Sunday, May 01, 2005

Speculations Concerning Half-Learned Lessons.

I was thinking about a few things I learned and read over the last year: among them were a few seminar talks about the Wasserstein metric space of probability measures, which seems a handy idea; for the Analysis group here at UM, I believe the motivation for learning this area was to understand a result of Lott and Villani concerning analogues of lower curvature bounds in metric measure spaces.

This manner of philosophy seems to me quite interesting: one pursues an equivalent condition in a more general context without the a priori tools to do so. In this case, the goal is a condition similar to lower Ricci curvature bounds, and the original condition requires, at the very least, a Riemannian manifold structure. In contrast, a metric measure space needn't even have a manifold structure (consider a metric tree, for instance) and may be a rather 'floppy' object.

Fascinating, how one may be able to speak of 'curvature' things in places where curvature does not exist.

I was also browsing a book of E. Giusti's (Minimal surfaces and Functions of BV, to be exact) and he, like many others from the school of De Giorgi, formulate functions of bounded variation in a different way from how it is done in Lebesgue theory (cf Royden's Real Analysis, for example). So doing, a BV function is equivalently a function whose distributional derivative is a finite (signed) measure.

This is almost too tempting. Up to renormalization, it seems that we may use the Wasserstein theory to study derivatives of BV functions and apply these results to first-order PDE. And if some hearsay is correct (I remember hearing once that the Wasserstein space has some notion of a "tangent bundle" though it is not a manifold), then perhaps we may consider second-order PDE.

Concerning Cacciopoli sets (sets of rectifiable boundary, which uses this language of BV functions), it would be interesting whether we may write down an evolution equation (PDE) for the parametrization of the boundary. Now this is really a stretch, but what if we could use this framework to study isoperimetric questions?

Yes, I'm speculating here. It's times like these when I wish I had a better understanding of PDE, the Calculus of Variations, Riemannian Geometry, and another half-dozen subjects which would make such speculations either rigorous or moot.

I'm also fairly certain that these are not original ideas. A quick Google search indicates that these techniques may already be in practice, but where to start looking ..?
F*ck it. I don't know enough .. not enough to try problems in metric analysis and geometry or in these Italian-flavored mathematics. I hate being a kid sometimes, and the only solution seems more work and more lessons, if only toward some better insights into these problems or others.

Oh well. Back to work.


Anonymous said...

I don't know much about the mean curvature flow, but Evans and Spruck wrote a series of nice papers on this subject. Instead of looking at parametrizations (which tend to develop singularities) they represent the boundary as a level set of a function, and then evolve that function. I don't think that they cover general Caccioppoli sets, though.

A philosophical aside: it's more useful to get a good grip on simple things than a poor grip on complicated things.

janus said...

Thanks for the Evans-Spruck reference. Now that you mention it, it would make more sense to use the boundary as a level set (the parametrization technique sounds messy ..)

Concerning your philosophical aside, you make a good point. Too often I find myself jumping into complicated situations, when what I should have done was concentrate on simpler contexts, first. What can I say? The complicated things lure me.

Unfortunately, this problem isn't isolated to my mathematical pursuits. \: