ARR!.. imagine computers as research collaborators?

sometimes i wonder if i should be a mathematician at all .. lately i've been reminiscing about the dreams my younger self had: among them was to be a successful novelist, perhaps in science fiction.

so when i read news articles like this one ..

"Some might argue that computers will never be able to match human ingenuity but it is difficult these days to argue they can't at least mimic many of our skills.

Take the eDavid painting robot. The computer-controlled arm - adapted from a welding machine - chooses from five brushes and 24 colours to create impressive artworks on canvas.

It works by snapping a photo of its subject matter and then making the necessary calculations to turn the image into a drawing or painting in a wide variety of styles.

Its creators admit that it has no awareness of what it is doing. But it is able to make decisions about things like shading and brushstrokes as it goes, tweaking its moves based on how the picture is evolving, rather than just creating a pre-determined image."

~ from "The quest to turn computers into creative artists" @bbc_tech

.. then i immediately begin to imagine the possibilities:

if the strength of a computer lies in being very efficient with a finite, fixed set of tools, then imagine if we could get computers to prove simple lemmata for us, just by giving them suitable hypotheses and a fixed set of axioms and existing lemmata ..

yes, it is hard enough to build a robust proof-checking program .. and some researchers have spent years of their lives focusing on a single, specific verification .. but i'm not talking about a universal engine:

i liken it to writing programs that play go or chess well. it's not that the computer can think on its own, but rather that it can traverse through the decision tree of possible games very efficiently. in fact, a competitive program mightn't even run through all the possibilities, but simply the games that grand masters have played before.

so imagine coding in all the basic, rigorous proofs from mathematics (e.g. the proof of the triangle inequality in Euclιdean space) and adding shortcuts into the space of all proofs. i wonder what lemmas a computer could tell me ..?

being a mathematician, sometimes i have mathematical daydreams.

a shot in the dark (but no updates yet: stay tuned)

(yes, it's been a while .. and no, i don't have time yet to write about what's happened since the last two posts ..)


[sighs]

i'm almost convinced that there is no good way to teach a first course in proofs to undergraduates. sometimes i even wonder if it's something that can be "taught" .. in the sense that if the student really wants to learn and to understand, then (s)he has to commit to a minimum amount of time for self-study and development.

it's like teaching someone how to be paranoid: as a skill, it only develops with time, experience, and stimuli ..

*sighs*

ye gods, i hate asking for money.

---

it's clearer to me now that there is a "rat race" to academia in general and to the sciences in particular. more and more i envision a future where i'll never stop writing grants and there will always be another meeting to sit in, another memorandum that i should have read (but have skimmed over, at best).

for a while i've wondered if i was cut out to be a mathematician, but i've made my peace with it now. it's been long enough that i wasn't going to cut it, then i would probably be doing something else by now.

i'm starting to wonder, though, if i'm cut out to be a professional mathematician.

the research is fine and the teaching, though time-consuming, is also fine and often enough fulfilling (if not enjoyable). as for the grants .. and the applications .. and the meetings, and so on;

i can see why many faculty "give up" upon earning tenure.

these professional aspects of the job were never advertised to me, as a ph.d. student; maybe the advisor was deliberately putting it in the background, if only so that we could have a greater focus, when working together. as a postdoc there seemed more and more of it, when discussing the nature of work with my colleagues.

who knows? maybe i've just always been naive;

my colleagues, near and far, seem quite able to maintain research as their primary focus and if anything, shape their other duties to complement this one singular priority. more and more i find this admirable.

maybe i'm just too new to this position, that these are all just growing pains, and that these shall pass with time and enough patience and a little humor. i don't know and it's hard to say.

i'm not giving up. it's just that i can see why others do.

ANH: from end to start, for now.

so it feels like ages since i last thought about a blog post of any kind. it seems like there's so much to saybut at the same time, none of it is really worth mentioning. that's always the difficulty of beginning a story at the beginning ..

.. so, being lazy at the moment, i'll not. i'll begin at the ending instead, which is today.

---

so today i gave a lecture about metric spaces to my students. it's a first course in analysis and the textbook [1] happens to cover the topic, which to me sounds like a license to expound on it for 75 minutes.

so i showed them the discrete metric on any set, and how the unit circle would look if the set were the euclidean plane. i showed them the L-infinity norm, how the unit circle looks like the usual unit square, and how short the proof is for its triangle inequality. this is in contrast to how the proof of the triangle inequality goes for the usual L-2 distance, which uses Cauchy-Schwarz and in turn, a nod to Pythagoreas's theorem.

i thought it was cool. it would be the kind of lecture that would have inspired me as a student .. but i don't know. i'm getting to know the students in my class, but i'm still learning all the time.



[1] we're using baby Rudin.

ARR! more machine now, than man .. twisted and evil.

"On the one hand, today’s computers feature programming and writing tools more powerful than anything available in the twentieth century. But, in a different way, each of these tasks would be much harder: on a modern machine, each man would face a more challenging battle with distraction ... Kafka, Kerouac, and Wozniak had one advantage over us: they worked on machines that did not readily do more than one thing at a time, easily yielding to our conflicting desires. And, while distraction was surely available—say, by reading the newspaper, or chatting with friends—there was a crucial difference. Today’s machines don’t just allow distraction; they promote it. The Web calls us constantly, like a carnival barker, and the machines, instead of keeping us on task, make it easy to get drawn in—and even add their own distractions to the mix. In short: we have built a generation of “distraction machines” that make great feats of concentrated effort harder instead of easier."

~ from "HOW TODAY'S COMPUTERS WEAKEN OUR BRAIN @newyorker "

on how our choices can haunt us later.

if i sit and think about it, then it feels i have a lot to say about the last two weeks, of this new job, at this university.

i don't know where to begin, though;
if i start now, then it will all come out as chaos.

maybe i've been writing too many lectures lately, and habit urges me to put some order or narrative into it. after all, life is simply a sequence of events; any additional order or structure on it is an inherently human contribution.

my guess is that it will take months for me to make sense of it all: these experiences, mistakes, small joys, and frequent setbacks. (i don't know.)

---

as for something small to share ..

.. during the first lecture of multivariable calculus, on a whim i decided to pronounce the letter z as zed, just like how they seem to do in europe and the u.k.

as a result, now i feel compelled to be consistent and remember, from now on, to refer to the vertical axis (in 3 dimensions) as the 'zed-axis" .. or else risk being caught as a pretentious snob!

ARR!.. apparently i still have more trigοnometry to learn.

well, i learned something new today:

It sounds cumbersome now, but doing multiplication by hand requires a lot more operations than addition does. When each operation takes a nontrivial amount of time (and is prone to a nontrivial amount of error), a procedure that lets you convert multiplication into addition is a real time-saver, and it can help increase accuracy.

The secret trig functions, like logarithms, made computations easier. Versine and haversine [1] were used the most often. Near the angle $\theta = 0$, $\cos(\theta)$ is very close to $1$. If you were doing a computation that had $1-\cos(\theta)$ in it, your computation might be ruined if your cosine table didn’t have enough significant figures. To illustrate, the cosine of $5$ degrees is $0.996194698$, and the cosine of $1$ degree is $0.999847695$. The difference $\cos(1^o)-\cos(5^o)$ is $0.003652997$. If you had three significant figures in your cosine table, you would only get 1 significant figure of precision in your answer, due to the leading zeroes in the difference. And a table with only three significant figures of precision would not be able to distinguish between 0 degree and 1 degree angles. In many cases, this wouldn’t matter, but it could be a problem if the errors built up over the course of a computation.

~ from "10 Secret Trig Functions Your Math Teachers Never Taught You" @sciam

in other news: it's been more than two weeks into this new job, and i still feel disoriented. often i feel exhausted, too.

on the bright side: i finally found an expensive apartment and signed a lease .. after a month of searching (and simultaneously teaching, for the last 2 1/2 weeks).

[1] these are defined, respectively, as $\textrm{versin}(\theta) = 1-\cos(\theta)$ and $\textrm{haversin}(\theta) = \frac{1}{2}\textrm{versin}(\theta)$. suggestively, "ha" mean half.

ANH: first day unease.

today [0] i taught two classes, each of which were 75 minutes long. in each i felt like i was saying obvious things [1] and wondered if i was boring the students into a desperation of some kind, that i just stop talking and dismiss class early.

i don't know why, but i hate being boring ... well actually, i do know:

first, i don't want to enforce the stereotype that maths is hard and boring;
on the other hand, it takes a while for me to get anywhere interesting.

a lot of times i struggle with writing lectures because i can't convince myself of really, is this it? come on! there has to be something interesting in this whole topic..!

last night, after deliberating on and off, i finally put some lecture notes down on paper .. at about 1am. then i promptly fell asleep, glad to be rid of the task.

i don't think i succeeded. i don't think i "get" the students yet, and i don't think they "get" me.


[0] that is, on friday: i finished this post later that night.

[1] which, of course, they were .. to me. that's not a statement of arrogance; any seasoned calculus instructor would probably tell you the same.

ANH*: life comes first, then blogging.

.. yes, it's been pretty quiet on this blog. i guess you could say that i've been busy:

i'm teaching two courses this fall,
with plenty of preparations to do;

i'm starting a new job at a new department,
which involves figuring out how things work here;

i'm still trying to find an apartment, which is incredibly frustrating ..!

*sighs*
maybe things will calm down soon.


^* this is the pre-amble that will probably lead to a year-long series, where i'll comment (read: complain) about starting a new academic life in a new department .. this time, as an assistant professor on the tenure track.

as longtime readers of this blog may recall: i get weirded out by the term "professor." in fact, during my postdoc i told my student not to call me that, due to inaccuracies. now that it's part of the job title, i suppose i can't really escape it anymore ..

.. anyway, expect this to read like a "how-NOT-to" type of guide ..!

lastly, for star wars fans it's probably clear that ANH is short for a new hope; for the longest time i thought i'd fade out from academia like obi-wan kenobi, but apparently that didn't work out .. or rather, that did work out.

not quite ARR! numbers and symbols, as viewed by a man of letters.

these are some excerpts from the pleasures and sorrows of work by a. de botton, one of my favorite authors. he has a way of revealing the sublime aspects about everyday life.

anyway, this is his take on science ..

"Gone were the days of geniuses in their observatories and workshops, single-handedly rerouting scientific history. We had entered the sober era of the collaborative laboratory, where astrophysicists and aeronautical engineers banded themselves together for decade-long assaults on minor mysteries, resisting the media's attempts to raise any one of their number into a contemporary Galileo. A company might limit itself to perfecting the performance of silver-zinc batteries in zero-gravity conditions, rightly sensing the foolishness of expanding to address further puzzles in satellite electrics. A scientist might spend a lifetime examining the properties of titanium at high temperatures or the behaviour of hydrogen at the moment of ignition. The sum total of one's contributions to mankind might end up in an issue of the Journal of Advanced Propulsion Methods."

.. and this is his take on maths, viewed from the non-technical viewpoint:

"Noting my puzzlement, Ian told me that he was calculating the force of gravity at work on the cable, and that in his equation $l$ stood for the length of the span, $w$ for the effective weight per unit of length, and $T_H$ for the constant along the line. He explained that transmission engineers were unusually blessed in having at their fingertips a highly precise, efficient and universal vocabulary with which to convey even the most labyrinthine electrical scenarios, so that from Iran to Chile, $\psi$ referred to electric flux, $\mu$ to permeability, $\mathcal{P}$ to pereance, and $\alpha$ to the temperature coefficient of resistance.

I was struck by how impoverished ordinary language can be by contrast, requiring its user to arrange inordinate numbers of words in tottering and unstable piles in order to communicate meanings infinitely more basic than anything related to an electrical network. I found myself wishing that the rest of mankind would follow the engineers' example and agree on a series of symbols which could point incontrovertibly to certain elusive, vaporous, ad often painful psychological states -- a code which might help us to feel less tongue-tied and less lonely, and enable us to resolve arguments with swift and silent exchanges of equations.

There seemed to be no shortage of feelings to which the engineers' brevity might be profitably applied. If only a letter could have been identified, for example, with which elegantly to allude the strange desire one occasionally has to elicit love from people one does not even particularly like ($\beta$, say); or the irritation evoked when acquaintances seem to be more worried about one's illnesses than one is oneself ($\omega$); or the still vaguer sense one can sometimes have that different periods of one's life are in coexistence, so that one would have only to return to one's childhood home to find everything the same as it once was, with no one having died and nothing having changed ($\xi$). Possessed of such a notational system, one would be able to compress the free-floating nostalgia and anxiety fo a typical Sunday afternoon into a single pellucid and unambiguous sequence ($\beta + \omega | \xi \times 2$)" and attract sympathy and compassion from the friends around whom one would otherwise have grunted unhelpfully."

ARR! in which maths could bring about the end of the world (unless algebraιc geοmetry saves us)?

predictions are always hard, especially when it comes to the future. this one, however, concerns actual maths for once:

"Our conclusion is there is a small but definite chance that RSA and classic Diffie-Hellman will not be usable for encryption purposes in four to five years,” said Stamos, referring to the two most commonly used encryption methods.
..
RSA and Diffie-Hellman encryption are both underpinned by a mathematical challenge known as the discrete logarithm problem. That problem is computationally difficult to solve, ensuring that encrypted data can only be decoded quickly with knowledge of the secret key used to encode it in the first place. Breaking RSA or Diffie-Hellman encryption today requires using vast computing resources for significant periods of time.

However, it is possible that algorithms able to solve the discrete logarithm problem quickly could exist. “We rely on that efficient algorithm not being found,” said Jarved Samuel, a cryptographer who works for security consultancy ISEC Partners and presented alongside Stamos. “If it is found the cryptosystem is broken."

~ from "Math Advances Raise the Prospect of an Internet Security Crisis" @mit:techreview

related to this, algebraic geometry might actually be useful for something .. soon, which means that i'll never heard the end of it from a few of my colleagues!

anyway, another excerpt from the article reads:

Stamos called on the security industry to think about how to move away from Diffie-Hellman and RSA, and specifically to use an alternative known as elliptic curve cryptography (ECC), which is significantly younger but relies on more intractable mathematical challenges to secure encrypted data.

The U.S. National Security Agency has for years recommended ECC as the most reliable cryptographic protection available. In 2005 the agency released a toolkit called SuiteB featuring encryption algorithms to be used to protect government information. SuiteB makes use of ECC and eschews RSA and Diffie-Hellman. A classified encryption toolkit, SuiteA, is used internally by the NSA and is also believed to be based on ECC.

ARR! statistics about .. well, mathematics.

i like the way this guy thinks:

"... but I think we're starting to see a new kind of metamathematics, where people use statistical methods to study the structure of mathematics itself.  This is mathematics as actually done by people, so it involves issues of taste and style.  These are subjective things.  But I suspect there are some features of math that are fairly independent of who is doing it.  Maybe some theorems are 'important' in a fairly objective sense - important crossroads that most travelers tend to stop at.  And someday we may understand why."

~ from "The network of mathematics" @johnbaez:g+ (via mathbabe)

reading this reminds me also of the flysρeck project, regarding the use of formal proof:as expounded on their wiki:

" How does a formal proof differ from a traditional mathematical proof?

Traditional mathematical proofs are written in a way to make them easily understood by mathematicians. Routine logical steps are omitted. An enormous amount of context is assumed on the part of the reader. Proofs, especially in topology and geometry, rely on intuitive arguments in situations where a trained mathematician would be capable of translating those intuitive arguments into a more rigorous argument.

In a formal proof, all the intermediate logical steps are supplied. No appeal is made to intuition, even if the translation from intuition to logic is routine. Thus, a formal proof is less intuitive, and yet less susceptible to logical errors."

the way i understand it, traditional proofs are like pseudocode whereas formal proofs would be real computer programs that you can "compile" against the standard axioms .. although it would probably resemble machine code.

i think this comparison also highlights their comparative advantages nicely:

the formal proof mightn't be readable, but at least you know it runs and witness how it does ..

.. whereas getting the idea for the formal proof would probably require some basic principles for why it could conceivably work, in which case one would probably have a traditional proof in mind.

going back to the idea of a network of all mathematics, i suppose that a formal proof would be checking the existence of a continuum from a statement (a given node on that network) to the fundamental axioms (or "roots" of the network).

ye gods: that would be a really complicated network!

disruptions, distractions.

so i was having dinner with my sister [0] when she asked me what i was thinking about lately. (i took this to mean mathematics.)

at first i couldn't think of anything, which was really weird,
especially since i was working on something that very morning;

i guess i've been really distracted lately.

---

to put things into perspective, the last five posts ..

(namely this, that, also this and this, and that too)

.. were dead-man's switches .. at least in erdös's sense [1]. they were compiled about a week in advance. i was away on the spur-of-the-moment holiday because .. well, the girl i'm dating was on holiday and we wanted to spend more time together. [2]

at first i thought i'd not bring any maths with me, if only as a romantic gesture .. but then i looked up my flight and it was 3 hours long: a long time, and i didn't have anything in mind to read at the time [e];

well, a laptop's not that heavy, and neither is that manuscript ..

so on the flight to rome, i idly decided to edit my $\LaTeX$. i sorted out one lemma, then another, and then in the middle of another proof, something didn't make sense.

closing the laptop, i tried to rework the proof in my mind .. only to perceive a gap. there was only an hour left in the flight to think of a patch before she and i would meet up again in the airport baggage claim .. and i started panicking ..

because the manuscript is supposed to be ready already, dammit! argh!

thus the plane touched down, decelerated [3], and arrived to its gate while i struggled without any viable ideas. dejected, i re-activated my cell phone, shuffled out of the plane and into another unfamiliar airport, and wondered ..

what kind of mathematician makes that kind of mistake, misses a gap like that?

sighing, i looked for the right baggage carousel from my flight ..

this would prove to be a distracted trip; in the mornings before we got up, i'd fumble with details in my head and feign sleepiness when she asked what was wrong.

.. to be continued?

[0] i.e. a mathematical sibling.

[1] according to this article and other accounts i've read, erdös would consider a friend "dead" if he stopped working on mathematics, so the term "dead man's switch" isn't that far off. (i wonder, then, if erdös considered "death" to be a permanent condition!)

[2] i'm leaving finland in 2 weeks. she and i haven't spoken about the future yet .. at least in much detail. suffice it to say that it's a rather turbulent time in my life. i'm pleasantly surprised that i can focus on anything mathematical, at the moment!

[e] in case you were wondering, she and i were on separate flights. now that i think about it, i don't know if she's aware of how much and often i work. (personally i don't think i work all that much, but experience with family and previous girlfriends seems to suggest otherwise.)

on an epilogical note: the laptop did help after all. i had merely a dumbphone and her smartphone wasn't connecting properly to the hotel wifi, so it came in handy with looking up various cool locations to visit.

[3] "de-celerate" shouldn't be an actual word. acceleration, being a directional quantity, can take on both negative and positive values.

ARR! accounting for asymmetry .. socially?

i'm not completely convinced of this study, partly because there is no discussion of how the researchers made their observations and recorded their data:

"The brain, Cacioppo demonstrated, reacts more strongly to stimuli it deems negative. There is a greater surge in electrical activity. Thus, our attitudes are more heavily influenced by downbeat news than good news.
..
Here's the tricky part. Because of the disproportionate weight of the negative, balance does not mean a 50-50 equilibrium. Researchers have carefully charted the amount of time couples spend fighting vs. interacting positively. And they have found that a very specific ratio exists between the amount of positivity and negativity required to make married life satisfying to both partners.

That magic ratio is five to one. As long as there was five times as much positive feeling and interaction between husband and wife as there was negative, researchers found, the marriage was likely to be stable over time. In contrast, those couples who were heading for divorce were doing far too little on the positive side to compensate for the growing negativity between them."

~ from "Our Brain's Negative Bias" @psytoday

i mean, how does a researcher get access to a couple's daily life so that they can objectively measure how much time they spend fighting? (this is not to say that the study is bogus, but only points out how little i know about how to conduct social research.)

a particular MOOC that's been coming up in my feeds.

i'm curious to see how this particular course on mathematical philosophy unfolds. at first glance it reads like a discrete mathematics course that i took for a computer science major^* but with more "relevant" and "edgy" material ..

such as the mοnty hall problem and ) arrow's voting paradox (also called an "impossibility theorem").^**

another interesting aspect is the topics in game theory, such as the von Neumann-Morgenstern axioms.

looking through the syllabus, i would rather call this a "mathematics literacy" course. the title of 'mathematical philosophy' suggests it as part of philosophy of mathematics, from whose recurrent themes strikes me as a wholly different thing!


* not that i ever completed the major. it was my final semester for my undergraduate degree, i had a 3-credit course left called "operating systems," but by all accounts i would have had to spend 20 hours per week, just coding. i decided to take a topics in ΡDEs course which, in the long run, was a wiser and more useful choice. (-:

** it's funny how we become aware of certain facts. i actually learned of the latter result from an acquaintance, while couchsurfing .. about 9 years ago? similarly, the vNM axioms came up when one of my students from a proofs class wanted to write his term paper on economics and game theory.

on writing: when you can no longer trust yourself ..?

// from sunday, 28 july 2013 @12:19 EEST:

currently i'm editing this one manuscript for the .. who knows: 27^th? 41^st?.. $n$^th time and it's getting to the point where it's hard to spot errors, however significant or small. i blame this on the general principle that the more exposure i have to this thing, the more familiar and not-out-of-the-ordinary it seems. (in other words, i've lived with this thing long enough that maybe, unconsciously, i believe it to be true.)

so i've decided on a different tack, which hardly seems efficient. i don't know any other way, though [1].

---

each time i think about proofreading the manuscript again, (1) i pick a random lemma, (2) i take out a few blank pages and try to prove it on my own, and (3) becoming aware of what is tricky about it, i focus on those parts in the writeup ..

.. and inevitably find gaps. they're getting smaller and less significant, though, so maybe it's some measure of progress?

at any rate, i had in mind (perhaps too optimistically) to send the manuscript to a big shot in the field today; let's see if i can actually keep to my word!

epilogue (29 july 2013): i didn't keep my word.
on the other hand, i found another gap and patched it.


[1] i've been given the advice before to read every line and check if it is true. in principle it's the best way to guarantee that a manuscript is as correct and consistent as possible, but i always fall short (and by a big margin): i guess i'm just that impatient .. \-:

a template .. of a slightly unusual kind.

disclaimer: as usual, this has nothing to do with actual maths.

recently, every work email i write begins with:
".. thanks for writing, and apologies for the late reply .."

i don't know why, but it always takes me a week to reply to anything ..

ARR! point & counterpoint.

so i've been reading about MOOCS again. here are a few excerpts i found:

---

"What you can do over the Internet this way is deliver information, but that's not education. Education, as any real teacher will tell you, involves more than just transmitting facts. It means teaching students what to do with those facts, as well as the skills they need to go out and learn new information themselves."

(as pointed out in "The MOOC Racket" @slate)

".. are we sure the only way to teach people what to do with facts is face-to-face? This seems like something that could at least conceivably be taught to more than one person at once. I can remember lots of professors teaching me what to do with facts via lectures in extremely large auditoriums, which is not that different than a lecture you watch online."

(a counterpoint via College Professors Are About to Get Really Mad at President Οbama @nymag)

---

ye gods, this issue is confusing, especially when one accounts for the perspectives of the given pundits. for one thing ..

.. the first point comes from a university professor, who has probably developed an expertise in little-known fields (or at least poorly popularised) over years of study in academia. for him, relevant professional information typically arrives through academic channels and processing the information is a careful, length process of some depth. (think of the peer review process: ouch!)

the second point comes from a editor/journalist who has developed a different expertise in a widely-recognised occupation, probably by way of on the job training and less formal study [1]. relevant professional information probably comes through many diverse channels and rapidly so; the process of response probably requires similar speed (in order to remain relevant).

there are also tacit yet important questions here:

for a young adult, is college necessary for a future successful career?
if so, then what should (s)he learn at university?

honestly, i have no idea. there are too many types of careers out there for a simple answer. the issue gets even murkier when you account for advances in technology, even at the scale of a generation or two.

for example, it seems that there is a lack of available workers in the skilled trades, and the current infrastructure of civilization relies crucially on the fruits of their labor.

on the other hand, what if 3-D printing becomes robust enough, and available through a sufficiently diverse selection of materials, so that plumbing, welding, and soldering no longer require the work of human hands?

this sounds like science fiction, of course. i'll not discuss the likelihoods of certain events occurring .. mostly because i cannot even guess, much less quantify the time-dependent sample space of modern civilisation.

on the other hand, i would like to point out that they are real possibilities: take, for example, the history of the Luddites or how human computers were replaced by digital ones. now that i think about it, i wonder how many more travel agents there are nowadays, with the popularity of flight search engines and all ..? [2]

at any rate, the main problem is that we don't know what "workers of the future" need to know how to do, because many of those future jobs don't exist yet. (explain, for example, the notion of a web developer to someone in the 1970s.) at best we can only make decisions about how to help young adults now, with well-defined criteria ..

.. such as economic ones, i.e. whether they should be obligated to put themselves into tens of thousands of dollars in debt before the age of 30?



[1] this is not to say that the second writer knows any less than the first, nor is he any worse at his job. honestly, you cannot compare such experiences. if the second writer is a success, then i would guess that has to do with a lot of deliberate and systematic effort on his part. he may even have studied many journalists he has admired, read very carefully their work and took notes, which is clearly a kind of study, but not the formal kind you see in universities.

if this is his approach, then i applaud the guy. deliberate practice of this kind, regardless of the circumstance, is often necessary to succeed in many areas in life.

[2] actually, there may in fact be more travel agents than ever before. travel for pleasure has become more and more accessible; on the other hand, there is still a large population out there who cannot (or will not) deal with a computer .. or even afford a computer or high-speed internet. \-:

well, how big is $n$ again?

as found off a friend's feed:

courtesy of phdcomics (so don't sue).

(deceptive) cadence ..♫

you know, there is something very soothing about coding in $\LaTeX$ .. provided that you already know what you're putting into code, that is:

it's like being in the zone, letting the fingers go on autopilot. (i guess i mean "flow" in the sense of psychology.)

---

on a related note, maybe i was judging too quickly or harshly earlier.

so far, this and last afternoon have been rather productive. on the other hand, my typing on a computer doesn't necessarily imply that i am getting any work done.

from experience, $\LaTeX$ is as aesthetically pleasing as it is deceptive!


♫ the title was stolen from an NΡR classical music blog of the same name, which i recommend. (-:

quick post: round&round, and other visualisations (*UPDATED*)

apparently, this is π;
(courtesy of mkweb).


a realisation of a strange attractor
(courtesy of chaotic atmospheres)



a somewhat self-similar sculpture by carl jara (via thisiscolossal)



i immediately wondered what the π₁ of this sculpture was ..(-:
(art by henrique oliveira, via thisiscolossal)


lastly, something more "real" ..?


this is apparently a visualisation of what wifi networks would look like.
(photos by nickolay lamm @mydeals)

"it is easy to be logical; it is almost impossible to be logical to the bitter end" ..▲

today i drew a paper calendar of the next four weeks,
filled in various plans and obligations, both professional and personal,
and came to a single conclusion;

if everything is to work out,
then this week has to be mathematically unproductive. )-:



^▲ this quote is taken from A. Camus's the myth of sisyphus, an excellent book.

ARR! ... YEAH! (or: spot on!)

finally!

"Alan Turing, the Enigma codebreaker who took his own life after being convicted of gross indecency under anti-homosexuality legislation, is to be given a posthumous pardon.

The government signalled on Friday that it is prepared to support a backbench bill that would pardon Turing, who died from cyanide poisoning at the age of 41 in 1954 after he was subjected to "chemical castration".

~ from "Enigma codebreaker Alan Turing to be given posthumous pardon" @guardian

then again, i don't know how sure of a bet this is. anyone know what a third reading is?

"Lord Ahmad of Wimbledon, a government whip, told peers that the government would table the third reading of the Alan Turing (statutory pardon) bill at the end of October if no amendments are made. "If nobody tables an amendment to this bill, its supporters can be assured that it will have speedy passage to the House of Commons," Ahmad said."

---

on a (barely) related note, i've been posting a blitz of ARR! posts in the last week and a half. this blog was never meant to be an aggregate maths news website, so i apologise if the content lately has seemed rather .. commercial(?) in nature.

related to this: i've recently experienced a few notable changes in my life ..

.. with more to come, like moving across the ocean ..

.. so my focus isn't 100% on maths, at the moment. to be honest, it makes me feel like i'm not a "real mathematician," even though that sounds ridiculous. (it just goes to show you how insidious "impostor syndrome" can be.)

somewhat oppositely:

ten years ago i entered a ph.d. program. five years ago, i defended a dissertation. it's been a long enough time and i've dodged enough bullets that, perhaps, it's safe for me to consider myself a mathematician and that i'll be around for the long haul, after all.

my point is that research life has settled down and i know that, if i work hard enough, i am capable of good work. as a result, the level of "mathematical drama" in my life has been toned down.

i guess this is a round-about way of saying:

yes, my life is boring now, compared to my student days ..
but i like it that way! (-:

irrelevant, but ..

odd: i cannot seem to spell square correctly, today. every time i start typing it out, my fingers move to the keys

..s ..e ..q ..u ..e ..?

maybe my unconscious is trying to tell me something, via psychography?..

.. that i should take a sequence, perhaps a bounded one,
and choose a convergent subsequence ..? (-;

ARR! the good fight.

first, the student side: this is so cool. the idealist in me wishes for all education to be free, and in some places it's still possible;

it's nice to see that part of that might still be kept alive.

"A unique aspect of the Cooper Union case is that several of the students fighting for the cause have already graduated, and remaining undergraduates still won’t have to pay tuition while they’re students. The reason the students have been sleeping in the president’s office for the past two months is because they fear what will happen to the school after they leave, how a decision to charge tuition might affect the character of the incoming classes and the direction of their alma mater."

~from "Can Cooper Union Find A Way To Continue Free Tuition And Its Social Mission?" @fastco

now, the faculty: when i was in the middle of my ph.d. i recalled a fellow student say the worse the teacher, the more students have to work and the more they have to learn. this was in response to seeing how neurotically i was writing my lectures, so that my students could learn in the most effective way possible.

thinking about it now, i must have been the calculus version of a helicopter parent ..! \-:

anyway, that came to mind when i read the following excerpt. apparently some students do react that way (in extreme cases).

"I had a teacher in college whose lectures were so incredibly clear that it made me think physics was the easiest thing in the world. Until I went home and tried to do the problem set. He was truly amazing, but sometimes I think he was TOO good. I didn't struggle to understand his lectures--but maybe I should have."

~ from "Do the Best Professors Get the Worst Ratings?" @psytoday

there's more:

"When you measure performance in the courses the professors taught (i.e., how intro students did in intro), the less experienced and less qualified professors produced the best performance. They also got the highest student evaluation scores. But more experienced and qualified professors' students did best in follow-on courses (i.e., their intro students did best in advanced classes).
..
To summarize the findings: because they didn't teach to the test, the professors who instilled the deepest learning in their students came out looking the worst in terms of student evaluations and initial exam performance."

on a related note: over the course of my career thus far, i've learned from various occasions to be wary of those instructors whose evaluation scores are too high, almost perfect.

lauding your first-year mathematics instructors is a little like thinking that your parents did a perfect job raising you while you are being raised through childhood. i don't deny that students have good, discerning taste .. but it's hard to accurately judge how someone is conveying lessons to you when you don't completely understand the lessons in question!

not quite ARR!.. but a logarithmic spiral, embedded in spacetime.

photo by arvin rahimzadeh (@ thisiscolossal)

ARR! muscles in polar coordinates ..

.. or, should i say, cγlindrical cοordinates?

"One of the major discoveries that David Williams brought to light is that force is generated in multiple directions, not just along the long axis of muscle as everyone thinks, but also in the radial direction.
..
The basics of how a muscle generates power remain the same: Filaments of myosin tugging on filaments of actin shorten, or contract, the muscle – but the power doesn’t just come from what’s happening straight up and down the length of the muscle, as has been assumed for 50 years. Instead, University of Washington-led research shows that as muscles bulge, the filaments are drawn apart from each other, the myosin tugs at sharper angles over greater distances, and it’s that action that deserves credit for half the change in muscle force scientists have been measuring.

~ from Biceps bulge, calves curve, 50-year-old assumptions muscled aside @u-dub

courtesy of the university of washington (so don't sue)

despite this being a mechanical and biological process, the funny thing is that the totals are apparently hard to measure:

"“The ability to model in three dimensions and separate the effects of changes in lattice spacing from changes in muscle length wouldn’t even have been possible without the advent of cloud computing in the last 10 years, because it takes ridiculous amounts of computational resources,” Williams said."

it makes me wonder: what basic but subtle aspects of nature have we been missing, all this time?

thoughts, while idle: clarκe's descriptions of geometries.

so i took this past weekend off, joining friends and a special someone to idle away the days in a beach resort town. it was good fun, though i believe now that the idea of a beach is often more attractive than the actual experience.

during those sun-soaked days, i couldn't really concentrate on anything. i tried a little, but having friends around all the time turned even those peaceful periods an hour or two into unusable blocks.

then again, it's a holiday; the point is not to be working!

i did start rendezvοus with rama by arτhur c. clarκe, which made for good episodic reading; the story stretches its telling ..

.. at least until page 100, where i stopped last night..

.. and the plot isn't too intricate that you have to pay attention all the time.

i make it sound simple, but it isn't. the beauty of this book is how it points out the subtlety of human perception towards mathematical possibility.

to give you an idea, the back cover of this book reads:

Rama is a vast alien spacecraft which enters our Solar System. A perfect cylinder some fifty kilometres long, spinning rapidly, Rama is a technological marvel, a mysterious and deeply enigmatic alien artifact. It is Mankind's first visitor from the stars and must be investigated ...

yes, we all know what a cylinder is and we have all (probably) driven as far as 50km on the highway. combining those two bits of information together, however, must make for a rather unique sight. even the large hadrοn collider at CERN is only 27km in circumference, and i don't know if anyone has ever caught the entire thing within a single frame of vision ..!

a view of the LHC, courtesy of alienshift

anyway, here is an excerpt from p. 74:

"Because they were now standing on the edge of a fifty-metre cliff, it was possible for the first time to appreciate the curvature of Rama. But no one had ever seen a frozen lake bent upwards into a cylindrical surface; that was distinctly unsettling, and the eye did its best to find some other interpretation. It seemed to Dr Ernst, who had once made a study of visual illusions, that half the time she was really looking at a horizontally curving bay, not a surface that soared up into the sky. It required a deliberate effort of will to accept the fantastic truth."

moreover, clarκe does an excellent job of suggesting that, though space and its contents may be absolute, our perspectives of it can be rather relative:

from p. 42:

"And now, Karl Mercer told himself, I have to make my first decision. Am I going up that ladder, or down it?

The question was not a trivial one. They were still essentially in zero gravity, and the brain could select any reference system it pleased. By a simple effort of will, Mercer could convince himself that he was looking out across a horizontal plain [1], or up the face of a vertical wall, or over the edge of a sheer cliff. Not a few astronauts had experienced grave psychological problems by choosing the wrong coordinates when they started on a complicated job."

[1] the first two times i read this paragraph, i thought that the word was 'plane' and not 'plain.' (-:

the how is done, but what about the why ..?

so i've been working systematically on the same problem all this week, and i think i've come up with a good proof. in that sense, it's been a satisfying week.

this morning, though, i looked at the result that the proof implies .. and debated whether it is worth publishing.

to its credit, the topic is mainly about fractals but not exactly the self-similar kind. nevertheless the diagrams should be pretty to look at.

it's not too technical either. most of the work lies in building the right lιpschitz functions, actually.

then again, it's about these objects called (metrιc) derivatiοns that come up in analysιs and geοmetry of metrιc-measure spaces. i've been working with these things for a while, but my feeling is that few people care about them .. or about metrιc spaces in general.

maybe i should advertise it as a gmτ result, of some kind. one corollary is that certain kinds of fractals cannot arise as flat chaιns (in the sense of whitηey), yet their weak tangeηts are flat.

i don't know how interesting that is in gmτ, though;

maybe i should just shelve the result for now, and think about something else. for one thing, i promised one newly-met colleague that i'll think about systems of ρde's, next week!

à la pοincaré: the 4-hour day.

tuesday, 9 july 2013 @ 11:44 EEST:

my new plan: i'll concentrate on being as productive as i can, but specifically during the hours that i know that i am usually productive (even if it means that i am working far fewer hours than i usually do).

this includes first thing in the morning ..

today was already productive:
i constructed a whole class of counter-examples!

.. and probably late afternoon into early evening, just before i usually go running or join friends for climbing, actually.

odd: why would the "bookend" times be the most productive?

maybe i should also keep a "journal" of what i did during those hours, and how long it took? this is in reaction to how ineffective my to-do lists are: they are always overloaded and i can never check everything off, which eventually gets depressing.

the tricky thing is how to be productive in the hours in between. one possibility is that if i'm going to waste time, then i may as well enjoy myself, like reading novels or blogging or taking photos of the city, but another possibility is to fill it with drudgery, such as writing a review for this article that is 4 months late .. etc.

on the other hand, maybe a few hours of real work is enough; i mean if it was good enough for poincare .. [0]

---

wednesday, 10 july 2013 @ 08:38 EEST:

ups and downs again:

two days ago i knew what worked .. or at least, i thought i did;

yesterday i was convinced that a colleague's "conjecture" [1] is false and even had a rough proof in mind,

but just now i found a gap in my argument,
so i'll spend today working on a patch.

despite having been privately wrong [2] this has been a pretty exciting week, mathematically speaking. the uncertainty of it all is .. not intoxicating, but like a high. not knowing what exactly is true is all the more motivation to figuring out what it could be .. especially if it keeps on switching, due to increased effort.

so work is fun again and i can obsess about it with new abandon.



[0] to be fair, i wonder what poincare would consider "work." my guess is that the man had high standards, so a lot of things he wouldn't consider work, i would!

[1] i like to reserve the word 'conjecture' for those unproven claims that are particularly nontrivial .. and no, i don't have a definition of nontrivial, but just a sufficient condition: if, by telling someone related to your field of the basic notions, by stating the problem, and by showing why it is hard or what it would imply, that that someone become interested, then it's nontrivial.

[2] 'privately wrong' is doing something stupid in the privacy of your own workspace. 'publicly wrong' is having many people know about it. i've been both, and the former is very, very much preferable to the latter.

ARR! strange occurrences in science

// initially written: last week (or before)

ok: so what exactly is a measurement, then?

"Just like position and momentum, quantum theory predicts that the polarization along two different axes cannot simultaneously be known with certainty (see Nature). The team adopted a strategy in which the polarization is initially probed using a series of ‘weak’ measurements — detections that barely disturb the system but must be repeated several times to record the same information that a single ‘strong’ measurement can detect. They found that, on average, the polarization measurements disturbed the system by only about half as much as Heisenberg’s original formulation of the uncertainty principle dictates."

~ from "Proof mooted for quantum uncertainty" @nature

speaking of science, maybe this explanation of mercury's liquid state is well-known .. but admittedly, i was ignorant of it.

"Relativity states that objects get heavier the faster they move. In atoms, the velocity of the innermost electrons is related to the nuclear charge. The larger the nucleus gets the greater the electrostatic attraction and the faster the electrons have to move to avoid falling into it. So, as you go down the periodic table these 1s electrons get faster and faster, and therefore heavier, causing the radius of the atom to shrink. This stabilises some orbitals, which also have a relativistic nature of their own, while destabilising others. This interplay means that for heavy elements like mercury and gold, the outer electrons are stabilised. In mercury’s case, instead of forming bonds between neighbouring mercury atoms, the electrons stay associated with their own nuclei, and weaker interatomic forces such as van der Waals bonds hold the atoms together."

~ from "Relativity behind mercury's liquidity" @rsc

---

lastly .. judging from how i wrote this post, it's safe to assume that those monday roundups aren't returning anytime soon.

let me clarify: i still think that rounding up articles is a good idea, but a weekly time constraint feels slightly artificial to me. i'd much rather collect items with a common theme and let the ideas percolate into something coherent. (i don't know if i'll do this weekly; it really depends on how much interesting stuff appears on the blogosphere.)

that said, i'm open to taking requests;
if you want my opinion on a topical article, then send me the link.

that said, the MoAR label will become just ARR!, where AR refers to ARticle, R to Roundup, and ! to indicate that i have an opinion!

off the cuff: snippets before the weekend.

// snippets from thursday, 4 july 2013

i'm trying to unlock a door, figure out blueprints for objects that, in their full generality and obscurity, requires believing in statements like the axiοm of choice and their implications [1]. for the right skeleton i've recast someone else's conjecture for bones and soft cartiliage.

i have examples. i know that these things exist, at least in crude forms, but how sophisticated must the designs really be?

---

i don't know if unlock or build is the right word, mathematically speaking. i don't know what the right word could be.

all i know is that i'm stumped at the mathematics, and all i can do now is discuss the frustration of those mathematics.

[1] apparently "axiom of choice" is also the name of a band.

unfocused .. (updated)

maybe i should take a vacation: a short one, maybe a week.

i haven't been able to focus for a while. today i was in the office and rather than sticking to a list of things that i really should do, i spun the same ideas on the same problem that i've failed to solve for .. well, years.

..

it's one of those transitional times in my life again:

1. in six weeks i'll start a new job in the states, one that could be for life .. that is, if they like me enough;

2. i recently solved an open problem that's plagued me from the latter years of my ph.d. it was the kind of problem that if i had a spare day, then i'd just attack it with any idea i had .. even if it meant that the day would be lost to compulsion, folly, and frustration.

so i can't seem to get excited about any new projects. there's plenty of things to do, of course, but .. i can't convince myself to do anything.

sometimes i'll catch myself staring through the window, not thinking about anything in particular.

// added 2 july 2013 @ 03:46 EST

so perhaps i spoke too soon:

when i woke up today and made coffee, i decided not to attack the problem again but instead, work out a related formula that has stumped me before.

it turns out that it follows pretty easily from basic principles .. at least, easily after a good night's sleep (and/or subconscious hacking).

now that i think about it: it would make sense that i had proven it before, forgotten it, and just now re-discovered it. i don't know, i can't remember, and my notes are not archived well enough for me to check it readily.

[shrugs]

well, at least it's true. it doesn't solve that aforementioned problem, of course, but it does narrow the gap so that there are fewer things to try.

..

related to all of this, i keep forgetting about the happy clarity that mornings can provide. let me make this clear: though it might be nice to be part of the 5am club, i'm happy enough enjoying the evenings and getting a few hours in the morning before heading to the office. (-:

sometimes maths is just about numbers.

well, i've been dating this girl ..

---

this morning we were talking about work. she happened to have a few blueprints(?) in her bag of what her firm was doing and how many simultaneous processes the proposed factory would be using.

it seemed very complicated to me .. like an electric circuit diagram on steroids!

then an idea came to mind .. maybe it can be my turn! .. so on her computer i showed her one of my preprints. her brow furrowed.

the basic idea's isn't that bad, i tell her.

for things that behave like mass distributions, the small-scale behavior can sometimes determine the large-scale. if, for example, there aren't enough directions when you zoom in at any given point, then the mass has to be distributed either like dust [1] or be restricted to fewer dimensions, like a line in space.

then there was a brief pause.

wait, she said, it takes 27 pages to explain that?!?

i shrugged.
well, the latest version is now 29 pages [2].

we then talked about how, at a certain point, maths no longer consists of calculating things .. not exactly, anyway.


[1] i.e. a fractal with non-integer hausdοrff dimensiοn, but not necessarily self-simιlar. at the time i was thinking about purely unrectifιable subsets of the plane.

[2] thinking about it, maybe i should have said something like: well, define "dust" ..?

*NOTE* the new label ½ × 2 is meant to tag those (few) posts which are about how mathematics, for inexplicable reasons, relates to my relationships. the idea is: once you've been with someone long enough, you are no longer 1 person. rather, you are one-half of two people. (-:

backlog.

i think i must be the world's worst research collaborator. despite good intentions, i never seem to get back to my co-authors in any reasonable time-frame.

[sighs]

also, forget that bit about a lack of hurry;
i'm starting to think that i have too many projects.

MoAR #25: inter alia, the human kind of asymmetry.

well, it's been 25 of these roundups. my guess, as before, is that you readers can probably do your own news searches, and with greater effectiveness to fit your own tastes.

that said, i'll still be posting articles i find interesting, but just not in roundup form anymore.

---

1. not the isomorphism that i expected, but ..

.. apparently some processes are universal, whether organic or digital:

"On the surface, ants and the Internet don't seem to have much in common. But two Stanford researchers have discovered that a species of harvester ants determine how many foragers to send out of the nest in much the same way that Internet protocols discover how much bandwidth is available for the transfer of data."

~ from "Stanford researchers discover the 'anternet'" @stanford.edu

---

2. the 'flame' kind of war

when i read this excerpt, the first thing which came to mind was:

why is this even a question? we haven't stopped teaching spelling, grammar, and vocabulary to students, just because they now have word processors, right?

The standard algorithms should be avoided because, reformists claim, mastering them is a merely mechanical exercise that threatens individual growth. The idea is that competence with algorithms can be substituted for by the use of calculators, and reformists often call for training students in the use of calculators as early as first or second grade.
..
That the use of standard algorithms isn’t merely mechanical is not by itself a reason to teach them. It is important to teach them because, as we already noted, they are also the most elegant and powerful methods for specific operations. This means that they are our best representations of connections among mathematical concepts.

~ from "The Faulty Logic of the ‘Math Wars’" @nyt

admittedly, though, a lot of times i just opine when i feel like it, and drop things when i don't ..

These professors maintain that college-level work requires ready and effortless competence with the standard algorithms and that the student who needs to ponder fractions — or is dependent on a calculator — is simply not prepared for college math. They express outrage and bafflement that so much American math education policy is set by people with no special knowledge of the discipline.

to be honest, i don't know who really is in charge of mathematics educational policy, but i strongly suspect that most college faculty opt out of any involvement with it. call it a professional opportunity cost: unless you rake in a lot of external funding from it, there's not a lot of incentive to work on educational problems instead of research ones.

---

3. wait .. what?

there are plenty of shocking items of news in the world lately, like:

This year, a pilot scheme was introduced to enforce the rules.
..
When students at the No.3 high school in Zhongxiang arrived to sit their exams this month, they were dismayed to find that they would be supervised by 54 randomly selected external invigilators.
..
By late afternoon, the invigilators were trapped as students pelted the windows with rocks. Outside, more than 2000 people had gathered, smashing cars and chanting: ''We want fairness. There is no fairness if you do not let us cheat.''
..
The protesters claim cheating is endemic in China and that sitting the exams without help puts their children at a disadvantage.

~ from "Chinese students and families fight for the right to cheat their exams" @smh

the last excerpt warrants interpretation; i think the protest is specific to why their city was chosen for enforcement, instead of other cities. the article goes on to point out that last year, the education department received 99 identical exam papers .. but then again, there's no comparison as to how other cities' stats stack up.

imagine, for example, if other cities had been recorded at 500+ identical copies.

considering the population of certain cities in china, that's no longer a large number of students;
as a result, without additional data i wouldn't rule out political favoritism yet ..

nonetheless, this kind of news is alarming. i've heard that eastern culture favors the group over the individual, and that the socioeconomic inequity in india and china is huge, but .. really?

well, moving towards a more abstract direction ..

The researchers are trying to avoid a situation where we outsmart ourselves, and create a system that can in turn invent its own technologies, which could "steamroll" humanity – not because it’s evil, but simply because we couldn’t foresee the long-term ramifications of how we programmed it.
..
"Think how it might be to compete for resources with the dominant species," Price says. "Take gorillas, for example – the reason they’re becoming extinct isn’t because humans are actively hostile towards them, but because we control the environment in a way that suits us, but is detrimental to their survival.

~ from "The men trying to save us from the machines" @pcpro

considering how well we understand turιng machines yet still face computer crashes regularly, in this day and age, this might not be a bad approach in general. it would be rather .. depressing if all of humanity were swept away, due to a computer glitch!

to a lesser, more realistic extent: the access and operations for my savings and checking accounts are probably automated within some banking computer system. if enough errors pile up, then .. [cringes].

---

4. to humans, does time lack symmetry?

i would have thought that we humans are good at accounting for symmetry. then again, i'd probably also fall prey to this ..

But when asked to predict what their personalities and tastes would be like in 10 years, people of all ages consistently played down the potential changes ahead.
..
Thus, the typical 20-year-old woman’s predictions for her next decade were not nearly as radical as the typical 30-year-old woman’s recollection of how much she had changed in her 20s. This sort of discrepancy persisted among respondents all the way into their 60s.

~ from "Why You Won’t Be the Person You Expect to Be" @nyt

the main mechanism might be memory, which resides clearly in the past, not the present, and is arguably inherently faulty.

anyway, towards a more familiar setting .. here's a thought that makes sense, but never came to mind when i was teaching:

You would think that since you have been a student and survived you would be able to recognize their misconceptions and guide them to enlightenment. But even if you could remember what it was like to be a student, that moment for you was characterized by a similar hit-and-run-don’t-leave-a-calling-card confusion.

Everybody gets hit by the bus, everybody gets knocked out, everybody survives, but nobody remembers what they got hit by. So you can’t tell them what to watch out for. Teachers can’t understand their students’ confusion even though they once experienced the exact same confusion!

~ from "Teach Like You Don’t Know" @desmondrawls

a good point, but the comparison (i.e. hit-&-run) as well as the phenomenon seems a bit forced. it's not like the act of learning is some version of achilles and the tortoise, right!

maybe the point is the instability of learning. everything is fine for students, as long as they're following diligently and carefully .. but once they trip up, their confusion can be utter confusion. /-:

---

5. lastly, a bit about online education.

i know that i've ranted on and on about MOOCs a lot, but this is the first article that mentions any behind-the-scenes kind of details:

So, what hasn’t gone as planned? Certainly some things do not translate from a traditional classroom course to a MOOC. Our team realized quickly that we needed to do a better job cross-linking material on the course site. For example, if we mention the syllabus, we must link to it. Some students, we have learned, want a great deal of guidance.
..
We also underestimated the misunderstandings that can arise from idiomatic and discipline-specific language. We began the course by asking students to complete a Personal Benchmark Statement, only to discover that we needed to provide a definition of “benchmark.” A longer glossary of terms became a featured part of our site.

~ from "Inside a MOOC in Progress" @the_chronicle

interesting: in terms of the latter gaffe, in a multivariable calculus class it once took me a week to realise that i'd been calling a $\partial$ a 'del' without actually having told the class of the pronunciation.

(for the record, it was one very brave student who asked in the middle of class; subsequently a collective 'oh' erupted through the lecture hall.)

on the lack of hurry lately.

from: saturday, 22 june 2013

all i did today was make coffee, read a little, and sort out photos from yesterday. i didn't do any maths.

today is midsummer's day (or juhannus) and yesterday was midsummer's eve. both are official holidays in finland and arguably the most important ones. celebrating it has the same, easy feeling as thanksgiving in the u.s.: you don't have to believe in anything, other than the fact that the longest day of the year is upon us, and one observes it during the closest weekend to it.

---

i used to work through holidays or, at the very least, try to do so as much as possible .. where "possible" depended on whose company i kept. living in finland these last two years, though, has tempered that desperate feeling.

i'd like to think that i'm a tireless, relentless researcher, but that's hardly true. for instance, i found out that there's only so much time in a single day that i can spend thinking, reading, attacking a certain line of thought.

the truly productive hours are preciously few, in fact: usually two hours in the morning and two in the late afternoon or early evening. maybe the former is fueled by an early surge of coffee and dopamine, the latter by the mere suggestion that the end of the day's work is near and sh-t, i'd better get something done! [1].

it is hard to ignore the regular workday, even if one isn't beholden to it. during the academic year the department is often quite empty by 4pm, if only because that's when kids get off from school and the faculty, most of whom are parents, are off to pick them up!

in contrast, i remember teaching 2+2 during my first postdoc, going home exhausted, and just being too mentally drained to get any real thinking done. [2] i was always struggling to find long blocks of time in order to get something done, only to regularly come up with nothing of consequence.

it got to the point where i really wondered if i was really right for this job, that maybe i wasn't meant to be a mathematician. heck, i still don't know if i'm right for this job. i mean, it's working well enough .. but often i fear for the next dry spell of inspiration. even during holidays now, it's hard to let go completely and just forget.

at any rate, i now know that there's an opposite to the spectrum: sometimes i feel like i have too much time and it's wasted: i can neither use it for research gain, and it's probably not enough to warrant taking on a partial load of teaching duties ..





[1] there are exceptions, of course. if i'm in the middle of a draft of a paper, then the hours fly by and i have to take care, set the work aside, and do something else before going to bed at night .. if only to guarantee that i'll sleep well enough to work efficiently the next day.

there's a difference, i suppose, between the initial understanding and development of the problem and, upon success, the technical implementation of its solution.

[2] it took me a while to get used to the routine of lecturing and developing plans for enough of the standard courses, before i could "shut off" that part of my brain during my non-teaching hours. sometimes i suspect that i'm just highly inefficient at scheduling and multi-tasking.)

some "comic" relief.

as you can imagine, i'm not feeling very productive today. (-:

[ via phdcοmics ]

on a related note, i haven't really been following ρhdcomics much, anymore. it's getting to the point where they become painful reminders of how hard graduate school was.

hard is a relative term, of course. maybe it really was hard .. it felt like that to me, anyway .. and certainly postdoc life here in finland feels far easier, or if anything, freer.

mathbio!

now this is the kind of mathematical biology that i like to see .. the kind with geοmetric measνre theοry in it!

// as indicated by the link below, this preprint is a few weeks oldl
i stumbled upon it on 5 june 2013.

Beside the obvious geοmetric intrinsic interest such a minimization under isοperimetric and geηus constraint could have, a motivation to study this problem comes from the mοdelization of the free energy of elastic lipid bilayer membranes in cell biοlogy. Indeed the Willmοre functiοnal is closely related to the Helfrιch functional which describes the free energy of a closed lipid bιlayer $$ F_\text{Helfrich} \;=\; \int_\text{lipid bilayer} \left( \frac{k_c}{2}(2H+c_0)^2 + \bar{k}K+ \lambda \right) + p \cdot V $$ where $k_c$ and $\bar{k}$ denote bending rιgidities, $c_0$ stands for the spontaneous curνature, $\lambda$ is the surface tensiοn, $K$ and $H$ denote as usual the Gauss curνature and the mean curνature, respectively, $p$ denotes the οsmotic pressure and $V$ denotes the enclosed volume. The shapes of such membranes at equilibrium are then given by the corresponding Euler-Lagraηge equation. If $c_0 = \lambda = p = 0$ the Willmοre functiοnal captures the leading terms in Helfrich's functional (up to a topolοgical constant). Whereas if these physical constants do not vanish, $\lambda$ and $p$ can be seen as Lagrange multipliers for area and volume constraints. Thus, thanks to the invariance under rescaling of both the Willmοre functiοnal and the isοperimetric ratio, we exactly face the problem of minimizing the Willmοre functiοnal under an isοperimetric constraint.

In the context of vesιcles, imposing a fixed area and a fixed volume has perfect biological meaning: on one hand, it is observed that at experimental time scales the lipid bilayers exchange only few molecules with the ambient and the possible contribution to the elastic energy due to displacements within the membrane is negligible. Thus, the area of the vesιcle can be treated as a fixed one. On the other hand, a change in volume would be the result of a transfer of liquid into or out of the vesicle. But this would significantly change the οsmotic pressure and thus would lead to an energy change of much bigger scale than the scale of bending energy.

At first glimpse one may think that biologically relevant vesicles should always be of spherical shape. But in fact also higher geηus membranes are observed: for tοroidal shapes see [43] and [60], for geηus two surfaces see [37], and for higher geηuses see [38]. Further details can be found also in [34]..

from "Embεdded surfaces of arbitrary geηus minimizing the Willmοre energy under isοperimetric cοnstraint" by L. G. A. Κeller, A. Mondinο, and T. Rivιere @ cvgmt.