in which a conjecture has been a theorem for (at least) three days.

wow; i just learned about this today.

Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group 𝖧1

Jih-Hsin Cheηg, Andrea Malchiοdi, Paul Yaηg

(Submitted on 29 Jan 2015)

In this paper we prove that isοperimetric sets in three-dimensiοnal hοmogeneous spaces diffeοmorphic to ℝ3 are tοpological balls. Due to the work in [MMPR13], this settles the Uniqueness of Isοperimetric Dοmains Cοnjecture, concerning congruence of such sets. We also prove that in three-dimensiοnal homοgeneous spheres isοpermetric sets are either two-spheres or symmetric genus-one tori. We then apply our first result to the three-dimensiοnal Heιsenberg grοup 𝖧1, characterizing the isοperimetric sets and constants for a family of Riemannιan adapted metrics. Using Γ-cοnvergence of the perimeter functiοnals, we also settle an isoperimetric conjecture in 𝖧1 posed by P. Paηsu.

[arXiv link].

exposure: what does maths look like, to non-mathematicians?

sometimes i wonder if the general public expects maths professors to do scholarly research in the same frequency (or smaller) as literature teachers who are also authors. [1]

it would make sense that to teach the inner workings of a particular subject, it would be helpful to create or to know how to create such academic works.

i say "smaller" because the layperson can more likely imagine what literary work looks logs, but have probably never seen what a maths research article looks like ..

( possibly to be continued.. )

[1] far be it, of course, for me to expect society to bother thinking about academics and academia, unless it's an issue of budget cuts to education ..

setting traps, of the non-lethal kind ..

one of the most frustrating tasks is writing the problem set for the first homework assignment:

there's usually very little content from which to ask any interesting [1] problems;

for the student who lack self-awareness, it's best to have a few hard, interesting problems early on, if only so that they are not deluded into thinking that this will be an easy class [2] ..!

the problems shouldn't be impossible, either, otherwise done students will panic, expect everything to be hard, and won't be able to do as well as they could otherwise!

[1] from experience, if i think a problem is interesting then often it will be too hard for the students ..

[2] the trouble with teaching maths is that we always teach topics that we know with great certainty .. which means, without experience, it is very difficult to tell what is easy for the students and what is not. for one thing, i'm recently becoming aware that students never know linear algebra as well as i'd want them to know it..

in which, this semester, i must hide my heretic ways.

so i just wrote my first lecture of 2015 .. and it reads like algebra, if only because it's supposed to read that way.

it's a first lecture in an undergraduate course on complex analysis. i've never taught it before.

something worries me; i wonder if i'm the right guy to teach it.

---
i'm not worried about screwing it up, not like last fall's numerical analysis ..

.. more on that, later ..

.. but now i have to come to terms with the subject. you see, i've deemed the subject unnecessary and consistently tried to avoid all aspects of complex analysis in my mathematical life.

yes, i know; it's the mathematical version of being a bigot or committing heresy!

perhaps it's just my first exposure to the subject [1]. perhaps it's the nature of the research problems i've chosen to study over the years. perhaps i generally take a pessimistic viewpoint in maths and a statement of the form ..

.. "every differentiable function at a point is infinitely differentiable at that same point and can be represented as a power series" ..

.. just sounds too good to be true, that there must have been some mistake. ye gods: what kind of dark, forbidden magic have we wrought?

---
now, of all things, i have to teach it.

with luck, my students won't inherit my own prejudices on the matter. i'll even attempt to sell them on its good points, if only for the sake of being a responsible mathematical role model.

on a related note, my first lecture can be summed up as such:

if you take the usual coordinate page, view it as a 2-dimensional linear subspace of R^4, rotate it appropriately, and project it down to its original domain, then you can make sense of square roots of negative numbers, provided that you treat those image vectors as 2x2 matrices.

---
notes:

[1] it's fair to say that my first course in it was uninspiring and largely i skipped all of the lectures except the exam periods. how i did "well" in that course is beyond me ..

.. but to be fair, i went to my favorite coffeehouse three times a week, armed with the textbook, ordered a large coffee, and would work through the problem sets from scratch. my barista friend would see me there regularly; she once asked me if i really liked that book or class, because i was reading it so much, to which i guffawed.

on fabricating data (but in a good way)

yes, it's been a while;
no, i'm no happier;
yes, I still wonder if i'm cut out for the academic life...

... but let's set aside those irreconcilable issues for now.

* * *
at the moment, i'm having far too much fun devising homework problems for my Numerical Analysis students.

one problem gives them some data points for a unknown function. their task is to show that, given a few hypotheses about the function, why the data cannot possibly come from Newton's method!

ARR, MoAR!... in which i don't know what to say ..

.. except that something has to change; this shouldn't happen in a country that calls itself a democracy.

Urban teachers have a kind of underground economy, Cohen explained. Some teachers hustle and negotiate to get books and paper and desks for their students. They spend their spare time running campaigns on fundraising sites like DonοrsChoose.org, and they keep an eye out for any materials they can nab from other schools. Philadelphia teachers spend an average of $300 to $\$$1,000 of their own money each year to supplement their $100 annual budget for classroom supplies, according to a Philadelphia Federation of Teachers survey.

~ from "Why Poor Schools Can’t Win at Standardized Testing" @theatlantιc

ARR, MoAR!... a digital version of synesthesia?

as the washingtonpost calls it: "paris with an echo"

idle movements.

ε

ARR, MoAR!.. on the downside of passion.

this article is about how programming, despite the call to arms about learning how to code, is a low-status job.

when i read this post, though, it funded more like the plight of teachers:

.."that we allow “passion” to be used against us. When we like our work, we let it be known. We work extremely hard. That has two negative side effects. The first is that we don’t like our work and put in a half-assed effort like everyone else, it shows. Executives generally have the political aplomb not to show whether they enjoy what they’re doing, except to people they trust with that bit of information. Programmers, on the other hand, make it too obvious how they feel about their work. This means the happy ones don’t get the raises and promotions they deserve (because they’re working so hard) because management sees no need to reward them, and that the unhappy ones stand out to aggressive management as potential “performance issues”. The second is that we allow this “passion” to be used against us. Not to be passionate is almost a crime .."

~ from "How the Other Half Works: an Adventure in the Low Status of Software Engineers" @Michael0Church.

ARR, MoAR!.. on risk-aversion.

from "Don't Send Your Kid to the Ivy League" @NewRepublic:

So extreme are the admission standards now that kids who manage to get into elite colleges have, by definition, never experienced anything but success. The prospect of not being successful terrifies them, disorients them. The cost of falling short, even temporarily, becomes not merely practical, but existential. The result is a violent aversion to risk. You have no margin for error, so you avoid the possibility that you will ever make an error. Once, a student at Pomona told me that she’d love to have a chance to think about the things she’s studying, only she doesn’t have the time. I asked her if she had ever considered not trying to get an A in every class. She looked at me as if I had made an indecent suggestion.

like any news article on education, one should take this report with a reasonable amount of skepticism ..

.. but being a university educator myself, there's some truth in it. generally my students are uncomfortable when i ask them problems in the exam that don't match up with their textbook problems (even though they are usually combinations of the same problems). the risk of a new obstacle, of not having seen something on which they will be evaluated .. it seems to really affect them.

-----
for instance, last semester i think i spooked most of my linear algebra class with one geometry problem on each exam. [1]  at some point several students asked for practice geometry problems.

"everyone's worried about the geometry problem," one of them admitted. i tried to point out that it was only one of at most five problems and that i generally curve the scores ..

.. but (s)he didn't seem convinced.

[1] e.g. "Determine, if it exists, an equation for the sphere passing through the following four points." (i even reminded them what the equation of a 2-sphere in 3-space was!)

highly irrelevant, but ..

.. once, at a party full of mathematicians, a friend was trying to formulate this one lemma. [1] he drew a shape and said ..

.. "ok, so this is a triangle .."

.. but at that point i had one too many and suddenly blurred out ..

.. "no. that's just an approximation of a triangle!.."

.. at which time everyone around just just burst into hysterical laughter.

terrible, unrepentant mathematicians, we were .. :-)

[1] yeah, we were that far in. i truly suspect maths is a language, because many drunken mathmos i know still revert to their mother tongue, after one too many ..

ARR, MoAR!.. on picking-&-choosing.

now i don't feel so bad about not caring about number theory problems.

In one letter he even displayed contempt for the problem. His friend the German astronomer Heinrich Οlbers had written to Gaμss encouraging him to compete for a prize which had been offered by the Paris Academy for a solution to Fermαt's challenge: "It seems to me, dear Gaμss, that you should get busy about this." Two weeks later Gaμss replied, "I am very much obliged for your news concerning the Paris prize. But I confess that Fermat's Last Theorem as an isolated proposition has very little interest for me, for I could easily lay down a multitude of such propositions, which one could neither prove nor disprove."

~ from "Math's Hidden Woman" @pbs

so i suppose that even the best of us should pick and choose the tasks best suited for ourselves. i wonder, though, what Gaμss thought of the Rιemann hypothesis ..?

---

also, to explain the title of the cited article, Gaμss isn't its main subject .. but the French mathematician Marιe-Sophιe Germaιn.

it's quite an account! i wonder sometimes how many women in history have kept to the academic shadows because of a lack of social tolerance and the societally-induced hardships upon them.

if the best minds of their time, such as Gaμss as well as Hιlbert (in the case of Emmy Nοether) could see the potential of these scholars, then you'd think that others would be willing to listen .. \-:

on an unrelated note, Ctrl-F is amazing.

.. not that this is of any real importance to post about ..
..
.. but i don't know what i'd do without the "find-&-replace" command that is standard on most text-editors now. ("copy-&-paste" has been indispensable for checking long chains of estimates, too.)

the next generation, part 3*: possibilities.

you know, today's meeting was pretty fun.

---

the student, having had a few days to think about a few concrete aspects of the problem, was a lot more comfortable showing me things that he thought about, telling me claims that he suspects are lemmas .. and why he thinks so.

what really helped, i think, is that it became clear to us that there was plenty we could learn, just by computing explicit configurations.

the student seemed to feel both awed and excited, that these were strange, interesting, yet accessible things for him. i think he realised today some scope of what was possible for him, that he started to believe in himself.

this isn't to say that i didn't guide the discussion, but i felt like the back &-forth today is suggestive, and this laissez faire style might actually work.

---

* .. and no, you didn't miss part 2; i haven't posted it yet.

training the next generation (a first post in a potential thread)

so i finally have something ready to write about [1]: i'm advising a former calculus student of mine on a summer (undergraduate) research project .. and (for me, anyway) it's a scary thing.

many of my colleagues are old hands at this, i know;
so if i seem naive, it's because this is the first go and i don't know any better.

i worry, for good reason: he's studying a topic i know little to nothing about. in particular, it's not clear to me how easy or hard the problems i pose to him really are .. and as a result, i don't know how much frustration i'm throwing his way.

it depends, of course, how much the student is willing to work. if i give him a badly-posed problem, then a good work ethick can actually be bad .. in the sense that, by working with abandon for too long a time, he burns out and gets turned off by pure maths in the future.

in case it's not clear, i'm encouraging the student to make his own conjectures;
of the established theorems whose proofs he can easily understand,
i'm suggesting him to try his own variants.

in other words .. and for better or worse ..
i'm insisting that i don't give him orders;
he'll have to train himself to think like a pure mathematician,
but i'll be there if he needs advice or guidance.

i was worried about his technical chops before .. until i realise that if his proof-writing skills require work, then this is potentially the best way that he can practice them: by working with a topic that interests him.

let's hope these aren't another example of famous last words ..!

[1] as you can see at the end of this post, i'll be tagging these thoughts with the handle "Σ:nextgeneration" .. and the usual disclaimer follows: unlike other maths blogs out there, i'm not out to train or educate maths-inclined people out there, at least not directly. instead, i'm going to show you, through my mistakes, what not to do.

to be more creative, try becoming a more boring person?

first of all, apologies to my readers for the dearth of posts in the last few weeks months and especially the lack of posts with any personal depth in them. i won't go into detail about it today, but this change in my life from a "gun-for-hire" postdoc to a "lifer" prof has bent my mind awry and i'm still learning how to cope with the job. sometimes it just feels .. crippling.

more precisely, it's not the actual job that's hard, but the stress and overthinking of this faraway goal called tenure. the more i think about it, the more it feels like i'm getting my ph.d. again.

all of that said, i'm going to go the lazy route again: i'll pass someone else's well-written piece to you (instead of writing my own).

---

when i was younger and a newer hand at research, i'd make a startling insight in my work. immediately aftewards, i'd lament why it took me so long to figure it out .. especially when the outcome appeared very simple.

i've become less critical of myself over the years, but the question still remains:

what if there were ways to become better at solving problems?
is it more than just a pipe dream to improve oneself?

below are excerpts from an essay i found, through one social media engine or another. what struck me about it was how i've unconsciously kept some of these habits and gotten more insights, in the last few years.

"A 2004 study published in Naturε examined the role of sleep in the process of generating insight. They found that sleep, regardless of time of day, doubled the number of subjects who came up with the insight solution to a task. (Presented graphically above.) This effect was only evident in those who had struggled with the problem, so it was the unique combination of struggling followed by sleep and not sleep alone that boosted insight.
..
"There’s a good reason for this: mind-wandering fosters creativity. A 2012 study (results pictured below) found that any sort of mind-wandering will do, but the kind elicited during a low-effort task was more effective than even that of doing nothing .. This, too, is congruent with my experience. How much insight has been produced while taking a shower or mowing the lawn? Paul Dιrac, the Nobel Prize winning physicist, would take long hikes in the wood. I’d bet money that this was prime mind-wandering time."

~ from "The Science of Problem Solving" @rs.io

of course, this could all just be a manifestation of confirmation bias (or if you will, the fallacy of the consequent). what i do recall, however, are periods of correlation: i was highly uncreative during those times when i was sleeping very little and had no time to exercise.

lastly, the tl;dr at the end of the essay is suggestively helpful. i'd recommend newcomers to research at least to try a few of the habits listed, if only to see what works (or not) for you.

sometimes negative signs matter ..?

after browsing this article, suddenly the difference between mεasures, sιgned mεasures, and vectοr mεasures come to mind.

The answer to this question takes us to the heart of quantum mechanιcs, to the part that popular explanations usually mangle. Quantum mechanιcs wasn't the first theory to introduce randomness and prοbabilities into physics. Ironically, the real novelty of quantum mechanιcs was that it replaced prοbabilities — which are defined as nonnegative real numbers — by less intuitive quantities called amplitudes, which can be positive, negative, or even complex. To find the prοbability of some event happening (say, an atom decaying, or a photon hitting a screen), quantum mechanιcs says that you need to add the amplitudes for all the possible ways that it could happen, and then take the squared absolute value of the result. If an event has positive and negative amplitudes, they can cancel each other out, so the event never happens at all.

The key point is that the behavior of amplitudes seems to force prοbabilities to play a different role in quantum mechanιcs than they do in other physical theories. As long as a theory only involves prοbabilities, we can imagine that the prοbabilities merely reflect our ignorance, and that a “God’s-eye view” of the precise coοrdinates of every subatomιc particle would restore determinism. But quantum mechanιcs’ amplitudes only turn into prοbabilities on being measured — and the specific way the transformation happens depends on which measurement an observer chooses to perform. That is, nature “cooks prοbabilities to order” for us in response to the measurement choice. That being so, how can we regard the prοbabilities as reflecting ignorance of a preexisting truth?

~ via "Quantum Randomness" @ AmericanScientist

ARR, MOAR!.. on writing.

the following excerpt by s.a.pιnker isn't terribly representative of his original article about writing, but i liked it anyway and thought to share it:

For example, everyone knows that scientists overuse the passive voice. It's one of the signatures of academese: "the experiment was performed" instead of "I performed the experiment." But if you follow the guideline, "Change every passive sentence into an active sentence," you don't improve the prose, because there's no way the passive construction could have survived in the English language for millennia if it hadn't served some purpose.

The problem with any given construction, like the passive voice, isn't that people use it, but that they use it too much or in the wrong circumstances. Active and passive sentences express the same underlying content (who did what to whom) while varying the topic, focus, and linear order of the participants, all of which have cognitive ramifications. The passive is a better construction than the active when the affected entity (the thing that has moved or changed) is the topic of the preceding discourse, and should therefore come early in the sentence to connect with what came before; when the affected entity is shorter or grammatically simpler than the agent of the action, so expressing it early relieves the reader's memory load; and when the agent is irrelevant to the story, and is best omitted altogether (which the passive, but not the active, allows you to do). To give good advice on how to write, you have to understand what the passive can accomplish, and therefore you should not blue-pencil every passive sentence into an active one (as one of my copyeditors once did).

(for more articles of this kind, visit edge.org.)

are the dollar signs worth it?

so i spent most of the day LaTeX'ing, which went well enough.

the best thing about writing up ready results in LaTeX is that one feels smart, that maybe all those days of banging one's head against these damned technical lemmas [1] were worth it after all.

..
.. on the other hand, there's something else I should write up, but i can't convince myself to do so:

whenever i think about the proof, I don't feel smart. instead, it just seems .. trivial.

[1] technically, the plural of lemma is "lemmata."

my younger self used to be so smart ..

yesterday i printed out some research notes i wrote in february ..

.. all 24 pages of them, just to prove one lemma. [1]

ye gods; the ideas still make intuitive sense, but the proofs are painfully technical .. much more so than i recall. [2]

they're not even a complete set of notes! for the lemma to have a complete proof, i'll have to read another set of 5 pages of notes for a sub-lemma, as well as a short 3-page section of an old preprint of mine ..

no wonder why i couldn't convince myself to work hard in april. after that and then that trip to spain in march, i had little to no energy left.

anyway, it's time to write up the result(s) ..

.. because what currently exists are the notes, not a preprint. there's a lot of exposition and simplification before i have anything remotely readable!

[1] at this point maybe i should promote it to "theorem" status. (before i thought it was .. well, obvious, and the proof would be short.)

[2] one series of estimates involved 4-5 indices. i was comparing intervals at different dyadic scales, as well as the same scale, which explains two indices .. but the intervals aren't nested in one another, which requires two more indices. lastly, these estimates were "fibrewise," so an additional fifth index kept track of which fibre was which.