Tuesday, December 27, 2005

a list of links, and a new preprint on the arXiv.

I've built a new webpage of bookmarks on del.icio.us recently, and many of them are tagged by the term "mathematics."

Have a look at what I've saved, if you like. Here's the link to the mathematical pages, if you like.



I've been out of the game for a while (as a spectator) and have forgotten the current state of affairs of the Isoperimetric Problem in the Heisenberg group. However, there's a new article in preparation with another partial solution; some of the abstract is omitted.
Area-stationary surfaces in the Heisenberg group H^1 [link]
Authors: Manuel Ritoré, César Rosales

We use variational arguments to introduce a notion of mean curvature for surfaces in the Heisenberg group H^1 endowed with its Carnot-Carath\'eodory distance. By analyzing the first variation of area, we characterize C^2 stationary surfaces for the area as those with mean curvature zero ..

.. we prove our main results where we classify volume-preserving area-stationary surfaces in H^1 with non-empty singular set. In particular, we deduce the following counterpart to Alexandrov uniqueness theorem in Euclidean space: any compact, connected, C^2 surface in H^1 area-stationary under a volume constraint must be congruent with a rotationally symmetric sphere obtained as the union of all the geodesics of the same curvature joining two points.

As a consequence, we solve the isoperimetric problem in H^1 assuming C^2 smoothness of the solutions.
I might not understand the hypotheses of "area-stationary surfaces," but it seems plausible that one main difficulty of the Heisenberg Isoperimetric problem -- the symmetry -- may have been resolved. Once I get out of this holiday/post-prelim funk and summon enough will to set aside my thesis work, I'll have to read it carefully ..

If the article is correct, then what remains is regularity of the boundary of the extremal set. Immediately I wonder why the C2-smoothness is used by the authors, other than as a convenience. With more work, can it be done without such an a priori assumption?

I should read the damn paper and be done with it .. if only I could convince myself that
mathematics ≠ thesis work.

Monday, December 26, 2005

Lipschitz thoughts, and the spectre of Rectifiability

In the last few days I've accumulated a good share of stir-craziness. Like a laptop running Windows XP, I feel as if my brain is on "Stand By" mode.
To return to my usual work habit, all I must do is

  1. flip up the screen,
  2. tap a key,
  3. type the magic words (my passphrase)

and after a few process cycles, everything will run smoothly, good as new. However, it's not quite time to work yet, and I'm still in this mental holding pattern of "Stand By" mode.

If now is the time to think about mathematics again, then it's time to think about my thesis problem and to read Hirsch's Differential Topology for clues about the diffeotopy extension problem.

It means looking towards which direction of research is next, and what I must prepare or explore.

It also means addressing the matter of my prelim from 20 December: what exactly happened and why. I'm not ready for that.

So it's not time to think about "work mathematics" quite yet. Let me stay on vacation and think of "play mathematics" instead.



What didn't happen during my prelim were notions of rectifiability.

It was the one topic which caused me the most unease, though the framework is elegant. In Mattila's exposition, the essential message is that there are 7-8 equivalent notions of rectifiability in Euclidean space. But as you'd guess, the proofs are somewhat technical.

Time ran out and neither member of my committee asked about any of that, which surprised me to no end. So I have no idea how well (or unwell) I understand this notion. I've only my own guesses, which are unreliable and sketchy, at best.

But I do want to know, if only because it seems like a topic worth investigating and generalising to the context of metric spaces. Recently L. Ambrosio and B. Kirchheim have written a few articles on this general context (one in Acta) ..

.. and as per my usual tyronism, it amazes me that one can say anything about this and metrics spaces: they seem to me too "floppy" and general and too abstract for me.

Let me explain myself. Perhaps I am too stuck on my visual intuition, and I find it hard to imagine such and such a metric space with such and such a measure. Conformal modulus is still a new tool to me, and possible pathologies of curves and their images don't come easily to my mind.

I've warmed up enough to my advisor's suggestions, and now Lipschitz manifolds seem to me rather interesting and somewhat elusive objects.

In the Euclidean case, rectifiability of sets in Rn is inexorably related to Lipschitz functions and parametrizations.

From what I've been told, if you enforce a strong enough condition of rectifiability [1] on a given set, it will give rise to a bi-Lipschitz parametrization, i.e. a Lipschitz manifold structure.

Now that's interesting, if only because images come easily to mind. It is like some geological means of detecting a topological character.

I envision walking across a field, and picking up a rock. With only the flat of my palm I study its exterior and see how flat it gets at this spot or that. If it's almost flat, in some measure sense, then guess what?

It arises from a rough sort of manifold. Huh.

[1] It's something related to tangent plane approximations, called Reifenberg's condition.

Saturday, December 24, 2005

holiday thoughts, whence running.

so i had succumbed to inward temptation today .. not only did i drink three servings of coffee [1] and eat a half-dozen brownies and several handfuls of roasted peas (an asian goodie), but i thought a little about mathematics today.

i had gone running, the thoughts wouldn't shut off [2], and somehow a theorem of Federer and Fleming came to mind: it states that the Sobolev Embedding Theorem for p = 1 gives rise to an isoperimetric inequality, and vice versa; better yet, the optimal constants are the same. from the pedestrian viewpoint [3] it does give a definitive correspondence between function theory and geometry, which is still not obvious to me.

the thought and a few fancies about Federer-Fleming lasted the three miles i ran .. not very long and not very deep thoughts, but they linger still ..



[1] .. one of these servings arose from breakfast at the diner. there the coffee was weak, but i had 2-3 refills of the stuff. i suppose that it evens out, then.

[2] .. after running cross-country and track in school, i can't not pay attention to minute details when running. listening for biofeedback and watching for race opponents has been ingrained in me over 5-6 years of youth competitive running, and now i can't turn it off: i can't stop thinking or processing sensory data when i run.

[3] .. assuming that this pedestrian had a little knowledge of Sobolev spaces and Hausdorff measures, that is. q:

Tuesday, December 20, 2005

i'll say no more, but ..

well .. i passed.

it was terrible, i looked like a complete idiot, forgot all the mathematics that i should have known since undergrad, couldn't integrate a damn thing, but i passed.

now all i feel is revulsion and hate. funny how the one thing you want is the very thing you can't bear after you have it.

an acquaintance of mine warned me that these latter years were the bleakest for him, pools of depression and gloom and things like that. i didn't realise that he was that right.



EDIT, AS OF LATE TUESDAY NIGHT: thanks to my friends who were around East Hall for their own reasons tonight, listened to me whine and rant, and bantered with me until the evening hours became infeasible.

I feel like a person again .. still a dolt who should have failed his prelim but didn't .. but nevertheless, a dolt who happens to be a person.

Saturday, December 17, 2005

when Death smiles upon you ..

My god. I am so f*cking stupid, but I mean that in a good way.
If it clarifies the situation, I'm laughing at myself .. half-hysterically and half-nervously, of course. But regardless of context, it is laughter.

It took me "far too long" [1] to realise that a particular claim follows ..

.. not from a big theorem, a moderate proposition, or even a simple lemma. It follows from a change of variable, via integration. That's all.

Well, that's embarrassing, but I suppose I should take the advice I've always given my students:

one stupid mistake now means one less stupid mistake later [2].

All righty. Back to work, then.



[1] .. and no, I'm not going to define that precisely. I am embarrassed for a good reason, you know.

[2] The advice was meant to be reassuring, for when my students do poorly on the quizzes I give them.

Wednesday, December 14, 2005

article post

"Natural Born Mathematicians" - an excerpt:

"We are born with a core sense of cardinal number", says neuropsychologist Brian Butterworth, author of The mathematical brain, reviewed in this issue of Plus. "We understand that sets have a cardinality, that is, that collections have a number associated with them and it doesn't really matter what the members of that set are. Infants, even in the first week of life, notice when the number of things that they're looking at changes.

Read more here. What do you think?

Monday, December 12, 2005

weary, and miles to go ..

Up to one key detail, I think I finally understand what I'm going to talk about tomorrow in Hyperbolic 3-Manifolds class. It took far more hours of turning analytic arguments into geometric reasoning than I'd have liked, and in the end, I was only rarely successful ..

Now all I have to do is write the outline of the talk in such a way so that, sleep-deprived and weary, I have enough information to go on auto-pilot and say what I mean to say. I think I know myself too well; there will be some hitch in the plans, and as a result I won't get any sleep tonight.



On a partially related note, I think my coffee consumption is spiralling out of control. I have half a mind to go "cold turkey" during winter holidays .. that is, after my prelim has come and gone.

It might be a disaster, otherwise. I do want to pass this thing, after all.



On an even more tenuous note, I don't think I understand rectifiability in Rn after all. Sitting in Geometric Measure Theory class today I had trouble following the motivation and intuition, and then I wondered if I adhered too closely to Mattila as the gospel truth.

Too often I forget to be distrustful of books. Illusions shatter: better they do now than next week.

It still feels depressing, though. This was one thing I thought I actually knew, too .. \:



Lately I've forgone the notion of finishing the term strong and well, competent and carefully. It's enough just to finish: do what needs to be done, and deal with the malaise and self-recriminations later.

After all, there will always be self-recriminations. More fuel to the fire, is all.

Sunday, December 11, 2005

quick post: a Mobius Band.

I've been in the office for most of the day, and fool that I was, I brought my laptop with me. As a result, I've been distracted by this Tangled 'Web and have gotten very little done .. and there is so much to do ..

Anyways, the little thing I discovered today: there is an electronica band out there called "Mobius Band."

I'm not kidding; someone took the pun to a new level! [2]

You can listen to a few of their songs via audio-stream on their myspace page. They're not bad, but no news yet about whether there's an umlaut (ö) on their spelling of "Mobius." q:



[1] Having hoped that I would write my entire talk for Hyperbolic 3-Manifolds class in one day, I've now cut my losses and am trying to finish two of five sections, and do the rest after I grade quizzes and team homeworks and when I get home.

Meanwhile, there's the problem of finding time to read about rectifiability of sets in Rn so that I can talk about them competently in 9 days' time (read: my prelim exam).

Then there is that last thing my advisor asked me to prove .. ): You'd figure I'd be smarter than this .. or at least more efficient with time.


[2] I haven't been this startled by a band name since someone told me about "Jedi Mind Tricks," which happens to be a hip-hop rap group!

Saturday, December 10, 2005

a book and its audience.

In the spirit of procrastination, earlier I stopped reading a section from Mattila for a moment, and turned the book over. This line caught my eye:

Essentailly self-contained, this book is suitable for graduate students and researchers in mathematics.

Wow. Then I remembered something else, pulled out Evans-Gariepy, and flipped to their preface. It was as I remembered it.

This book is definitely not for beginners. We explicitly assume our readers are at least fairly conversant with both Lebesgue measure and abstract measure theory. The expository style reflects this expectation. We do not offer lengthy heuristics or motivation, but as compensation have tried to present all the technicalities of the proofs: "God is in the details."

It feels like a pleasant difference from the usual disclaimer of many books I've encountered, which would read something like

This book is suitable for beginning graduate students and advanced undergraduates.

I suppose nothing is wrong with such an assessment, except that it damages my ego and my self-esteem. It's the same philosophy with the terms "clear" or "obvious" or "easy," relative to proofs and arguments you see.

If I didn't know any better, I could swear that either (1) many authors underestimate the content of their books, or (2) I'm not as intelligent as I would like to be. Being that my "beginning" years of graduate school are fast ending, the bar of expectation has been raised and it is now a good question whether I can vault over that bar.



On a more objective and less self-deprecating note, I wonder how authors determine the "level" of the books they write.

Many books do arise out of lectures from a class taught to graduate students, which explains why the reading level is set for graduate students. The exposition is meant to be organized, and the key results are understood to a level which omits most of the mess from thought experiments and previous endeavors.

In short, many good books are polished. It's easy to read such books and forget that the viewpoint is retrospective, and a generation ago, the path of argument could have been rather confusing and unclear. Unless the choice is deliberate, most books and lecturers may never show you the "wrong paths" that were taken to argue this or that, and depending on relevance and clarity, it's purely a judgment call.

It's like how some authors rewrite history to suit a particular image or ideology: to oversimplify, nations win wars because our side has "just cause" and our enemies are "evil."

In some mathematics books, it feels more like one timeline of reality or possibility is presented: the prevailing and practical one, up to some consensus. It's not so much rewriting history, because there could be alternate realities; we just choose not to study them.
But enough of philosophy. I'm curious to hear what you guys think: how do and how should authors label the "level" of books they write? Should they account for the factors below, or are there missing factors?
  • How specialised the topics are, or how obscure a field.
  • The treatment of the topics, and amount of technical details.
  • Better sales, varying by the size of the audience
Or could it be there is no good scale?

Friday, December 09, 2005

"You'll do fine."

O, teach me how I should forget to think!
Romeo, Romeo & Juliet

Everyone's been telling me not to worry so much about my upcoming prelim exam and that I'll do fine. Even my advisor said so during our meeting today ..

.. so I suppose it can't hurt to listen. q:


Thinking about it, worry doesn't really help very much. It didn't help, for example, when I was taking and re-taking my Analysis Quals during my first year.

To this day I think I was overworried about doing well, and after doing poorly once, I plunged into a vicious cycle of worry and fret and failure. I only passed Analysis when I didn't study too hard for it, and at that time I was focusing on the Topology Qual.

It could have been that measure theory and complex analysis had to gestate in my mind for that long (one year) before I could fully process that information and implement it in an exam. Nonetheless, the worry was of very little help.

Maybe my concern is complacency and laziness. One could say that I am a high-inertial person: if there is no push for me to do or act towards something, then I'll likely not.

I can think of countless times when I posited goals or plans one day and forget them after a day or a week. Completely ineffective and intolerable, that! What good is my word if it cannot be trusted, or my promises if they never bear fruit?

If you think me an idealist, I'd have to disagree. I'm not trying to be a moral person, but a consistent one.

I suppose this is really my attempt at making sure that I am not myself, or rather, not the part of myself that is fallible and inconstant. Said otherwise, perhaps I'm trying not to let my humanity get in the way of progress.

As for what that means, I don't know and I don't think I want to know, either. Let me worry simply about doing well on this prelim, and leave the philosophy to January when I have time to ponder it. \:

Sunday, December 04, 2005

on studying, remembering, and what to remember.

After a studying session for my prelim, I've observed a few things:
  1. I'm making slow but steady progress. This might be illusion or delusion, but it honestly feels like I'm gaining some degree of intuition from the arguments that I've been reading. I can only hope that my memory serves and I don't forget it all, but perhaps that can be corrected by how I've been processing this information.

  2. I've given up on remembering the technical lemmas which are used to prove big theorems. Instead, I've been trying to remember the big theorems ..

    (.. which aren't too hard; they are big, after all.)

    and what is the "right way" to prove them; this also allows me to build intuition on how to work with certain concepts and properties. If I remember the "right way," then I'll have to collide into what smaller claims I'll need to prove it, and those are the very technical lemmas that I would fumbled to remember. It's the difference between a priori and a posteriori knowledge, I suppose.

    Perhaps I am slow-witted and should have thought of this long ago. I've learned it now, though, and it suggests that I might not be a wholly hopeless cause .. (:
In particular, I never realised how cool the Area Formula for Lipschitz functions is. From the approach in the Evans-Gariepy text, the idea is to dissect the domain into Borel subsets, where the given Lipschitz function, when restricted to one of these sets, is "essentially" bi-Lipschitz with controlled constant.

One proceeds to argue that the Lipschitz images are comparable in measure to images from linear transformations (relative to these Borel subsets), and then we apply this linearity with impunity and without apology.

Fine ideas, if I've quoted them correctly (I've been wrong before), and it seems reasonably general (though the use of linearity worries me).

What strikes me most is the "optimism" or "courage" behind this type of argument: subdividing the domain as necessary, Lipschitz functions behave like bi-Lipschitz functions or linear transformations in the sense that we want them to. How does one learn such bravado, and learn to get away with it?

So let me say that I'm getting better at remembering, if only because I'm learning to reduce to the essentials. This could mean that I'm becoming a better student, but that doesn't mean that I'm becoming a better mathematician.

Oh well. It's something, at least.