article post: autism, genetics, and the technical fields.

[excerpts from the article, selectively chosen with bias q:]

He believes the genes which make some analytical may also impair their social and communication skills. A weakness in these areas is the key characteristic of autism.

.. In addition, students in the natural sciences have a higher number of relatives with autism than do students in the humanities, and mathematicians have a higher rate of autistic spectrum conditions compared with the general population.

..Professor Baron-Cohen said the rise in autism may be linked to the fact that it has become easier for systemizers to meet each other, with the advent of international conferences, greater job opportunities and more women working in these fields.

happy rant: vector bundles.

More from Tuesday. Apparently it was one of those days when I had thoughts of many sorts, and didn't bother that much with sorting which were mathematical, which were research-related, and which were whimsical.

Man. Vector bundles are so damn cool. If vector bundles were people then they would be really cool people and I would give them high-fives if I saw one of them on the street.

Tangent bundles are fine and all, but it always felt constraining to insist that the vectorspace structure of the tangent space be governed by some unseen chart parametrization.

So let the vectors in the additional structure point where they like. Why not? Pick up a normal bundle instead, and you can build the same objects in a more intuitive way. Take the Möbius band, for instance: it makes more sense to view it as a line bundle (1-dimensional fibres) than a non-orientable surface. Orientation's always bothered me, to tell you the true.

Let the vectors in this linear structure take a different dimension from the manifold dimension. In fact, when I think of it, the first Heisenberg group feels more like a vector bundle with 2-dimensional fibres than a Lie group or a smooth manifold .. what, with its vectorfields {X1, X2} and all ..

.. for the record, I didn't actually check whether the Heisenberg group had that structure I "felt," but I think it does. The vectorfields (which I never wrote down, here) are realised as a Lie algebra from the group action by left-translation, so that same left-action should give the vector bundle chart changes ..

.. but don't hold me to that. I'll check it when I have the chance. In the meanwhile, it's back to reading Hirsch for tomorrow's advisor meeting.

differential topology and inner dialogue

I wrote this yesterday morning-into-afternoon. It's short but says something.

Two weeks ago it was Tuesday and I was sitting at the Caribou Coffee down the street from my apartment, with a legal pad and a copy of Hirsch's Differential Topology on the table. [1] I was reading about isotopies of diffeomorphisms, tubular and collar neighborhoods of submanifolds, and beneath it all, this notion of vector bundles.

Today I find myself doing the exact same thing -- reading the same parts from Hirsch -- and strangely enough, it doesn't bother me.

The difference, I think, lies in attitude and manner. That fortnight ago, I had the mentality of "Oh, sh!t. How am I going to learn all of this?!?" and part of me [2] rolled his eyes at that state of affairs, as if saying, "F*ck, man. You still haven't sorted out this smooth Schoenflies issue yet, have you?"

But now I can say to that older self: "F*ck off, son. [3] If we were able to do it then, it would be done already, wouldn't it? We're doing it now and it will get done. So, cool it.

"What's your rush, anyways? This bundle stuff is damned cool, and when's the next chance we'll study these smooth things? After this, we're jumping back to coarse, Lipschitz stuff and PL technique, maybe. Let's enjoy it as it goes, yeah?"

Little pissant that he was, my past self had nothing to say in response.

and yes, I'm still reading Hirsch as of now. With any luck, this stuff will be sorted out tomorrow and I can return to getting more research results.

---

[1] I highly recommend the book, if you're looking for something a step beyond a first look into (smooth) manifolds and tangent spaces (cf. Guilleman/Pollack). Also, for the record there was the de-facto cup of coffee on the table, as well. q:

[2] More and more I believe that I have virtual multiple personalities; it's either that or I'm too good at naysaying to let myself off the hook.

[3] Even though that past self is chronologically older than my present self, the past self is frozen in age and maturity, so in a way, my past self is also "younger" than my present self. Weird, I know.

never mind: i am an idiot.

F*ck. I am a f*cking idiot and ignorant of obvious facts, but in a good way. You see, in an earlier post I had written that

.. I'm stuck on a small but important part of the very last argument.

It concerns Sobolev functions, as well as functions of bounded variation (BV for short). The theory of BV functions is nice and well-developed, but to be honest I never thought that I'd actually use them. It's like having your bike stolen: you know it's possible but you never expect it to happen.

Well, surprise of surprises. I was overthinking it, and my instincts were right: there's no need to apply the machinery of BV functions, after all. In fact, an elementary fact about Sobolev functions was all I needed:

Sobolev functions on the real line have absolutely continuous representatives.

Maybe I should have listened to myself more carefully when I was giving that Student Analysis Seminar talk! q:

more about thesis work.

I learned the difference between a "differentiable manifold" and a "differential manifold," today. It's a subtle one.

Thesis work has been going all right. I wouldn't rightly call it research, because it's retracing the work of others, in order to assess how "my" result [1] sits in the body of knowledge. It's hard work trying to understand this stuff, but it makes me uneasy that I haven't been contributing any ideas of my own into the mix.

On the plus side, I've been doing all sorts of fun differential topological stuff: reading Milnor's paper on exotic differential structures, and now learning about isotopies of diffeomorphisms and gluing manifolds. I might even learn about vector bundles, which is a topic I hadn't expected to learn unless I "defected" to geometry.

Sometimes it's nice to be able to bake your cake and eat it, too. q:

---

[1] More accurately, I should say that it's Fred Gehring's result. The good ideas are his, and at best I've proven a few corollaries.

works and days.

On Tuesday night I wrote a post of some length about the first talk for Student Analysis Seminar this term and how it went -- my impressions, hopes, and malaise, all of it. Then I posted it, and ten minutes later, it disappeared.

Fortune of fate, I guess. Perhaps it's better it happened, because now I can say very briefly what I wanted to say:

1. Sh*t. I'm not ready to be one of the "older students." I'm still a newbie and nobody should trust what I say when it comes to rigor and proof.
   
   
2. The audience was comfortable asking questions, and I'd like to thank John and Marie for spearheading the inquiry. I can see why my advisor prefers it when students ask questions and make comments.
   
   
3. I only made it through page 3 of 6 of my notes, and barely got through definitions and several questionable examples. I wonder if it means that I should talk again, next week .. \:
   

---

Thesis work plods on, and my own research feels intractable. I'm stuck on a small but important part of the very last argument.

It concerns Sobolev functions, as well as functions of bounded variation (BV for short). The theory of BV functions is nice and well-developed, but to be honest I never thought that I'd actually use them. It's like having your bike stolen: you know it's possible but you never expect it to happen.

Well, there they are: in the work of mine and my coauthor's, and still that one step remains elusive. I'm convinced that it must be true and we're this close .. now if only I can say exactly why there should be no atoms in the derivative measure ..

[sighs]

Time marches on, and it's either time to get back to work or to go home: one or the other ..

the race is on.

In less than a month, there's already three hits on the arXiv about the Isoperimetric problem on the Heisenberg group: two by DGN in just a week's time, and one from December by RR. If I didn't know any better, this problem may be settled before I obtain my Ph.D!

It seems that cylindrical symmetry is now subservient to questions of boundary regularity -- but I won't know for sure until I finally sit down and read what's been going on.

That might take a while, though. After all, this is supposed to be the term I will get lots of research done .. the "Semester of Janus," if you will ..

.. and yes, I did steal that from the "Summer of George" of Seinfeld fame. q:

article post: "quants" and "math"

For the record, today I seem to have attended classes, a seminar, and a meeting, and have done no work yet. I was honestly expecting to do a quick e-mail check, and then get to work .. but evidently plans change. (;

---

Through some digging just now, I found another article which deals with "mathematics" but I can't decide whether they actually mean statistics or applied mathematics. I suppose that for everyday life, the difference doesn't truly matter, but for those who care, it's a question of what you're doing with the data you're given.

I don't know. Here's the article; you can decide for yourself. But just to bias your reading, I found the following passage interesting:

How do you convert written words into math? Goldman says it takes a combination of algebra and geometry. Imagine an object floating in space that has an edge for every known scrap of information. It's called a polytope and it has near-infinite dimensions, almost impossible to conjure up in our earthbound minds. It contains every topic written about in the press. And every article that Inform processes becomes a single line within it.

Each line has a series of relationships. A single article on Bordeaux wine, for example, turns up in the polytope near France, agriculture, wine, even alcoholism. In each case, Inform's algorithm calculates the relevance of one article to the next by measuring the angle between the two lines.

It reminds me of sequence in infinite-dimensional spaces, and if they discuss a notion of angle, does it mean that they must use Hilbert spaces? Being a poor analyst, perhaps I can't be expected to know the intricacies of the real world, but if a boy can dream, then certainly a boy can guess .. q:

It also reminds me of this article a friend of mine (who studies library science and works as a sys.admin. for a college library) of this auto-indexing theory by Salton, Wong, and Yang; the idea is that one organises documents as vectors, where the entries are keywords. Now apply metric space theory, and you have a method of nonlinear sorting if linearity does not help.

A vector space model for automatic indexing (PDF available)

Anyways, that's their attempt at geometry. They also make some big claims, which I would rather they quantify .. but being reality and not mathematics, I may as well be asking for the moon, and maybe the rings of Saturn, while I'm at it .. \:

By the time you're reading these words, this very article will exist as a line in Goldman's polytope. And that raises a fundamental question: If long articles full of twists and turns can be reduced to a mathematical essence, what's next? Our businesses -- and, yes, ourselves.

.. This industrial metamorphosis also has a dark side. The power of mathematicians to make sense of personal data and to model the behavior of individuals will inevitably continue to erode privacy. Merchants will be in a position to track many of our most intimate purchases, and employers will be able to rank us not only by productivity, but by wasted minutes. What's more, the rise of math can contribute to a sense that individuals are powerless, a foreboding that mathematics, from our credit rating to our genomic map, spells out our destiny.

Wonderful. Shall we all become discretized and take our robot forms, now?

But just look at where the mathematicians are now. They're helping to map out advertising campaigns, they're changing the nature of research in newsrooms and in biology labs, and they're enabling marketers to forge new one-on-one relationships with customers. As this occurs, more of the economy falls into the realm of numbers. Says James R. Schatz, chief of the mathematics research group at the National Security Agency: "There has never been a better time to be a mathematician."

Whenever I see the words "mathematics" and "numbers" I cringe, because it probably means that I have to explain myself to more people about

1. why the mathematics I study contain very few numbers,
   
2. why what I do has no immediate practical applications,
   
3. or if my work does have such applications, then why I am not pursuing them ..
   

The world needs more lessons in when to use the words "mathematics," "applied mathematics," "mathematical modelling," and "statistics."

I would rant more, but that would head into careless opinions and my personal, illogical biases, so let me stop here. The article is actually quite long and makes the usual bold claims ..

.. and after all, I have work to do. Reading this article carefully can wait.

unproductive and uneasy.

It's been a while.

Aside from a few half-hearted posts here and there (quotes by Bohr and by Einstein, and that silly Rudin link) I haven't written anything of substance since 27 December. It took me a moment to do the mental arithmetic [1]: about two weeks' time, really.

I could say that I've had little to write about: the semester is in its infancy (tomorrow finishes a full 5 days) and I've done little maths, so to speak. [2] On the other side of the coin, I haven't been much of a person, either. Withdrawn and with little to say except jests and curiosity, there hasn't been much to report there, either.

I don't feel like the same person that I was, before my prelim.

Maybe it's some residual disillusionment from that exam not being some watershed moment or an irrevocable disaster. It wouldn't define me, at this age and capacity, and it won't .. which, ironically enough, is why it will.

As a pessimist, I've been proven right. If the prelim has taught me anything, it's that I'll never quite understand anything as completely as I would like. More than that, it means that I won't ever live up to the mathematician I would like to be.

That guy, a possible me, an ideal mathematician: he doesn't exist anywhere but in my head. He will remain there forever, trapped by the constraint of being too unflawed.

I've reached a few conclusions and confronted a few inner demons, but those are inner demons and I won't share them here.

---

[1] To be wholly honest, I had to count the days off my fingers. I've never been good at telling time, I'm afraid.

[2] I suppose that's technically not true. An issue of regularity has been bothering me lately, which arose from joint work with a friend of mine and has thrown a wrench into an otherwise decent result.

Mathematics is so damned frustrating, especially when you have to prove something. q:

wait! where did the quotes go?

Yes, there were two posts full of Bohr and Einstein quotes before. I've put them somewhere else for better keeping.

Here's a link, if you insist on seeing them again. q:

you've got to be kidding me.

There is some deal (or false advertising) in which one can obtain Rudin's Functional Analysis book for free. The link is here, if you want to investigate it.

It's too bad that I already have a copy of Rudin .. I'd otherwise be severely tempted.