Deformation Theory for Boundary Value Problems

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It is the aim of this project to construct a deformation quantization associated to the algebra of all operators of order and class (type) zero in Boutet de Monvel's calculus for R-n+ and, more generally, for a compact manifold with boundary. Part of the project is to determine the continuous field of C*-algebras associated to this deformation, thus extending Connes' notion of the tangent groupoid to the case of manifolds with boundary. Eventually we hope to be able to combine this with the techniques of Nest and Tsygan to derive a new approach to index theory for boundary value problems. In a parallel study we want to consider the corresponding problems for the Heisenberg calculus on foliated manifolds. This project is of major interest in itself. It continues the investigation of Boutet de Monvel's calculus from an operator-algebraic point of view. At the same time it may be seen as a test for the approach to index theory developed by Nest and Tsygan.