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Analysis on singular spaces and index theory

Abstract : This thesis is set in the general context of extending the theory of elliptic operators, well-understood in the smooth setting, to so-called singular domains. The methods used rely on operator algebras and tools coming from non commutative geometry, together with suitable pseudodifferential calculi that are often built from a groupoid adapted to the particular geometry of the problem. The first part of the thesis deals with the general investigation of a particular class of such groupoids, called Fredholm, that provide a very good setting for the study of elliptic operators. One of the major results proved here is that this Fredholm property is local, in the sense that it only depends on the restrictions of the groupoid to sufficiently many open subsets. In the same spirit, we study with C. Carvalho and Y. Qiao groupoids whose local structure is given by gluing group actions, and consider in particular a groupoid suited to the study of layer potential operators. This part concludes with a well-posedness result for a boundary value problem on a domain with a rotational cusp. The second part deals with equivariant operators on a compact manifold, acted upon by a finite group. We answer the following question: given an irreducible representation of the group, under which condition is a differential operator Fredholm between the corresponding isotypical components of the Sobolev spaces? In a joint work with A. Baldare, M. Lesch and V. Nistor, we introduce a corresponding notion of ellipticity associated with some fixed irreducible representation, and show that it characterizes Fredholm operators.
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https://hal.univ-lorraine.fr/tel-02970044
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Rémi Côme. Analysis on singular spaces and index theory. Mathematics [math]. Université de Lorraine, 2020. English. ⟨NNT : 2020LORR0096⟩. ⟨tel-02970044⟩

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