**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.*sighs*)Oh well. Back to work.

## 2 comments:

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.

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.

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