Thursday, April 21, 2005

Lessons, and some Phenomena of Note.

Today I met with my advisor, and oddly enough, it went smoothly. For once, I don't think I blanked out at something obvious, but I do not think this phenomenon will last. After all, there's no reason to believe that my level of understanding in mathematics should be, say, a nondecreasing function. (;

It still feels strange to say that I have an advisor. It almost feels like I lucked out, though by no means is it easy. There is indeed much to do and much to learn, and from the looks of things I fear it will be a long time before I'm ready to attack research problems in this area of geometric analysis.

On a side note, today I did learn about two separate and interesting phenomena in analysis. I apologize for any forthcoming lack of rigor.
Measure Theory
In the context of collections of pairwise-disjoint balls { B } whose union has finite measure, then their dilations { λB } for a scale λ ≥ 1 cannot have "very much overlap." In particular, one may form a (weighted) counting function from characteristic functions of these balls, and
its exponential integral must be finite.

Looking at it intuitively, my prof described it as having infinitely many leaves on the ground in autumn (which may overlap in some way), yet poking a pointy stick into the ground always gives you only a finite number of them.

Partial Differential Equations
One may study the Poisson equation Δu = f for a given p-integrable source function f, and it is a fact that u must lie in W2,p, the Sobolev space of second order weak derivatives. This is amazing, because the Laplacian operator Δ only uses the unmixed second derivatives (d / d xi)2 and says nothing about the mixed partials!

Moreover, the approach is itself interesting (and a little off the wall): one argues this by using a substantial fact about Sobolev spaces of differential forms -- in particular, a function (or in general, a k-form) can be expressed as the sum of the exterior differential and codifferential of two k+1 forms of a k-1 form and a codifferential of a k+1 form. Now using a characterization of Sobolev spaces of differential forms, this immediately gives the right regularity for u.

Amazing: to use differential forms in a case where none are seen (PDE, no less) and they grant you the regularity that you might heuristically assume.
Simply amazing.

1 comment:

Anonymous said...

Of course, you meant "the sum of the exterior differential of a k-1 form and the codifferential of a k+1 form". I am somewhat puzzled by this application of the Hodge decomposition to the Poisson equation. The L^p-Hodge decomposition is usually proved using the boundedness of singular integral operators on L^p. But once you know that the SIO's are bounded, the above-mentioned result about Poisson equation follows immediately, because the mixed partials of u are given by Riesz transforms of \Delta u.