admittedly ..

i'm disappointed in my theorem.

it's not as strong as i'd like it to be. then again, i have no choice in the matter. mistakenly i thought that it would cover a larger generality than at first advertised, but i was wrong.

a version of the theorem is still true: there was one missing necessary condition that i only found, after spending a few weeks, searching in vain for an example that would have proven that this new theorem is strictly stronger than previous ones.

even now, i don't know how strong this theorem is, regarding measurabΙe differentiabΙe structures and these strange objects called derιviations ..

---

so there's still a theorem, but it irks me that i didn't see that additional hypothesis until recently. i thought that i was smarter than that.

the good thing is that i caught it before it came to submission and to print,
but the bad thing is that i have shιtty intuition ..

[sighs]

chocolate, coffee, maths.

an interesting headline:

How eating chocolate can help improve your maths
(via telegraph.co.uk)

the improvement, however, doesn't sound specific to maths, though ..

For the study 30 volunteers were asked to count backwards in groups of three from a random number between 800 and 999 generated by a computer.

The findings show that they could do the calculations more quickly and more accurately after they had been given the drink.

However, the same was not true when the group was asked to count backwards in groups of seven, which the researchers described as a more complex task, requiring a slightly different part of the brain.

The findings also show that the volunteers did not get as tired doing the calculations if they had been given the cocoa drink, despite being asked to do them over and over for an hour.

(cont'd ..)

i think i'll stick to coffee. there's something nice about waking up and doing maths in the mornings while drinking coffee ..

.. because it really feels like i'm slowly becoming smarter, as one hour passes and then another.

conversely [1], the drowsiness (before the caffeine kicks in) can also be helpful. sometimes i find myself trying ideas that -- were i fully awake, alert, and judgmental -- i would never attempt.

it's led to a lemma or two, which is empirical enough for me. q-:

[1] in a conversation once, i was explaining something to a non-mathematician acquaintance.

when i said "conversely," he pointed out that i was already talking. (he thought that conversely meant referring to something regarding conversation.) so then i explained what conversely meant ..

.. to his embarrassment. in retrospect, i should have just nodded, said "on the other hand," and mentioned it to him later.

lines are circles, from a certain point of view.

so i found this today, while wasting time on the internet:


for a mathematical observation, this is quite emo. someone should tell this guy about 1-point cοmpactifications.

sure, you drift apart,
maybe a fixed distance, many farther as time goes on ..

..but if you wait an eternity,
then eventually we'll meet up.

the same complaint can be said for the song "a line allows progress, a circle does not" by bright eyes. q-:

misreading (or: running out of coffee)

early on in my mathematical studies, i was told that "lie" in 'lιe group' was pronounced lee (after the mathematιcian sophus lιe), and not lye (as in "liar").

maybe i haven't drank enough coffee today, but

earlier i read the term "lie theory" and immediately thought of a troupe of actors, teaching apprentices how to cheat, spread rumors, and other means of telling lies ..!

it starts to sound worse when you imagine these apprentices, not satisfied with these vices, then go on to learn various forms of killing. q-:

anyways, back to writing;
there are many sobοlev functions that require extending, in several different ways ..

in which i should stop writing, so that i can start .. er, writing.

i feel like i've been living under a rock, lately. at the expense of ignoring what my friends and colleagues have been up, i've been focusing on finishing this damned preprint ..

[sighs]

every time i think i'm done with it, another idea or question or doubt comes up .. maybe it's time to quit: leave the write-up alone, imperfect as it is, and come back to those accruing ideas another day.

there are other tasks pending,
such as other joint papers to finalise ..

the joys of more writing!
[sighs again]

maybe i'll finish the introduction tonight, and then it will be ready for private circulation.

---

admittedly, i had meant this post to be optimistic and positive.

my plan for june and july is to read more maths, think about some new and strange ideas ..

[shrugs]
.. maybe they'll lead to new projects? (-:

Thoughts about a class about proofs (Part 3)

two weeks later, i still think about that intro-to-proofs class.

i'm still without firm conclusions,
only guesses and impressions.

---

one comment from a student evaluation was rather poignant:

apparently my lectures were very clear,
they could follow my proofs easily,
perhaps a little less so with the textbook ..

but when it came time to prove things by themselves,
they quickly became stuck.

"stuck." i don't know why, but that really bothers me ..

.. it's not in anger or resentment, but close;
it's the exasperation of being at my wit's end.

---

having structured the course as much as i could [2], i'm at a loss of what more i could have done, barring the extreme:

if i had enough time and willpower, i suppose i could have resorted to the socratic method, that is:

nudge them along with directed questions,
convince them to open their books, revisit previous examples,
guide them in building their own argument.

then again, i had 25-30 students. in addition to time and willpower, i'd have needed a few clones of myself to cover all of them.

so, at other times, i wondered if i was doing too much .. and paradoxically, if i still was doing enough.

sure, a student has the responsibility to learn,
endure the hard work if it becomes necessary for success ..

.. but it's still not clear to me if my students learned any mathematical self-sufficiency:

learning how to read a maths textbook [3],
dissecting proofs on their own;

developing a memory for definitions and theorems,
along with a sense of what is relevant, what's not;

looking out for tricks that recur,
learning how to be relentless in their own ideas to try.

---

as i said before though, maths isn't something that one learns immediately. after my own first course in proofs, it took many more years for me to become competent at maths ..

.. and even now, i sometimes think i have a proof,
only to realize that i made a gap in reasoning:

"wait: that one little lemma .. isn't so little. in fact, i don't know how to prove it after all.

argh .. is there a counter-example?
do i just need another hypothesis ..?"

[sighs]

teaching this proofs course has been an exercise in patience. not being a naturally patient person, i guess i did all right.


[2] .. in light of cutting aside time for research and writing and conferences and the like. it would be different if i were hired purely as an instructor.

[3] i wouldn't be surprised if someone could write an entire lesson about such a skill, equipped with worksheets in a 6-week tutorial. this sounds glib, i know, but if it means that students become good at it ..

jargon into vernacular: also, time is not necessarily platonic, in the linguistic sense ..

sometimes i smile when a friend of mine uses the word "modulo" a party that includes non-mathmos, and inwardly i smile:

1. you poor mathmo: you don't realise that "modulo" means nothing to my barista friend;
2. my poor friends: you don't understand what we mean by "modulo," do you? it's such a useful word .. like osmosis in the sense that "i sat in enough talks that i learned about qc mappings by osmosis!"

admittedly i was once playing basketball during an REU program .. and when a friend of mine hit a lay-up, he immediately began playing defence against his defender ..

"don't bother," i tell him. "that dude is nilpοtent!"

this threw him into a fit of giggles, and we had to call a time out for his .. lack of self-control.

some people, i tell you. q-:

anyway, i say all of this because i happened upon this quote ..

"But perhaps most surprising is the team's suggestion that there is no "mapping" between concepts of time passage and movement through space."

.. which was from an article about how one amazon tribe has no independent sense of time .. in the sense, i suppose, that time is embedded in the way one thinks of a lifetime.

"To link number, time, tense, mood and space by a single causal relationship seems to me hopeless, based on the linguistic diversity that I know of," he told BBC News.

Dr Pica said the study "shows very interesting data" but argues quite simply that failing to show the space/time mapping does not refute the "mapping hypothesis".

Small societies like the Amondawa tend to use absolute terms for normal, spatial relations - for example, referring to a particular river location that everyone in the culture will know intimately rather than using generic words for river or riverbank.

These, Dr Pica argued, do not readily lend themselves to being co-opted in the description of time.

"When you have an absolute vocabulary - 'at the water', 'upstream', 'downstream' and so on, you just cannot use it for other domains, you cannot use the mapping hypothesis in this way," he said.

In other words, while the Amondawa may perceive themselves moving through time and spatial arrangements of events in time, the language may not necessarily reflect it in an obvious way.

"mappings," eh? perhaps it is a good thing, sometimes, to be a "geometric mapping theorist" ..!

writing annoyances.

i've recently developed a dislike for the "mathematical we." you know what i mean:

"with these hypotheses in mind, we now proceed to the proof of the main result."

perhaps i should improve my writing skills. "we" is particularly convenient, though:

• it avoids the passive voice, which i was trained in school to avoid. it's possible to use the pronoun "it," but the ensuing sentence becomes quite wordy. [1]
• "one" (the neutral, impersonal, third-person pronoun) serves this purpose, but it sounds awkward.

[sighs]

anyways, back to writing.


[1] actually, the sentence -- "it's possible to use the pronoun 'it,' but the ensuing sentence becomes quite wordy." -- is itself an example of how awkward it is to avoid 'we' or 'one' ..! q-;

Thoughts about a class about proofs (Part 2)

two weeks later, i still think about that intro-to-proofs class.

i'm still without firm conclusions,
only guesses and impressions.

---

when i was structuring the course, i added in two weeks of propositional logic and truth tables, because the syllabus i was given didn't include it. i was hesitant about this decision at first. it would mean that i'd lose 2 weeks of lectures.

looking back at it now, i wonder if i did enough.

there were a few lectures about "techniques of proof" --

for each type (direct, contradiction, contraposition, induction), i gave an example,

then in later lectures i gave more complicated examples that combined multiple types,

and then for the next two weeks, every time i proved something, i either

1. stated what kind of proof it would be;
2. verbally explained how one method of proof fit the situation better than another [1].

so yes, i think that's plenty to explain what a proof is ..

but when i think about it, though, i never gave a single lecture about problem solving. perhaps i regularly explained a strategy behind a proof, where it comes from, but i never formalised the approach.

to be honest, i don't know exactly how i would do such a thing.
it just seems like so much common sense, a few guiding principles.


[1] in one student's evaluation, there was a complaint that we didn't cover enough basic proof techniques. my best guess is that many students only wrote what i wrote on the board, and didn't record what i said in relation to it.

to be fair, i tried to write everything i spoke on the blackboard. that, of course, gets very tedious after a while. it probably explains why i used to be able to cover 5 pages of my own notes in a single 50-minute lecture; now i usually cover 4.

Thoughts about a class about proofs (Part 1)

two weeks later, i still think about that intro-to-proofs class.

i'm still without firm conclusions,
only guesses and impressions.

---

i'd like to think that my students have begun to see proof as a means of discovering truth in mathematics, much like how scientists think in terms of experiments in order to make new discoveries.

we know the world is out there,
that it obeys certain laws and patterns,
but how do we find out what they are?

i should emphasise: begun. many students tried very hard, but they were not often successful [0].

however, i think it very important that they tried. many of them improved substantially.

when i was a graduate student and more cavalier, i used to say:
"you know what half a proof is? half of nothing."

my older self would now disagree. maths is not something one learns instantaneously, nor is much else in life. even a video game take days of obsession to figure out and win.

that younger self did always underestimate the value of progress .. to the point where he (the old me) was stunned, after some years, that he earned a ph.d.

if you think of maths as a language, then it takes a lot of time to become fluent at it. if you think of a maths ph.d. as "mastery" (which i discourage you from doing) then it takes quite a few years .. on top of having completed an undergraduate degree for its preparation.

so i thought of my students' proofs much as i did compositions in a first course in spanish: there are always a few made-up words, errors in grammar, a lack of style from a lack of experience .. and occasionally you read a sentence that sounds .. real, that you could hear someone actually saying ..

[0] even induction proofs were tricky to some.

mathematicians amongst mathematicians.

thursday afternoon we're in a van, driving to a conference. i take out a paper and began to read.

one of my colleagues asks me, "janus, are you working?"

i grin. "refereeing a paper .."

.. which becomes a lively discussion on the annoyances of refereeing and reviewing papers.

today afternoon, we're back in the van, heading home. i'm editing a printout of my manuscript. a friend also takes out a printout, clearly formatted in amsart.

"say, are you editing a preprint?" i ask.

"actually, i'm refereeing this article," he replies ..

mathematicians are predictable: at one point during the ride back:

one of us was refereeing,
another was latexing,
another was reading a paper,
and i was editing that preprint that i've been constantly writing about here and there.

---

there have been some posts i've been meaning to write here, including:

• thoughts on how the "intro to proofs" course went,
• initial thoughts about leaving the country,
• life after one postdoc,
• how the conference went.

don't worry: i won't let the door hit me on the way out.

little by little, reality sets in:

---

it started a few months ago, when i was told that the department extended an offer to a new postdoc in analysis.

huh. so they found my "successor."

---

about a month ago, one of the departmental staff asked me if i was still going to use my office after the end of finals week.

never mind that he probably does nothing except teach;
what's this about janus doing research?

to my credit, i said that i still had a few projects to finish with faculty, over the summer. the staff didn't bring it up again.

---

yesterday i received an email from university computing services. unless i contact the network admin, my once-university email will be defunct.

i didn't realise that server space is so expensive,
that the university was managing so many users that they can't spare one more?

---

i knew, walking into this job, that it would all be over in 3 years. i didn't expect a reward for services rendered ..

.. but it would have been nice to exit gracefully ..

"words, words, words .. what is the matter ..?"

well, i was right not to predict 20 pages because,
for some reason, the preprint has now reached 25.

[sighs]

there's nothing wrong with 25, but i thought that this would be a short and easy paper .. well, "easy" by my own standards ..

that is: if i proved it,
then it can't be that hard. q:

admittedly, i added a table of contents, which is 3/4's of a page, and the references fill up another entire page.

it's still incomplete, so: back to work ..

this picture better be worth a thousand words ..!

maybe it's good not to have too many goals or make many promises.

there are too many interesting ideas out there,
there's never not enough time,
and every co-author eventually becomes impatient.

in other news, i'm still working on this preprint about measurabΙe differentιable structurεs. i think i've created a monster.

to make sure that the theorem is non-trivially new, i've come up with a new example to which the extant theorems do not apply. explaining it is taking much more time and space than i thought, though ..

.. in fact, it's become a new, separate section of the paper!

conceptually, it's not hard; it's a self-similar construction. in fact, it looks like this:

i'm embarrassed to say how long it took me to make this simple image. at any rate, fractals are definitely the analyst's best friend ..! (-:

EDIT @13:19: i uploaded the wrong graphic; the actual set isn't supposed to be symmetric. argh ..

quietude

it's been a quiet week here. thinking back on it, i should either have taken a few days off completely or worked proper, full days.

it was only some days ago that i realised that teaching forces natural breaks in the week, even the day. that said, i'm not used to working 8 hours at a clip, solely on research. instead i worked for most of the daylight, idled after 6pm. even then there were plenty of breaks.

some people are fine taking complete vacations for a week. maybe grad school trauma is still fresh in my mind, but i still can't do it. the last proper vacation i had, going on holiday with my girlfriend at the time:

we climbed in the mornings, to avoid the heat of the midday sun;
while stretched and napped (for hours!) i took out my notebook, listened to the sound of the waves, and work on math.

admittedly, it was one of the best vacations i ever took. the food wasn't bad, either; only in thailand can you have a decent papaya salad.

[sighs] i guess we mathematicians have quite a reputation ..

.. even from colonial days:

"Thomas Godfrey, a self-taught mathematician, great in his way, and afterwards inventor of what is now called Hadley's Quadrant. But he knew little out of his way, and was not a pleasing companion; as, like most great mathematicians I have met with, he expected universal precision in everything said, or was for ever denying or distinguishing upon trifles, to the disturbances of all conversation .."

~ from the Autobiography of Benjamin Franklin.