# Isoperimetric domains in homogeneous three-manifolds and the isoperimetric constant of the Heisenberg group ��1

In this paper we prove that isοperimetric sets in three-dimensiοnal hοmogeneous spaces diffeοmorphic to 3 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 ��1, 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 ��1 posed by P. Paηsu.

## Saturday, January 24, 2015

### 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 ..

## Saturday, January 17, 2015

### 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..

## Saturday, January 10, 2015

### 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.

## Friday, October 03, 2014

### 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!

## Wednesday, July 30, 2014

### 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

## Monday, July 28, 2014

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

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

ε

## Wednesday, July 23, 2014

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

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.

## Tuesday, July 22, 2014

### 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!)