Friday, March 07, 2014

the curse of time.

i don't know what exactly i mean by the title;
at best, i can only explain it by example.

a month ago I was absolutely delighted to convert one of my old proofs .. which initially relied on weak star cοmpactness .. into a purely geometric argument.

at the time i thought this was the coolest thing ever .. and i still think it was worth the effort .. but it's been a month and now i have to stop myself from saying ..

wait: shouldn't it be obvious..?

as another example: whenever i wrote an exam for one of my courses, i'd take it myself and multiply the elapsed time by 4; if the product was less than the length of the exam period, then the exam was too hard for the course.

now i'm considering changing it to a factor of 5.

either students today are slower or i'm getting faster .. and i'm almost sure that i'm not getting any faster.

Monday, March 03, 2014

whatchamacallit (or: i'm tired)

i don't know what to call it.. maybe a research bender, a maths hackathon, an intense weekend of work?

all i know is that this morning i woke up from 5 hours of fitful sleep, felt like old, wrung-out towel, made coffee, and sat down and proved a lemma that bothered me all of yesterday.

i sighed after i was done. it was only 9am, and there was a day of work waiting for me .. my day job, you could say.

based on years of experience, i'm not one of those mathematicians who can work all the time and still wake up excited every morning. i wish i was.

maybe i should take weekends off. lately i just feel tired all the time.

Saturday, March 01, 2014

a picture is worth .. a few hundred backslashes!

often enough it feels like i'm wilfully banging my head against a wall. in this case, it's converting visual intuition into rigorous proof. [1]

i must have made this claim before .. but i'll say it again:
every time i have a proof by picture,
it takes 4-5 pages of $\LaTeX$ to write out the details!
[sighs]

oh well: it is the weekend and i finally have time to hack out details for research. i shouldn't complain .. but what kind of "frustrated (over)analyst" would i be, otherwise? (:

[1] for those whom are actually curious about the details, i need a suitable partition of $n$-dimensional dyadic cubes so that most of the measure lies either in subsets (a) that are convex polyhedra or (b) whose translates, under suitable unions, form convex polyhedra.

Thursday, February 27, 2014

in the trenches .. or so it feels.

i should probably leave the office soon, if only because ..
there are students *everywhere* in the department.

i hear many young voices from the few office doors that remain open.

the math help room is full of people.

young men and women have taken over the couches in the middle, watching a tutor intently as she's explaining demοrgan's laws on a nearby whiteboard.
needless to say, 'tis the season of midterms .. and as time goes on, fewer office doors will be open. probabilistically, it's only a matter of time before someone stops by my door and asks me a maths question ..

that said, i should probably get out of here before i have to say "NO" a lot to many slightly desperate faces.

Tuesday, February 25, 2014

in which it is not easy to be an adult.

this morning, after finalising my calculus lecture for the day, i stopped by a local cafe and started working on a proof of a statement that i had jotted down as a claim, months ago.

it was frustratingly fun. i almost had the proof .. when i realised that, soon enough, i would have to reach campus and give my lectures.

then i started speculating:
what if i .. just didn't show up?
what's the worst that could happen?
.. but then i sighed and realised that, deep down inside, i respect responsible people that can be counted to show up when they need to show up, and that i want to think of myself as such a person. that said, there is only one way to get what you want ..

.. and that is to put in the effort, to work hard, in order to attain that goal.
i've told my students that before, and that achieving their goals is conceptually simple but emotionally nontrivial. those things said, that means that i have to do the same ..

life would be so much easier if one could pick and choose one's convictions, at moment and whim. it is far harder to be consistent.
it isn't easy. it takes a commitment and somehow we educators do it .. but like anything worthwhile, it shouldn't be easy .. and for me, it isn't.

on a mostly unrelated note, i really enjoy visits to that cafe. the coffee is incredible.

Friday, February 21, 2014

ARR, MoAR! On computers and proofs.

so today i learned what a "discrepancy" is:
"Adding up the numbers in a sub-sequeηce gives a figure called the discrepaηcy, which acts as a measure of the structure of the sub-sequeηce .."

~ from " Wikipedia-size maths proof too big for humans to check" @newscientist
as for how this came up ..
Erdös thought that for any infinite sequeηce, it would always be possible to find a finite sub-sequeηce summing to a number larger than any you choose - but couldn't prove it.

It is relatively easy to show by hand that any way you arrange 12 +'s and -'s always has a sub-sequeηce whose sum exceeds 1. That means that anything longer – including any infinite sequeηce – must also have a discrepaηcy of 1 or more. But extending this method to showing that higher discrepaηcies must always exist is tough as the number of possible sub-sequeηces to test quickly balloons.

Now Konev and Lisitsa have used a computer to move things on. They have shown that an infinite sequeηce will always have a discrepaηcy larger than 2. In this case the cut-off was a sequeηce of length 1161, rather than 12. Establishing this took a computer nearly 6 hours and generated a 13-gigabyte file detailing its working.

Thursday, February 20, 2014

often when meeting new people, i just tell them that i teach maths.
it's a more expedient answer during those times when you don't feel like explaining your research to someone who doesn't seem particularly adept at basic algebra, simply wants to be friendly, subsequently asks about your work, and only expected a simple answer (but as to why they expected such a thing is beyond me). [1]
that kind of answer gets misconstrued, if you're not specific. occasionally i'm asked if i teach junior or (senior) high school.

so the next time someone asks me what grade i teach, i think i'll say:
"oh, 13th through 16th, and the occasional 17th grader." (-:

[1] just now i re-read that sentence and realised how long it is. did anyone get it on the first attempt? (likely the grammar is incorrect; if anything, the style is poor.)

Monday, February 17, 2014

Seasonal Affective .. Re-order.

odd. every february i get new ideas to struggle with.. or what i like to think as creating form out of chaos.

i wonder if it's a seasonal matter, if the isolation of winter stirs deeper thinking and contemplation.

when i think about it, time seemingly and magically slows down when snow is falling: i perceive it so, at least.

rain doesn't fall the same way, nor do baseballs and rocks. i'm too slow to realise that the structure of snowflakes allows an exception; falling leaves, too.

the alternative would be that the laws of gravity have been suddenly shut off or gone on holiday, allowing solidwater to float briefly on air.

it makes me believe that improbable things can happen, urges me to try and create impossible things, if only to prove a point (via contradiction).

landscapes trans-form into their mollified versions, where sharp corners of peaks and cusps are gone. in contrast, jagged edges appear from broken sheets of ice, and once clean lines along rooves are interrupted by icicles.

what was rough is smooth; what was void is now full of matter, with sharp corners. this is a different world, an inverted world.

Thursday, February 13, 2014

a cynical rant on .. you guessed it: teaching!

i hate to say this .. but the more i think about it, the more it makes sense that good teachers slowly become negligent, if not bad, teachers.

at the moment i'm grading exams. for one problem, i'm quickly realising that most of my students weren't paying attention to me when i was patiently .. doing my best .. to explain how to deal with this one particular concept.

that's not the only exam problem where this phenomenon has come to pass.
if the students aren't paying attention to you, then what's the point of putting in the effort to teach them carefully?
coupled with the constant excuses of "i had to miss class because .." or "but my high school teacher told me .." it's enough to make you wonder how students ever learn mathematics at all, and if you were some sort of social mutation to whom maths was somehow natural ..
(to their credit, students rarely (if ever) complain further,
once they realise you are being fair with them .. well, in my experience, anyway.)
[sighs]

teaching well is as equally frustrating as doing good research .. the problem is that the former task has no real guarantee of working. it really depends on the students in your class, particularly their disposition. i've heard pundits made casual demands of educators, where
..if you're a good teacher, then you always get through to your students and they can succeed.
i believe that as much as if you work hard enough, then you can become a multi-millionaire. it is surely possible, but the odds are bad and most of the time, circumstances don't favor that outcome.

epilogue (as of 15 feb): on these student papers i seem to be writing "irrelevant" just as often as i write "false" or "incomplete" ..

on a related note, i wish i had some mind to talk about my research .. but this new professorial life makes it all but impossible to get anything done ..

Monday, February 10, 2014

on alter-egos..

all I want to do lately is have enough time to write up this one result. in fact this morning I woke up extra early to $\LaTeX$ a lemma and most of its proof, before having to start my day job of being. . well.. a "professor."

odd..

I didn't mean for it to come out that way, but there seems to me a sharp divide between the 'me' that teaches university maths and the me that does the research.. or tries to, anyway.

it's a little like how Batman has to play at being Bruce Wayne, if only to make sure that he can be whom he thinks he should be, for at least some of the time.