Deformation Theory for Boundary Value Problems

本项目的目的是构建一个与bouet de Monvel的R-n+以及更一般地，具有边界的紧流形中所有阶和类（类型）为零的算子的代数相关的变形量化。项目的一部分是确定与这种变形相关的C*-代数的连续场，从而将Connes的切线群的概念扩展到具有边界的流形的情况。最终，我们希望能够将其与Nest和Tsygan的技术结合起来，推导出一种新的边值问题指标理论方法。在平行研究中，我们要考虑叶状流形上的海森堡微积分的相应问题。这个项目本身就很有趣。它继续从算子代数的角度对布特特·德·蒙维尔的微积分进行研究。与此同时，它可能被视为对Nest和Tsygan开发的指数理论方法的一次测试。

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.