class in session, part 3: teacher (not student!) evaluations.

so today was my final lecture;
tomorrow will be my last day in spain .. for a while, anyway.

---

part of me wonders whether it was really wise to agree to a lecture series.  i don't know if the students who attended got much out of my ramblings about metric spaces, geometry, and differentiability.

originally i thought that it would be best to focus on the ideas and how the proofs are structured [1], so a lot of what i presented consisted of special cases of theorems and sketches of proofs.  looking back at it now, i don't think i left enough of a trail to follow what was going on .. and i admit that i went rather fast [2].

i skipped a lot, yet still didn't get very far;
in the end, maybe the compromise was for nothing.

i don't know.
they're over now; that's all i really know.

my colleagues tell me that they really enjoyed the course .. but from experience, if an expert thinks your course is interesting, then you have to ask: is this really good for the students, to whom the course was directed? [3]

the course shouldn't be about the educator who gives the lectures, and it shouldn't be about impressing colleagues either. it should be about the students who are to learn from the lectures.

that's the whole point of a lecture, after all.



[1] put another way, details are easily accessible, by way of the full proofs in the referenced articles.  ideas and intuition --- the kinds of things that only an expert can convey quickly and efficiently --- are harder to come by.

[2] that's my fault, really. i promised more topics than i could really discuss in depth, despite the fact that there were four lectures.  i underestimated how long it takes to motivate the topic (1 hr) and what is really required to prove standard lemmas that i use every day (1 hr).  in the remaining time i discussed two topics, each of which would be a full finnish seminar's treatment (2 x 2 = 4 hrs).

[3] i think it would be fun, one day, to teach a semester-long advanced topics course, but i realise now that it comes with a personal price.  if you teach nothing but advanced topics, then it disconnects you from the day-to-day reality of students, their experience, and their expectations.

there's a school of thought that says that "if the student is strong, then (s)he will rise to the challenge" .. but that's like saying that mathematics is only for the masochists .. which some would believe to be true, too.   i believe otherwise, that if the student is inspired and if (s)he is shown that (s)he can succeed, then (s)he will succeed.

mildly relevant: startup $\perp$ research?

Beware of research. If an undergrad writes something all his friends start using, it's quite likely to represent a good startup idea. Whereas a PhD dissertation is extremely unlikely to. For some reason, the more a project has to count as research, the less likely it is to be something that could be turned into a startup. I think the reason is that the subset of ideas that count as research is so narrow that it's unlikely that a project that satisfied that constraint would also satisfy the orthogonal constraint of solving users' problems.

~ from "how to get startup ideas" by paul graham.

well, in that case i'm set:
being a researcher, that means it's easy to leave the money-making to the mercenaries.

---

maybe i'm channeling cayley and hardy overmuch [1], but there is something relieving in doing work that is not immediately applicable to anything. for instance,

• you're not arming militaries with new, more dangerous weapons,
• you're not making poor people poorer and rich people richer,
• you're not coding an iphone app that just makes it more likely for tech-addicts to further ignore each other at the dinner table.

to do useless work is to do no harm; i can get behind that!


[1] “I have the highest admiration for the notion of a quaternion; but, as I consider the full moon far more beautiful than any moonlit view, so I regard the notion of a quaternion as far more beautiful than any of its applications.”

class in session, part 2: third down, incomplete pass .. (also: #1200)

so i punted.

today was already lecture 3 of 4 and i planned sufficiently poorly so that the $\LaTeX$/PDF notes for lecture 2 were still only half-complete.  despite this, they remain self-contained, readable, and just-barely-suitable for public consumption [1].

this pains me nontrivially [2].

if these lectures were part of a "real" course .. that is, where one would solve problem sets and get actual credit .. then the students would be lost.

this is not an exaggeration.  i can tell something isn't quite clear, if only because my collaborator was part of the audience, and she had a great many questions about some points i made.

so if she, an established researcher, could not catch everything then what chances would a ph.d. student have to be able to catch something .. especially if this is not their field of interest?

[sighs]
so yes, i gave up.

between a final push of research collaboration this week and writing up my own notes [3] for lecture 4, i see little-to-no time available for catching up with the $\LaTeX$ for lecture 2 and proceeding with lecture 3 from this afternoon.  so if is infeasible to do so, then why bother?

instead i posted onto my webpage some PDF scans of some notes from previous talks.  it's not a perfect solution, but it's better than nothing.  more than that, it's important to make something available for those students who may actually want to look at the details [4].

yes, probably none of the students will actually do this .. but if there is a nonzero probability that one might, then it is worth doing.

i also have other reasons for disappointment.  for example, i really wanted to have the lecture notes in wiki format.  my reasons involve symmetry:

on research and moving targets.

this week starts the second week of my visit to spain.

i think it's going well. we're discussing a lot of problems, setting up projects, seeing which approaches make sense and which could be wishful thinking. two weeks isn't forever, you see;

it's good to plan what can be done later, on our own.

---

regularly my colleagues have been asking me specific questions, to which i don't have any answers. this is completely natural, of course ..

.. it's not like they know the answer either,
.. but i still can't stand not knowing.

it reminds me of the film butch cassιdy and the sundance kid:

there's this one scene where the two main characters are practicing their shooting, and sundance is missing every shot.

when asked how he can hit anything, sundance starts walking to the doorway and simultaneously shooting .. only to hit the bullseye every time.

"I'm better when I move," he says.

research-wise, i don't think i've ever been able to solve an explicit problem when "standing still" .. that is, in one concentrated session.

(this includes dedicating a full 6 weeks to it and working on nothing else.)

---

it almost always occurs by accident.

sometimes it's when i'm doing something else.
other times it's some time later after i've finally give up (usually at least a week).

either way it just dawns on me that two separate topics are related in exactly the way that i need them to relate .. at which point, it's not that i feel particularly happy that i hit the target. there's a base amount of amazement that it actually fits together, sure, but most of all i feel stupid that it took me that long!

that's not exactly how it works, though: it probably takes a while to absorb all the information, how everything connects logically, and to step sufficiently far away from it all in order to see the truly essential parts.

i guess seeing something "easy" after months of fruitless activity is just a little disarming to me ..

.. then again, even after the answer presents itself, there is usually a lot of technical work before the proof is fully rigorous.

so usually after the writeup, i feel slightly better .. that yes, it does take more than a lucky break to prove theorems.

if the result is worthwhile, then usually it requires quite a bit of hard work .. \-:

computers have a "shutdown" button .. but if brains were only that easy.

i wouldn't say that i'm tired. my mind is active, even after having worked for most of the day [1] but focus is hard. despite it being a bad idea for many reasons [2], i feel like working.

lately it's becoming hard to "shut off" at night.

there are always ideas that come to mind. of course, most of them don't work and some of them are (admittedly) crazy .. but getting a new idea is addictive. moreover, trying them out is like gambling:

one doesn't expect to win anything;
then again, if one does happen to win .. (-:



[1] woke up at 7:30am, wrote up a final set of lecture notes over breakfast, thought about a random problem or two before heading to my guest office at 9:30, answered some emails and ignored others, met collaborators and discussed ideas and technicalities for projects until the time for lecture, gave a 1-hour lecture, back to collaboration, late Spanish lunch at 14:00, back to work for a few hours ..

[2] until recently i had been consistently good with not doing maths until going to bed. at some point i developed a kind of mental allergy: now i can't sleep right after thinking about technical details.

class in session, part 1: false start.

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i don't know why, but after the lecture i felt a tremendous sense of despair, as if i had done something very wrong. it's not like it was a complete disaster, and as a first lecture i guess it started very reasonably ..

.. and maybe that's it; maybe i didn't plan out the lessons well enough. being so used to the 2-hours afforded by finnish lectures and seminars now, i'm quickly running out of time.

right now i don't know how to fit in all the topics. it wouldn't be as much of an issue .. had the talks not been advertised as a short seminar course.

i don't know where the time went. maybe in principle it was good to motivate the main results .. but somehow i found myself 40 minutes into the lecture and i hadn't even defined the basic object of the "course."

[sighs]

there's a lot of work to do with the material .. and on top of that, i'm really here for a research collaboration. i guess being away from finland for a little while doesn't quite change too much: there's always so much work to do, and so little time.

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preparing for class, part 3: lecture notes.

that monograph thing that i mentioned earlier is just an idea, really;
i don't know if i should be the one who would write such a thing.

to clarify, it's not that i don't think a monograph should exist on the subject [2]. it's just that someone more senior and established in the field should do it [1] .. or rather, someone more trustworthy and with a better view of the Big Picture, both in regards to the field as to how it started and how it is now.

anyway, i don't trust myself to do it .. that is, to do a good job with it.

the last time i tried to write an exposition, it was about the p-Laplace equation and the first draft was such a mess that i despaired of it ever seeing print.

most of it is a re-telling of DιBenedetto's story, of course, but with an emphasis on techniques that can easily be extended to both non-smooth spaces and functions on them. for those of you who know the field, you probably know there are enough of these kinds of expositions so that a better one than mine exists and is accessible.
..
..

.. maybe i'm being too rigid in my outlook, though.

---

thinking about it, the world's gone sufficiently digital as to allow a continuum of document types. on one end, we have well-polished textbooks and peer-reviewed research articles .. but there are less formal ways, too.

• take, for example, this prιmer on differentιable structures on metrιc spaces. maybe it will see print in a journal, maybe not (though i hope it will), but as it exists now, i think it's a great resource. it serves the very useful purpose of a clear, concise exposition without suffering the troubles of formal publication.
• a similar resource is a technical report that the advisor wrote, years ago: i still favor it over the standard GMT references, when thinking about fΙat currents. the topics are based on .. come to think of it, a series of lectures he gave one august.

a final decision can wait, i guess. the publication of monographs seems such a formal affair to consider. i think i'll start small. i've already developed the practice of sharing my handwrit talk notes with audience participants, and occasionally posting scans of them on my homepage.

being that the contents of these lectures will be a little substantial, maybe it will be good to have them in a digital, searchable format and easily accessible on the web ..

.. so it sounds like a wiki is in order!





[1] then again, colleagues of mine have co-written a book about some topics on metric space geometry. one of them defended his ph.d. the same year as i did .. so if he can do it, then maybe i can too?

[2] to be honest, i think there are too many books and papers out there. this is related to job candidates essentially being required to publish a lot of papers .. and with teaching loads as they are, these days, odds are that not every paper will be a strong one. maybe it's not a bad idea to show restraint and buck the culture .. because if we don't, then this is only going to get worse!

article post: not running away from a problem ..?

from "uncertainty, innovation, and the alchemy of fear"
by jonathan fields (@ the99%), found on 13 may 2012.

-- ✂ -- --

.. "People who tolerate ambiguity may be able to work effectively on a larger set of stimuli or situations, including ambiguous ones, whereas intolerant individuals will avoid or quickly stop treating such information.”

Problem is, with rare exception, when faced with the need to live in the question, most people, creators included, experience anything from unease to abject fear and paralyzing anxiety. And there’s a neuroscience basis. According to fMRI studies, acting in the face of uncertainty lights up a part of the brain known as the amygdala, which is a primary seat of fear and anxiety. That sends a surge of chemicals through our bodies that makes us want to run.

-- ✂ -- --

interesting choice of words: "living in the question," as if the problem causes you to live in your own little world .. which does agree with my experience, admittedly.

quite a few times, i had to convince myself that i should just try something .. no matter how stupid: just the first idea that comes to mind, anything. if by chance it works, then the problem stops: great! if it fails, however, then typically it does so for a very good reason, so if i understand why, then the next attempt will be more likely to work, and so on ..

as they say, sometimes solving the problem doesn't matter so much as learning about the problem, and finding interesting things.

preparing for class, part 2: preparedness.

likely i'm deeply biased about mathematical culture and conventions, but i always thought that lectures and mini-courses belong to more senior and experienced people.

for instance, i recall now some lectures that colleagues gave in chennai, a few years ago, as well as an earlier summer school in barcelona. these were very fine lectures indeed;

in particular, i admired their self-restraint.

it was clear that the lecturers knew many things, but opted to discuss a handful of ideas with care, clarity, and motivation. there was no real grand-standing and strutting: the ideas were the stars of the show.

having been impressed by this approach, i think it best to follow the same traditions. the ideas should come first.

i already sent a title/abstract of these lectures. thinking about it, many of these ideas are very new .. new as in the last few years, so this is going to require a great deal of care. i don't have too many references to depend on, and those in the literature require ..

.. some experience to decipher, you could say. so unless i expand the topics further ..

.. which runs the risk of cramming too much into the space of a few hours [1] ..

.. then whatever i present will run the alternative risk of becoming a standard reference for these topics [2] .. especially as i'm thinking of assembling the lecture materials into some kind of a monograph ..!





[1] i've gotten a little too used to the length of finnish seminars, which is 2 x 45min. when you think about it, that harrowing week where i gave two seminar talks would contain the same content as 4 usual lectures elsewhere .. and in case you were wondering, yes: that week was spent beta-testing some preliminary lecture material for next month.

if you're surprised, then you shouldn't be. come on: do you really think i'm that un-lazy? (-;

[2] re-reading what i just wrote, it sounds arrogant and self-serving, but i don't know any other way to describe it. in an expositional article, the advisor once spent a chapter outlining one particular topic, but that was 5-6 years ago and due to brevity, it doesn't discuss how exactly one uses the theory. there are some new tools available now, too, which warrant their own exposition.

preparing for class, part 1: first day jitters (to come).

so i'm scheduled to give a few lectures this november, in spain.

originally in the invitation, i was asked to give a talk or two .. where two seemed like an important detail to consider. so i asked exactly how many talks they wanted and what topics the audience would find familiar. the organiser responded, then i wrote back, then he did ..

.. and one thing leads to another ..

.. so rather than rushing in my usual manner through highly technical details, i'll instead go through a half-expository route .. and probably promote my own (recent) results as well.

i think this to be a great honor;
that said, i'm deeply worried about it.