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Basic question: are there (co)homological or sheaf-based tools which might be useful in geometric measure theory?


Background: The jumping off point here is a simple analogy - geometric measure theory can be seen as a generalization of the theory of smooth manifolds, where we are generalizing by trying to reduce our regularity assumptions. The analogy is really to complex and algebraic geometry, which can also be seen as generalizations of smooth manifolds, where we either move the machinery into complex or algebraic territory (I know historically this is kind of absurd).

More to the point, there are a lot of tools which elucidate the breakdown or richness of the spaces of study. Specifically, smooth manifolds come associated with a whole set of sheaves which aren't terribly useful (since we can describe the theory richly in terms of more classical notions). Smooth manifolds have a cotangent and tangent sheaf, as well as a structure sheaf (which in the $C^k$ case is the sheaf taking values in the category of real algebras, where the target objects are just $C^k(U)$ for a given $U$ in the category of open sets on $M$). These objects aren't very useful, it seems, because they don't provide any more information in the smooth case than the traditional counterparts. For example, the cotangent sheaf is locally free, meaning there is an identification with covector fields. Another example is that the existence of partitions of unity on the manifold imply that the cohomology of the structure sheaf is vanishing.

However, having built this machinery, it is useful when trying to generalize. For example, when thinking about Riemann surfaces or complex geometry generally, the structure sheaf actually can have non-trivial cohomology which is helpful in understanding the structure of complex manifolds. Similarly, when generalizing away from smooth manifolds and into schemes, the algebraic constructions of the cotangent and tangent sheaves prove to be the right tools for generalization, and coherence/locally free is a useful characteristic when trying to prove that certain schemes are smooth.

Geometric measure theory starts of with a really weak version of an (embedded sub)manifold, namely rectifiable set. The process of building the machinery to do geometry here has this vague analogy to the process building machinery to do geometry in those other "generalizations" of smooth geometry, and it is this analogy I am trying to get at. One of the most useful tools in geometric measure theory is going from a rectifiable set to a current: given a rectifiable set $A$ sitting in $\mathbb{R}^{n+k}$, let $\Omega^n$ be the $n$ forms on $\mathbb{R}^n$. You can go from a set $A$ to a functional $A^*:\Omega^n \to \mathbb{R}; \omega \mapsto \int_{A} \omega$. This is a dualization of $\Omega^n$, and it identifies the rectifiable set $A$ with a current $A^* \in (\Omega^{n})^*$. Another similar tool is move from rectifiable sets to verifolds, where you are associating the set again with a function, this time a measure on the space $R^{n+k}\times G(n,n+k)$.


It seems like the basic game in geometric measure theory is the same as the game in all of these other types of geometry: moving from some underlying set to algebraic constructions on it, and these constructions feel remarkably similar! Are there any useful ways to think about this vague analogy?

My first thought, for example, was to try and associate something similar to a structure sheaf to the rectifiable set, where you are thinking about assigning sobolev spaces based on functions out of the set, or maybe conceptualize a verifold as a functor which goes from the category of open sets to the category (?) of sobelev spaces similarly to what can be done with manifolds. But I'm not sure how either of those would work, or if there is existing work which takes this approach?

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  • $\begingroup$ What would be a morphism of rectifiable sets? Does it make sense to consider real-valued "rectified" functions? $\endgroup$ – Leo Alonso Feb 23 at 10:53
  • $\begingroup$ Hmmm... Well any Lipschitz function from the ambient space to itselfs should work. $\endgroup$ – Juan Sebastian Lozano Feb 23 at 17:31
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    $\begingroup$ I'm not sure your motivation is that well founded. In particular, the singularities one introduces in (say complex) geometry are much tamer than those that occur in GMT. I think the more accurate analogy is between rectifiable sets and metric spaces. To me, and I could be off here, it's like you are proposing to study $L^2$ functions by using tools developed for meromoprhic functions. $\endgroup$ – RBega2 Feb 23 at 18:12
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    $\begingroup$ @JuanSebastianLozano Currents (which strictly speaking need an orientation) and varifolds are introduced as they are topological vector spaces with good functional analytic properties from the point of view of calculus of variations. I'm not sure that's really the same as what you are describing (though I don't know anything about sheaf theory). Going back to the analogy, I had before, one studies meromorphic functions because there are many interesting meromorphic functions, but one tends to study $L^2$ functions as the space of $L^2$ functions has a lot of good properties. $\endgroup$ – RBega2 Feb 23 at 19:40
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    $\begingroup$ I know that some people working in algebraic analysis do a lot with currents. Maybe that's somewhere to start? $\endgroup$ – Avi Steiner Feb 24 at 18:11

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