Deformation Theory for Boundary Value Problems
Final Report Abstract
Let us recall the summary of the original application: “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 Rn+ 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 not only 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 case for the approach to index theory developed by Nest and Tsygan.” We also recall the goals stated in the project application: “The two primary goals of the project will be (i) The deformation theory for operator algebras associated to Boutet de Monvel’s calculus on a compact manifold with boundary or the half-space Rn+ The new point here is that one has to study deformations of a non-abelian algebra, an aspect not covered in the literature. We shall start with the analysis of the algebra of regularizing singular Green operators, the probably simplest case exhibiting this difficulty. We shall then proceed to the ideal of singular Green operators, and eventually to the full algebra. (ii) The construction of continuous fields of C ∗-algebras related to these algebras of boundary value problems. The aim here is to construct and study the analogue of the tangent groupoid for manifolds with boundary. Concerning the long term goals of the project (i) We expect to be able to apply the results above to the index problem for Boutet de Monvel’s calculus, using the the results on cyclic homology of A and a suitable generalisation of the algebraic index theorems of Nest and Tsygan. (ii) In a next step one would like to study of the automorphism group of A. An information on this subject would lead naturally to a calculus of Fourier Integral Operators in the context of manifolds with boundary. An ultimate goal might be a generalization of the Atiyah-Weinstein index theorem in the spirit of Leichtnam-Nest-Tsygan.” While it actually took longer and turned out to be harder than expected, we eventually were lucky and could reach all our goals concerning the construction of the analog of the tangent groupoid and the index theorem for Boutet de Monvel’s calculus. Slightly different methods were used for the proof of the index theorem. In addition, we obtained results on the possible groupoid structure in joint work with Melo and Monthubert. Research on a possible extension of the Atiyah-Weinstein index theorem to manifolds with boundary has been started. The case of Heisenberg manifolds which was also mentioned has not been studied intensely in view of the fact that Raphael Ponge had meanwhile started to work very vigorously in this area and had already found an analog of the tangent groupoid.
Publications
- Boutet de Monvel’s calculus and groupoids I. Journal of Noncommutative Geometry
J. Aastrup, S. Melo, B. Monthubert, and E. Schrohe
- Index theory for boundary value problems via continuous fields of C*-algebras
Johannes Aastrup, Ryszard Nest, and Elmar Schrohe
- A continuous field of C∗-algebras and the tangent groupoid for manifolds with boundary. J. Funct. Anal., 237(2):482–506, 2006
Johannes Aastrup, Ryszard Nest, and Elmar Schrohe
