Non-local Differential Relations

Abstract

We introduce a "non-local" generalization of differential relations that can encode constraints involving values and derivatives of maps at a finite number of different points. The first part of this thesis lays the groundwork for later chapters: It provides a comprehensive theory of blowing up arrangements of submanifolds in order to build configuration spaces in the style of Fulton and MacPherson, but now equipped with a weighted structure. This is formalized using Loizides and Meinrenken's weightings, for which some novel results are given. As an application, we discuss configuration spaces of filtered manifolds. The second part of this thesis applies this framework to define a blown-up multijet space and prove that it satisfies a multijet transversality theorem that takes into account effects at the diagonal. We discuss in particular the relationship to Whitney topologies and polynomial interpolation. The final part of this thesis introduces non-local relations and uses the blown-up multijet spaces to discuss matters of integrability and stability of their solutions. We investigate the relationship between local and non-local constraints, give a breadth of examples and take first steps towards an h-principle approach by defining formal data.