Bertrand Monthubert

Abstract: We define an analytic index and prove a topological index theorem for a non-compact manifold M_0 with poly-cylindrical ends. We prove that an elliptic operator P on M_0 has an invertible perturbation P+R by a lower order operator if and only if its analytic index vanishes. As an application, we determine the K-theory groups of groupoid C*-algebras of manifolds with corners.

Abstract: We compute the $K$-theory of comparison $C^*$-algebra associated to a manifold with corners. These comparison algebras are an example of the abstract pseudodifferential algebras introduced by Connes and Moscovici \cite{M3}. Our calculation is obtained by showing that the comparison algebras are a homomorphic image of a groupoid $C^*$-algebra. We then prove an index theorem with values in the $K$-theory groups of the comparison algebra.

J. Aastrup, S. T. Melo, B. M. & E. Schrohe, Boutet de Monvel’s Calculus and Groupoids I, Journal of noncommutative geometry Volume 4, Issue 3, 2010, pp. 313–329

Abstract: Can Boutet de Monvel's algebra on a compact manifold with boundary be obtained as the algebra of pseudodifferential operators on some Lie groupoid G? If it could, the kernel of the principal symbol homomorphism would be isomorphic to the groupoid C*-algebra of G. While the answer to the above question remains open, we exhibit in this paper a groupoid G such that C*(G) possesses an ideal isomorphic to the kernel of the principal symbol homomorphism on Boutet de Monvel's algebra.

Abstract:In Connes (Non Commutative Geometry, 1994, II.5), a proof is given of the Atiyah–Singer index theorem for closed manifolds by using deformation groupoids and appropriate actions of these on RN. Following these ideas, we prove an index theorem for manifolds with boundary.

Abstract: We construct algebras of pseudodifferential operators on a continuous family groupoid G that are closed under holomorphic functional calculus, contain the algebra of all pseudodifferential operators of order 0 on G as a dense subalgebra, and reflect the smooth structure of the groupoid G, when G is smooth. As an application, we get a better understanding on the structure of inverses of elliptic pseudodifferential operators on classes of non-compact manifolds. For the construction of these algebras closed under holomorphic functional calculus, we develop three methods: one using two-sided semi-ideals, one using commutators, and one based on Schwartz spaces on the groupoid.

Abstract: We compute the K-theory groups of the C*-algebra of the groupoid of a manifold with corners, in which the analytic index takes its values.

Abstract: This article is a survey on the relations between pseudodifferential calculus on manifolds with corners and groupoids, based on a contribution to the conference ``Groupoids in Geometry, Analysis and Physics'' held in Boulder, USA in 1999.

Résumé : Nous construisons un groupoïde différentiable longitudinalement lisse associé à une variété à coins. Le calcul pseudo-différentiel sur ce groupo&iifférentiel sur ce groupoïde coïncide avec le calcul pseudo-différentiel de Melrose (aussi appel&eacutl de Melrose (aussi appelé $b$-calculus). Nous définissons également une algèbre de fonctions à décroissance rapide sur ce groupoïde ; elle contient les noyaux des opérateurs régularisants du (petit) $b$-calculus.

Résumé : Soit $G$ un groupoïde de Lie. Le calcul pseudodifférentiel sur $G$ définit l'indice analytique. Le groupoïde tangent associé à $G$ induit un morphisme de $K$-théorie dont nous prouvons qu'il coïncide avec l'indice analytique. Ce dernier peut donc se définir sans recours au calcul pseudodifférentiel.