Friday, March 30, 2012

a net is a net is a net, i bet!

i'm currently LaTeχing a revision of a manuscript, and now there are problems with terminology ..

i just realised that i'm using the term "net" in two different senses:
1. from topology, a net is a generalisation of a sequence: the index set no longer consists of integers, but an arbitrary directed set but not necessarily a poset.  (the usual example consists of subsets of a fixed set that are partially ordered by inclusion, but not necessarily totally so.)
2. an ε-net (with ε positive), on the other hand, is a notion from metric geometry: roughly speaking, it is a locally finite approximation of a metric space.
it's not that much of an annoyance: for each ε-net of a doubling space, i'm constructing an approximation fε of a fixed Lipschitz function f.
in other words, i'm building a net from a net of nets!
*grins*
..
*frowns*
*sighs*

sometimes we mathematicians have to be more inventive with terminology ..!

Tuesday, March 27, 2012

decisions, decisions vs. conventions, conventions ..

a few minutes ago [1] i was latexing and realised that i needed another name for a function .. yet i had already used f, g, and h ..

argh ..

i can't just call it f' ("f prime") either, because i already used (prime) for differentiation of functions on the real line .. and f0 just looks .. weird:

i mean, what's the subscript for?

[runs through alphabet]
[sighs]

i guess i'll use u;
it feels the least strange, to me.

the greek letter φ is close to f, but it looks too much like a smooth, compactly-supported function for my taste ..

odd, how some conventions become crippling.  to me, for instance,
• a and b are points or parameters (or very rarely, indices)
• c is a constant,
• d is the exterior differential,
• e is base e (and occasionally an embedding*)
• f, g, and h are functions,
• i, j, k are indices (with i sometimes the inclusion map [2])
• l denotes a line,
• m and n are natural numbers,
• o is a base point in a space*
• p and q are either points, exponents, or polynomials,
• r is the radius of a ball (occasionally a third polynomial),
• s and t are parametrisation variables,
• u and v are vectorfields,
• w is a weight function*
• x, y, and z are spatial variables.
as for uppercase letters,
• A is a matrix, sometimes a constant,
• B is a ball,
• C is a constant, subject to change, line by line,
• D is the total derivative map,
• E is the base space for a fibre bundle,
• F and G are mappings between spaces,
• H is used for homology,
• I is the identity map,
• J is used for jacobians,
• K is a distorsion function for quasiconformal mappings*
• L is a linear operator, or a space of integrable functions,
• M and N denote sobolev spaces of functions* (on metric spaces)
• O is an open set,
• P is .. an affine hyperplane?  (i rarely use this: huh ..)
• Q is a cube,
• R is the larger of two radii,
• S is a symmetric tensor,
• T is a linear operator between normed linear spaces,
• U is a unitary operator,
• V and W are vector spaces,
• X, Y, and Z are spaces.
and, of course, greek:
• α and β are multi-indices,
• γ is a curve,
• δ and ε are small numbers,
• ζ is an embedding [2]
• η is a standard, smooth mollifier,
• θ is an angle,
• ι is the inclusion map,
• κ denotes curvature,
• λ is an eigenvalue,
• μ and ν are measures,
• ξ are coordinates on a differentiable structure* (or a phase space variable)
• ο looks too much like an o, so it's still a base point,
• π is either a projection map or a homotopy group,
• ρ is the density function to an absolutely continuous measure,
• σ is surface area measure,
• τ is a dummy variable for integration,
• υ, i never use, though Υ is used for jets* (a la viscosity solutions for PDE)
• φ and ψ are test functions,
• χ is a characteristic (indicator) function,
• ω is a solid angle.

[1] .. and yes, clearly i'm blogging now. q-:

[2] i don't do complex analysis unless absolutely necessary.

Monday, March 26, 2012

a few snapshots, from a few weeks ago.

to my relief, the special session in early march was held in a rather nice building:

above left: the (usual) display of AMS book titles,
above right: a lobby busy with registrations, someone trying to concentrate ..

above: the lobby, less busy ..

.. and below is a photo of the emerging crowd,
upon hearing that there would be free lunch provided .. (-:

Saturday, March 24, 2012

thoughts about online education (wired.com article post)

somehow i suspect that online-available courses and degrees are not going away, despite recent outcries that some are scams.

it could be a good solution for access to education, amidst a trend of ever increasing university tuition costs. moreover,
"People around the world have gone crazy for this opportunity. Fully two-thirds of my 160,000 classmates live outside the US. There are students in 190 countries—from India and South Korea to New Zealand and the Republic of Azerbaijan. More than 100 volunteers have signed up to translate the lectures into 44 languages, including Bengali. In Iran, where YouTube is blocked, one student cloned the CS221 class website and—with the professors’ permission—began reposting the video files for 1,000 students."

from "The Stanford Education Experiment Could Change Higher Learning Forever" @wired.
this is putting your money where your mouth is:

if you truly believe that a more educated public is the path to a better society, then we should make more lessons accessible to everyone, locally and around the world.

it seems that, nowadays, the internet is the best way to disseminate that opportunity.

i wonder, though, about the wisdom of removing students from a physical classroom and whether a virtual presence can ever replace an actual physical attendance.

(to be fair, i'm an academic who's been (reasonably) successful in his career [1],
so necessarily i have some incentive in maintaining the status quo that has rewarded me.)

when i think of what a student gets out of a lecture, though, i'm hard-pressed to identify what would be so crucial in physically showing up to class.
the only thing that currently comes to mind is scope, at least if the lecturer uses a chalkboard.

it's harder to see the "whole picture" when all you have is a camera shot of the board.  on the other hand, being in the classroom allows a student to turn her head [π/2], look at the part of the lesson that came just before, remind herself of how the topic initially came about, the motivations ..

(this obstruction could be solved, however, by having better video cameras and a big computer screen, so that a viewer could "see it all.")
i can imagine, however, an analogous feature from having videos of lectures available on-line. despite a smallish projector screen in the classroom, where only a limited amount of information can be shown [2], students can rewind the lecture and catch something that they missed.

this is very crucial and incredibly helpful for students.

many of my students have told me that i go quite fast through the material, to the extent that they cannot take effective notes.  keep in mind that most students don't have any real time to think about the contents being discussed in lecture; often they are simply copying what you're writing on the chalkboard, and then they will read them later to understand [3] ..!

honestly, it almost seems that having students in the classroom is more a benefit for the lecturer than the students themselves (although they do get something out of it, too).  you see, body language is incredible effective; to me, its lack is almost like being blind [4].

it is impossible to gauge reaction from a video camera.  there have been plenty of times that my colleagues and i have turned to an absolutely confused room of students, realised that we were talking nonsense, and made a completely improvised, down-to-earth example on the spot [5].

sure, you could read through students' comments after posting the lecture online, realise that you gaffed up something and could have given a better example, and redo the lecture and re-post it.  then again, why go through that trouble if you only have to do it once during a live lecture, in front of students that react to you?

the fact is, we humans are physical creatures and interact best, face-to-face.  it's how we are hard-wired.  i mightn't be able to point out all the explicit strengths that come from the physical manifestation of lectures, but i don't think it should be so easily dismissed either.

and now .. for a random, nitpicking opinion:
".. Thrun acknowledges some harsh feedback from his students. “We made a lot of mistakes,” he says. “In the beginning I made each problem available only once. I got a flaming email from a student saying, ‘Look, you’re behaving like one of these arrogant Stanford professors looking to weed out students.’ I realized we should set up the student for success, not for failure.” KnowLabs tweaked the software to allow students to keep trying problems."
i don't know how i feel about that response.  when i think about it .. if you really want a student to master a particular concept, then the best thing to do is allow them opportunity to try until they succeed.  otherwise, will they really learn the material?

thinking about it more, though: if a student knows that there are countless chances to do something .. at least, up until the end of the term .. then the "natural" thing to do is to procrastinate, or at least, not to give your best effort.  if this is but one task, then it doesn't matter too much .. but if this is one of many concepts to master, throughout the course, then the student inevitably falls behind.

our students may physically be adults, but many of them come straight from high school and new to all the responsibilities of adulthood and learning on their own.

that said, drilling the point that there is only one opportunity to complete a task for the course emphasizes the point: don't slack off, because there is a penalty ..

.. but i've rambled on enough;
perhaps there's more to say, but not tonight. (-:

[1] by which i mean that i'm still employed, and found a tenure-track job.

[π/2] i learned recently that women are now the majority of university enrolled-students, so "her" might be a better pronoun to use, here.

[2] this is by far the most compelling reason that i prefer chalkboard talks to slides.  with slides, it is harder to track what came before; as for a chalkboard, it's there until someone erases it.

[3] not that it's something that i take pride in, but students have told me that they like my teaching style because i write out everything that i say (nowadays); it's how it even occurred to me that students have this problem.  their notes from my lectures "read like a book," apparently.  i suppose it is good that i'm clear, but .. you know, i did assign a textbook to the course, and i'm taking the material from there ..!

[4] this is why i hate talk to people on telephones.

[5] for example, one friend of mine was trying to explain to her class why
$$\#(A \cup B) \;=\; \#A \;+\; \#B \;-\; \#(A \cap B)$$ and why strict inequality can occur (here, $\#$ means cardinality of a set).  none of the students were having any of it, and then it occurred to her: she physically made the students break up into groups:
1. those that prefer Batman to Superman,
2. those that prefer Superman to Batman,
3. those that like S and B equally,
4. those that don't care for either.
when she asked the students which groups are included in the set of people that like S or B, everyone gave the right answer, right away.  she then pointed out that, unlike on their homework, they didn't count group #3 twice .. and then a collective "ah" erupted in the room.

you could argue that a really good instructor would have came up with an example like that in advance, but come on: how well do you read your students' minds? i credit my friend for being that insightful to explain something so effectively!

Wednesday, March 21, 2012

"simple" still means simple, but ..

finally: back in helsinki!

*sigh of relief*

i don't care if grοmov himself invites me .. to new york, france, anywhere;
i solemnly vow to stay put for at least one month!
last week i helped a friend stay put;
it's a lot easier than helping someone move.

i just went over to his house and made sure that he did not start to load all his stuff into a truck ..

- mitch hedberg.

to "celebrate" this occasion, i'm making corrections to a particular manuscript that i've set aside for close to 6 months.  in one part, i had written that:
"The proof of Lemma 4.3 is technical but the idea is simple."
i re-read the proof, then frowned: "wait: why is this simple" ..?

then i remembered: i drew a picture, felt satisfied,
decided to rewrite the statement as:
"The idea of Lemma 4.3 is simple, but the proof is rather technical" ..
.. and added another clarifying sentence or two.

to be fair, if you're not used to weak topolοgies, then the strategy doesn't look like it should work at all:
the point is that weakly-convergent yet geometric approximations of metric spaces give rise to isomorphisms of certain generalised differential operators (called derivations);

in the case of manifolds, this corresponds to how a diffeomorphism gives rise to a push-forward map between tangent bundles.

in the generality of metric spaces, though, the really cool part is that you can do this with certain embeddings. the bundles may degenerate, but the dimension won't!

Thursday, March 15, 2012

among other things, a bad joke.

earlier, a conversation [1]:
Niobe: By the way, Janus, this is Castor.
[Niobe points to Castor, who nods at Janus]
Niobe: Castor's a second-year.
Janus: Hello Castor.  You could say that I'm ..
[Janus thinks]
Janus: .. a ninth-year.
[Janus grins, Castor looks confused, Niobe shakes her head in dismay.]
aren't we all students, in some sense?

so i'm currently visiting the university of my ph.d. for a few days .. but nothing at all official.  in particular, nobody invited me ..

.. so there is no way,
none at all ..

that i'm being suckered into a talk #6 on week 6!

on a related note, i don't feel tired anymore.

i think that seeing old friends and following familiar routines has rejuvenated me, reminded me of who i once was.  this is not to say that i dislike finland ..

(in fact, it's been very enjoyable and fulfilling)

.. but i feel like a different person, there.  i guess you could say that i'm not completely used to "him" yet, and the "old me" still remains a better fit.

(more to come, perhaps ..)

[1] all names changed, of course.

Tuesday, March 13, 2012

less disparate bits, after a conference.

in most of my recent talks, an audience member always interrupts me with questions or comments.  most of the time it is a matter of clarification, which is suggestive that, apparently ..
1. i'm not good at details while at the chalkboard;
2. i'm not scary or intimidating enough to stop people from asking questions. q-:
i'm probably repeating myself here, but i used to hate questions.  they would unnerve me with their suddenness, throw me off my rhythm.  sometimes after answering a question, i'd completely forget what i was talking about, and it would take a few silent seconds to recollect.

(those seconds probably felt like strange, awkward minutes to those audiences.)

i guess i've gotten used to questions.  in fact, now i tend to expect them more than not, and a silent audience is becoming an exception and odd.  they're a good sign, i suppose: it means that somebody is listening, even if they don't quite understand me.

until recently i've never asked questions at the end of talks .. not publicly, anyway.  to be honest, questions don't come naturally to me.

more often than not, i'm thinking through what the speaker has just said, or trying to determine whether the discussion is related to anything i've seen before.

far be it for me to take crucial minutes of the speaker's time and ask for help to jog my memory!

so i tried a different tack at the special session: i tried to ask every speaker a question, at the end of their talk.

suffice it to say: i found it quite hard to think of good, relevant questions. it helped that my co-organiser and i were the ones to got to chose the speakers, of course, and clearly we opted for reasonably comprehensible people ..

.. but knowing my own experiences, i hope that i didn't throw anyone off their game ..!

perhaps i'll change my ways, perhaps not .. but it's certainly given me a new benchmark for attentiveness: if you're following a talk, then usually you can think of a good question for the speaker.

Sunday, March 11, 2012

disparate comments, just after "organising" a conference ..

being a host is not natural to me.
why do i feel like some kind of diplomat, in co-organising this special session?

well .. at least it wasn't a disaster.

i'm reaching a strange age:
1. soon it might be correct to call me a "professor" .. which is strange: i've spent 10 minutes of many a first day of class, telling my new students that it would be wrong to call me "professor" but that "doctor" or "janus" would suffice:

you wouldn't call a lieutenant a general, would you?

2. my friends and colleagues are picking up their first ph.d. students.  i wonder if that makes me some kind of unofficial mathematical "uncle" ..?

an odd thought: i wonder if they're "scared" of me,
in the same way that profs used to scare the younger me? (-:
(perhaps more, to come ..)

Wednesday, March 07, 2012

paper-folding (cool!)

i'm stuck; for some reason i can't get this kind of folding right ..

[ reposted from wheatpond.com ]

The Mιura-orι is a method for folding up a sheet such that it can be opened or closed in one smooth motion. A Mιura sheet has only one degree of freedom, and can be thought of as having only two states: fully open, or fully closed. Since reversing one fold in the sheet (that is, making a “mountain” into a “valley”) requires reversing all of the adjacent folds as well, the Mιura sheet feels as though it has a memory, and is very resistant to deformation.

Monday, March 05, 2012

not quite your everyday occurrence, but ..

two unusual things happened to me, today:
1. someone walked into my office, took a copy of federεr from the bookshelf, turned precisely to one page, and told me about an open problem that i could solve.  (he was friendly about it, though.)

2. i wrote to a fιelds medalist, which shook me to my nerves.

i kept re-reading the text of the email, making sure that nothing could possibly be taken as an offense or annoyance.

here's hoping that it works out ..

Sunday, March 04, 2012

the good from the bad (also: time off)

to borrow from an old joke:
we mathematicians need pen and paper, but we also need wastebaskets.

to quote from the Wired.com article: "how do we identify good ideas?" (via lifehacker):
"How can we sort our genius from our rubbish? The bookshelves groan with how-to guides for bolstering the powers of the imagination. But how can we become better at self-criticism? How can we get excel at the rejection process?"
jοnah lehrεr has a lot to say about the human brain.  every time i read him, i learn something interesting.  for example, here is something that might be (initially) non-intuitive:
"After writing down as many ideas as they could think of, both groups were asked to choose which of their ideas were the most creative. Although there was no difference in idea generation, giving the unconscious a few minutes now proved to be a big advantage, as those who had been distracted were much better at identifying their best ideas.  (An independent panel of experts scored all of the ideas.)"
of course, it would help to know what "best" means, here.  (this is discussed a little further in the article.)

it could just be my confirmation bias, but this would explain why i often struggle with an idea .. and after leaving it alone for a while, it suddenly becomes clear(er) why the idea will or won't work.  it only becomes more pronounced if, say, i "sleep on it." [1]

at any rate, it's nice to know --- though i'm being lazy about following something up --- that it could have some good side effects.  reading lehrεr today also affords me a scientific justification of ... well, why i should take a vacation.

i mean, if it's really going to make the mathematics better, then sure ..? (-:
"Taking a break is important. But make sure you do something that makes you happy, as positive moods make us even better at diagnosing the value of our creative work. After a few relaxing days of vacation, you'll suddenly know which new ideas deserve more time and which need to be abandoned."
every time i think i'm very busy, i run into a colleague that is even busier .. so experience tells me that it's not going that badly. [2]

still, i've been feeling tired lately.

i'm also traveling again on friday, with the goal of giving that aforementioned talk #5 for week 5.  though i'll be very happy to see old friends and make new ones, experience also tells me that the jet lag is going to cost me.

*sighs*

there is some lag time between that conference, a research visit, a wedding to attend, and coming back.  for once, i've settled on a plan of taking thursday (next, next week) off, as well as the following monday.

that monday happens to include a 6-hour layover in london, u.k.  as long as my bags are checked anyway, it means that i can wander the city freely for an afternoon ..

.. it's not like it would be easy to work anyway, so i might as well have a spot of fun!

[1] this is probably one of the main reasons why i've changed my mind about mornings.  it took me a while to realise that i really am more productive when i have a fresh start.

[2] i have a similar rule with whether or not i am getting old.  everyone ages, of course, but i know too many people older than me who .. upon hearing my saying that "i'm getting old" .. will not hesitate to give me a thorough tongue-lashing.

Thursday, March 01, 2012

the troubles of time zones, even when not traveling.

a friend of mine once suggested to me that jet lag is the difference in time that your soul takes to catch up with your body.

in retrospect, it reminds me of the storyline from eastern standard tribe, by cοry dοctorοw.

one oddity about eastern european time is that the end of the "usual" workday here is approximately the start of the workday on eastern standard time.
as a consequence, my american colleagues usually don't email me until i'm heading out of the office for dinner (or a road run).  instead, i get most of my messages at night.

the same thing happens for facebook updates .. which, i suppose, is a good thing: fewer distractions, right?
then there are ways to exploit the time difference.  if you are unlucky enough to ..
1. be on the job market,
2. live in europe,
3. prefer to return eventually to the united states,
4. and have a lot of work to do,
.. then typically you might find yourself completing a full day of work, and then doing your applications at night.

as you may imagine: productive or otherwise, this gets very old .. very quickly.

it's not that i specifically choose this routine myself.  it's just that the last-minute procrastination, so inherent in my nature, causes this to happen automatically.

well, at least this last round of applying and hiring is ending.  i feel like taking off for a holiday ..

in other news, the preprint i've been working on is still short:

at one point it grew to 15 pages, even creeping onto the first few lines of page 16 .. but now it's back to 13 pages.

13 (thirteen!) tight-knit pages, man.

i'm set:
i might submit it tomorrow!