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Geometric analysis on singular spaces. (English) Zbl 1421.58007

Brüning, Jochen (ed.) et al., Space – time – matter. Analytic and geometric structures. Berlin: De Gruyter. 349-416 (2018).
Authors’ abstract: We are interested in the analysis of Dirac and Schrödinger-type operators associated to certain stratified spaces known as Smooth Thom-Mather Spaces. These are topological spaces that consist of a smooth manifold as dense open subset to which manifolds of lower dimension are attached in a suitableway; they are briefly described in Section 2.2. Prominent examples are polyhedra, projective varieties, and connected orbit spaces of proper Lie group actions. They all come equipped with canonical metrics that are of a rather different character. We therefore discuss in Section 3 the geometric analysis of Dirac and Schrödinger-type operators on arbitrary Riemannian manifolds and the construction of resovents and heat semigroups. In Section 4, we describe the construction of certain geometric invariants which are expressed in terms of the spectral data of suitable self-adjoint extensions of these operators.
For the entire collection see [Zbl 1390.83006].

MSC:

58J35 Heat and other parabolic equation methods for PDEs on manifolds
57R35 Differentiable mappings in differential topology
58C25 Differentiable maps on manifolds
58J05 Elliptic equations on manifolds, general theory
58J65 Diffusion processes and stochastic analysis on manifolds
81S40 Path integrals in quantum mechanics
53Z05 Applications of differential geometry to physics
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