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.

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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?

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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.

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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.