Saturday, April 16, 2005

Love those Vector Fields ..

Edit (as of 18 April 2005): It's been brought to my attention that AIM stands for Applied and Interdisciplinary Mathematics. Apologies to all the AIM people out there.

Also, don't be fooled by the number of comments. One of them is mine.

Yesterday there was a seminar talk hosted by the Differential Equations people, but also invited were the AIM (Applied and Industrial Mathematics) folk and the Several Complex Variables crowd: a rare nexus of areas. In retrospect the talk wasn't applied at all, but being that many of the AIM folk do study differential equations, perhaps this isn't such an interesting intersection of interests, after all.

At any rate, the speaker addressed the question of (local) hypoellipticity of a particular differential operator (the sum of "squares" of prescribed vector fields) and recalled a result of Hoermander [1] and the bracket-generating condition on the vector fields. Solid, terrific stuff! This sort of study, among others, vindicates PDE from what numerical methods and applied ends have done to its image; it makes for interesting geometry and analysis, which I've ranted about, since my beginnings.

To summarize, the speaker recalls a key step in Hoermander's celebrated proof, in particular the use of sub-elliptic estimates: they allow control of derivatives to induce the hypoellipticity property. However, he soon demonstrates an example where a set of vector fields do not give sub-elliptic estimates, but through which hypoellipticity occurs anyway!

I was ready to be pummelled by an abstract construction, but surprisingly enough the example was simple to state: one takes the Heisenberg left-invariant vector fields and gives them a "contact" factor (a power of z or its conjugate, in terms of complex numbers) and supposed it does the job. The outcome looks like a strange "composition" of Grushin-type and Heisenberg-type vector fields, kin to what one would expect in the articles of Montgomery and/or Sussmann in the Sub-Riemannian Geometry book.

A strange coincidence.

In short I was floored by this talk, and excited, besides! Just when you think that everything's been done with Hoermander vector fields, someone breaks the dam and a new world of questions flood forth.

It did feel a little weird: looking around, I didn't see many others taking notes, as I was. I was also sitting next to a prof of some distinction, and I could swear that he was glancing at what I was writing [2]; so for a while I did my best to write only intelligent and insightful things on the page.

But that only lasted so long, before I resorted to jotting down ideas that I understood and reminders on what I didn't and what I'd look up at later times. When the talk ended and the announcement came for dinner with the speaker (not immediately, but in two hours' time), I thought a little about it before I realised something.

I was the only student left in the room.

Then I pictured myself later, in a hypothetical evening, the only student and silent amongst profs sharing pleasant conversation and insightful ideas. I imagined this going on for hours, until inevitably someone would ask me why I came to dinner. I wondered if I could explain myself properly, or more generally if I'd be any good at being such company.

Shuffling my papers, I decided against it. Besides, my fellow second-years would likely be planning an Ultimate Frisbee game soon enough, and so I went. Despite our love for certain things, we go where we feel most comfortable .. [3]

[1] "Hoermander" is spelled with an 'o-umlaut' instead of an 'oe,' but I don't know my ASCII codes that well. \:

[2] I'm still getting used to writing on legal pads, and I find it easier to use them when the pad is upside down (the short side with the free page edges is on top). The prof might have been staring because of that.

[3] .. even if it means that my team lost 3-8 and I couldn't break away from my defender the whole time, for example.


Anonymous said...

I used to take notes in my first two years of grad school, but then stopped. For one thing, I realized that I never looked at my notes afterwards. :-/ If I was indeed interested, I could easily get a paper or preprint with all the details of that talk. But I know professors and advanced grad students who diligently take notes of every math talk they attend. Even mine. 8-0

fragments of angry candy said...

AIM = Applied and INTERDISCIPLINARY Mathematics. Sheesh, ya think I'd go for something "industrial"? :)

I guess frisbee games are important. Still, interesting that you interpreted being the only student in the room as a reason to shy away from the dinner rather than evidence that you're way cooler than the rest of us and would be very welcome at the dinner, the professors happy to see your interest in their topic.

I was supposed to go to that talk but am unfortunately ruled by my stomach, and went to eat instead.

janus said...

first comment: now I'm wondering whether I took notes at any talks you gave. But you make a good point: I never browse through my seminar notes either (though I do so with lecture notes).

However, have you ever found that the speaker says something insightful in the talk that never got into the preprint? This could be fueled purely by my paranoia at missing details, though.

second comment. Sorry about missing the initials for AIM. The letter I is tricky!

Frisbee games are important. They offer exercise and keep you growing big and strong. (;

As for avoiding dinner, I don't know those profs very well and it could be argued that I should have been tending to other, more pending matters.

In the end I don't think I would have had much to add to the dinner discussion, and honestly, I was ruled by the fear that I would be the only student there and wouldn't fit in.

After all, nobody's ever accused me of having high self-esteem. \:

fragments of angry candy said...

third comment: I'm inclined to enlist myself as a Jasun's Self Esteem Advocate. On the other hand, you play the Eeyore rather well...

Thanks for the Munkres, by the way!