not knowing anything ..
.. rather, not being a specialist in any particular topic.

that's just another way of saying, of course,
that i don't know anything well.

---

there are some proofs that i've written of a few theorems, sure, but the sum body of work is not at all cohesive. it looks schizophrenic.

i can't shake off the feeling that for each paper that i will write hereafter, i'll have to get the hang of another theory. i don't mind learning new, interesting things -- after all, this is the academic's life -- but that future looks exhausting.

it's inevitable that there must be some pains. new papers should contain new ideas, or at least, new applications or perspectives on existing ideas.

that fact never bothered me before;
i don't know why i pause at it now.

maybe, as a student, i didn't prepare myself well enough, with a background sufficiently in depth. plenty of mathematicians work in a field equipped with "standard tools." how i managed not to achieve that .. well, the incidence is not surprising, but exactly how it happened is a mystery to me.

maybe i'm less clever than i thought i was,
or just lazier than i'd like to be.

teaching ends; exams begin.

today at 10am, i will proctor an 0DE exam;
at 1pm, we start grading:

4 class sections,
each with a class size limit of 70 students.

this might take a while.

my guess: if i get to thinking about research at all today, then it will happen no earlier than 8pm.

---

despite the end of classes, research is slow: my own fault. lately i seem unable to generate any good ideas, and i get distracted by,

among other things,
the m0nge-kant0rovich mass tran$p0rt problem.

article post: the teaching opinions of others.

i should reiterate that this is not a blog about teaching.

if i were a better teacher and if my job were mainly to teach, then sure. instead, this blog is about one man's frustration with mathematical research .. and, of course, the occasional lack of frustrations.

having said all of that, i still want to share this blurb from an article, which is about university education.

from "We preτend to τeach 'em, they preτend to 1earn"
by MAR6ARET WENτE @ the g1obe & mai1

In order to boost their high-school graduation rates, many provinces have mandated a no-fail approach. Nowhere is this policy more entrenched than Ontario, where schools are under intense pressure to get their numbers up. "Our hands are tied," said another caller, an Ottawa high-school teacher. "The government does not allow you now to give zeroes for work not done. If you give a kid 10 assignments and he does three, you can't give him a zero for the other ones. The government stance is that this is a behavioural problem, and you need to give them another chance to hand it in. If a student cheats on a major exam, you can't give them zero. The government doesn't tell you what to do the second time he cheats."

i don't know if it's true or not, but it sounds plausible ..
.. if only because it agrees with my embittered worldview.

something is wrong in the state of education today, but i can't say i know how to fix it. even if i did, i don't know if i'd have the energy and willpower to fix it, either.

---

in other news today, i'm endeavoring to flesh out some research ideas. as usual they are intuitive and nonrigourous, so it takes more discipline to carry them out than i thought.

goodbye, 0DE.

about a half hour ago i "taught" my last 0DE class --
(as review classes are not really teaching)

-- and i'm glad not to teach that course again in at least 8 months. i have nothing against people solving differenτia1 equati0ns in the privacy of their own homes and offices ..

.. come to think of it,
one particular 0DE has come up in my own work ..

.. but in terms of instruction, i'd rather teach something else.

---

linear algebra was fine, this term;
i'm teaching it again next term.

random post: band names.

i have heard of a-capella groups called the k1ein 4-gr0up as well as the n-harm0nics.

however, if i had a mathematics rock band, i would name it:

"Doktor FUNK-Tor & the Push Forwards."

heck, it even sounds like a band name -- one which makes no sense, but there are plenty of names like that.



** for the record, it happened once that i played percussion for an impromptu band of math grad students called barramundi. i essentially banged two wooden sticks together, to form a rhythm .. which was fun. **

viva italia! (or at least, this preprint server.)

i quite like the CV6MT preprint server @ $nS Pis@. in contrast, the arXiv just doesn't have that same cozy sort of feeling.

not only will they post preprints @ CV6MT, but they will also archive lecture notes from various summer schools held in italy.

i've said this before, but lecture notes are wonderful things.

sure, papers add to the existing knowledge of a branch of research, but too often their introductions begin at a rather technical level. this is fine for experts, but daunting for someone who just wants to have a brief look and learn a little.

some days ago, i discovered these lectures by de 1ellis, about mar$trand's theorem and other gems about ge0metric measure theory.

(for the record, i stumbled upon them while first reading this about metri¢ ¢urrents and these objects called we@k ja¢obians. one browse led to another, and soon i was poring over de 1ellis's publication list ..)

---

yesterday and today i've been browsing through these notes about optima1 transp0rt as lectured by ambr0sio.

they're from 2000, which preceeds vi11ani's book on the subject, but i think that's a good thing.

in 62 pages, the goals are modest: the intent is not to say everything about the subject, but simply to relate several formulations of mass transp0rt problems.

one formulation, due to εvans-9angb0, particularly caught my attention. it's listed as a PDE-type problem, but i see currenτs in it:

let f be a $igned mea$ure on a euc1idean subset X with zero mean value. find a (positive) mea$ure μ and a 1-Lips¢hitz function on Ω, X in Ω so that

(i) there exist smooth functi0ns u_h converging unif0rmly to u on X, equal to zero on ∂Ω, and such that ∇u_h converge in L² to a unit vect0rfield ∇_μu;

(ii) the following PDE holds in the sense of distributi0ns:

-div(∇_μu μ) = f in Ω

in other words, finding optima1 transp0rt plans, in some cases, is equivalent to showing that a given signed measure, with zero mean value, is the boundary of a normal 1-¢urrent.

anyway, i should get back to work, or get to sleep. tomorrow's a non-teaching day, and the early bird gets the research worm!

spring soon ends, summer soon begins, plans ensue.

i haven't written about research much, lately. i could blame the end of term and all its little teaching annoyances ..

.. looking ahead if i can finish all the topics in my syllabi,
planning out final exams with fellow instructors ..
preliminary preparations for review classes,

.. even preparing lectures is trying my patience. you'd think i'd be used to it by now.

but if i had my act together, none of these should matter. it's not easy being a responsible person. at the very least, it's never come easily to me.


besides, the research isn't going terribly well anyway. none of my ideas seem to be working anymore, which is no end of frustration.

---

but i meant to write about a little good news: for a conference in june, i will have a place to stay in barcelona for a few days; the conference organisers approved me for funding.

there remains, however, about 10 days of lodging to settle for the preceding summer school. that's going to cost a pretty penny, without funding ..

anyways, even a little good news remains good news.

review classes: how does one summarize several months into 1-2 days?

after friday: it's one more week of classes. among them will be review classes, and these are creatures of a different kind than the usual lecture.

i never know how to address review classes.

as an undergraduate i never liked them: to me they felt either redundant or limiting. nevertheless i attended the reviews.

study habits die hard, i guess.

perhaps i am too laisseζ-fairε. it is not my job to force the students to learn.

i offer them lessons, over the course of weeks and months; they can choose to listen, to study, or to do something else. there is always an assigned textbook; they can even read that and ignore me -- at least they would be learning!

trying to plan out a review seems like telling a student how (s)he should have learned the material, weeks ago. it has the feeling of "too little, too late."

say, didn't i do this or say that earlier ..
when i was teaching it the first time?

to be fair, i suppose we all need a reminder. there were many of those subtle points i attempted to convey, each lecture. it's hard to remember them all, even most of them ..

.. but on the other hand, can i really summarize all or most or a good dose of those comments in a last class or two? after all, it took me an entire semester to say all of them!

there must be a clever, happy middle.

---

then there are study strategies.

sure, some topics are important and others less important. some themes and types of problems occur again and again. that's usually a good sign of importance.

some topics are more difficult to put in the form of questions (that are easily graded) and they will likely not appear on exams.

i always felt these observations were obvious, but then again, maybe they are not. maybe students, when set to the mindset of studying for exams, simply ingest information and cease "thinking." maybe i should say these things anyway, to be sure.

there remains the problem of selecting what is particular hard or error-prone for my students, yet important and recurring.

but these are things i cannot fully know.
i teach these things: it's all the same to me.

"hard" is a word, a judgment, that only a student can say,
and i've never been good at mind-reading.

as always, i'll figure something out. here's hoping that i don't bore or frustrate the next generation of students ..

that was then; this is now.

a week ago or more, i gave a talk at one of the AM$ secti0nal meetings. the conference ended later that afternoon, and subsequently i wrote this.

i can't remember exactly when this was written. it was either that evening or the morning after, between my classes.

i didn't like [my talk]. colleagues did say good things about it, but in my own mind it could have been much better:

* too ambitious;
* too fast a pace (even though people assured me that it made sense) [1];
* even if i stuck with the content, the order and format of slides could have been improved ...

that was what i thought then.

i can say that i still don't like it, but some of my reasons are different now. moreover, i understand better what's really bothering me:

almost none of that talk was new.

sure, some of the background contained topics that i haven't presented before in talks, but the results are still my thesis results. i agonized over them a year ago, wrote them as well as i could ..

.. but that was a year ago;
what have i done, since then?

when was the last time that i had a new, good idea,
acted upon it,
made something out of it?

what i don't like about that talk isn't its mechanics or its flow or any of the thousand little subtleties which make it slightly better or worse. that's not the point. i've given good talks and bad talks, and i'll give bad talks again, whether i like it or not.

what i don't like about my last talk is what it suggests.

one decent result in 5 years [2]: it makes a thesis.
i can never be sure that it wasn't just luck ..

.. and i'm still not sure if i'm cut out for this line of work.

[1] it occurred to me later that it's pointless to ask your friends "did the talk make any sense?" of course they will tell you yes. it would require a habit of brutal honesty (or reacting to a truly horrible talk) before someone expresses a negative (albeit constructive) opinion.

of those who may genuinely mean it -- that this talk of mine made sense -- there is still the bias that, likely, they have heard some version of the talk before.

[2] that is, 5 years of graduate school, and yes, that may be a miscount. i spent ~2 years taking classes and exams: fine. that leaves 3 years.

i know recent ph.d.s which churned out at least a paper a year, in that same period of their lives. that's enough of them to know that it's possible; saying that "i was a student" is a reasonable excuse .. but still an excuse.

article post: cacao and maths.

most mathematicians i know have some sort of mild stimulant addiction. foremost is coffee, of course. then there are the chocolate lovers.

one colleague (who will remain nameless) of mine will choose a smaller dinner in favor of a larger, more chocolatey dessert!

that said, they may be on to something:

"how eating chocolate can help improve your maths"
(from telegraph.co.uk)

as usual, the article discusses basic maths skills like arithmetic. for once i would like to see a maths effect study in which one computes, say the h0mology of a given CW comp1ex, or estimates the m0dulus of a given curve fami1y ..

.. yeah, right.

when teaching becomes interesting, maintain self-control.

my 0DE lectures are becoming interesting [1].

today i talked about heavi$ide functi0ns and their 1ap1ace transf0rms.
on friday i will talk about the de1ta fun¢tion.

de1ta "functi0n" -- ha!
yeah, sure, it's a "functi0n" [2] ..

this will take some fancy pedagogical footwork.

this will also take some self control, because i can just imagine saying, "well, it is a me@sure, albeit 1ebesgue singu1ar .."

i'm already having enough trouble as it is. every other time i want to say "1ap1ace transf0rm," i almost say "f0urier transf0rm." as you can imagine, i like one much more than the other.

[1] that is, interesting to me. i'm explaining this material as best as i can, such as explaining heavi$ide functions as off/on switches, and why parts of the improper inte9ral suddenly become 0.

from experience, however, the students i teach are not as good as integrati0n as i would like. this probably means that i'm just as confusing as ever, but now with abstractions. still, i try.

[2] sometimes distributi0ns are also called "9eneralised functi0ns," a terminology that does not sit well with me. history, as usual, forces our hand.