Singularity structures for nonsmooth spaces

Abstract: In the geometric study of distributions, the notions of singular support and wave front set provide essential tools for the analysis of singularities and their propagation. From an operator-theoretic perspective, these notions can be captured by studying actions of smooth functions and pseudodifferential operators on a given distribution. Abstracting from this point of view, Dave and Kunzinger have introduced a unifying categorical framework of so-called singularity structures for Fréchet modules, which recovers the usual notions of singular support and wave front set for closed manifolds.

In this talk, we will discuss their approach and describe a generalization of their results to the setting of complete (noncompact) manifolds. We will give a refined description of Sobolev regularity in terms of tameness properties of an associated evaluation map on smoothing operators, and introduce a substitute for the algebra of pseudodifferential operators. Our construction is based entirely on the functional calculus of the Laplacian, with the aim of applying these techniques in the study of (infinitesimally Hilbertian) metric measure spaces. This is based on ongoing work with Günther Hörmann and Michael Kunzinger.

analysis of PDEsclassical analysis and ODEsdynamical systemsfunctional analysismetric geometryprobabilityspectral theory

Audience: researchers in the topic

SISSA's Analysis seminars