Hear the shape of a drum

March 12, 2006

There are just too many drafts waiting to be posted. I will try to post them all before April starts. Most of them are just interesting things I found elsewhere. I would very much like to add my comments – but this is taking quite sometime. So, I’ve decided to post their links and add my comments later.

To those who are wondering why I’ve vanished from the web this month , I’ll just inform them that my Keyboard is not working. That means everytime I’ve to write something, I should come to the IITK CC(Computer Centre). And I am not very fond of coming here everyday just to blog ! Nevertheless, I do have a list of drafts on various things happening around and I’ll post them one by one. And yeah, you should have noticed by now that the timestamp which appears in every post is NOT the time in which I posted it. It is the time when I started writing my post.

Ok, so here is an interesting article at Amar’s Blog titled Can one hear the shape of the molecule ?. This, I gather is inspired by an article titled Can one hear the shape of the drum ?
If you are still scratching your head(as I was :)), the definition at the beginning of the wikipedia link might be helpful.

To hear the shape of a drum is to infer information about the shape of the drumhead from the sound it makes

The answer to the above question unfortunately turns out to be No, people have proved that One cannot hear the shape of a drum(technical). Every subject has a list of questions which are damn simple to ask but have quite deep/difficult answers. This seems to be one such question in mathematical physics. This reminds me of a comment by Scott Aaronson that I came across.

What parts of mathematics do people feel have the property that the cutting edge research is understandable by a typical gifted high school student, both in terms of the theorems proved and the proofs themselves?

I like this question. I’ve always gravitated toward areas where the theorems and open problems — but not necessarily the proofs — can be understood by a high-school student. Indeed, I can get myself interested in a highfalutin theory if, and only if, I can see what HSSU (High-School-Student Understandable) problems it helps me answer.

Anyway, the person who posed this simple-looking question about drums was a mathematical physicist named Mark Kac. This is Kac for you !


You might like to read one of his interviews here. Or if you are lazy, some excerpts are available here. He says

I am reminded that his friend Eugene Wigner hit on it correctly by saying that he would gladly give a Ph.D. in physics to anyone who could really teach freshman physics. I know what he meant. I would attempt, I wouldn’t be very good at it, but I would attempt to teach a first semester course in quantum mechanics, and I would probably teach it reasonably well.

But I would not know how to teach a freshman course in physics, because mathematics is, in fact, a crutch. When you feel unsafe with something, with concepts, you say, “Well now. let’s derive it.” Correct? Here is the equation, and if you manipulate with it, you finally get it interpreted, and you’re there. But if you have to tell it to people who don’t know the symbols, you have to think in terms of concepts. That is in fact where the major breach between the two – how to say – the two lines of thought come in. You are either like von Neumann, and I am in that sense closer to him, or you are like Ulam, who when you say pressure, feels it. It is not the partial derivative of the free energy with respect to volume; it is really something you feel with your fingers, so to speak.

How true !

Update:(26/9/06) Saw an interesting comment by Gonzalo Gutierrez(English site) referring to his AJP paper “Can an ideal gas feel the shape of its container ? ” available here. And, I guess the other reference( Spectra of Finite Systems ) mentioned goes someway in answering Amar’s call to point him “to a readable version of Weyl’s proof” (see especially pg.19 of the above book/Pg.15 in the pdf file). I’ll probably read up this stuff this weekend before commenting further…