Weyl quantization and an index theorem for pseudodifferential operators

本文的目的是给出开流形上，特别是单连通双曲流形上伪微分算子的Atiyah-Singer型指标公式.该项目分为两个步骤。第一个是隔离一类具有FredhoIm指标的伪微分算子。第二个是实际证明指标定理，在这个研究基金的第一年，我们的目标是完成第一步。一个初步的研究强烈表明，这样做的困难，由于缺乏一般的谱理论对这些流形。这提示我们研究双曲流形的几何。在与各个领域的研究人员进行了讨论，以获得与我们的项目相关的信息后，我们设法分离出一类可能具有Fredholm指标的伪微分算子，并希望完成第一步。然而，它被推迟到第二年的赠款，以证明我们隔离的类是正确的， ...更多信息 e.由于技术上的障碍，这个目标过于雄心勃勃，不幸的是我们无法完成项目的第一步。在实现上述目标的同时，我们还在开发证明指数定理所需的基本工具。其中之一是辛流形的C^*-代数形变量子化的概念。在这方面，已经取得了重大进展。在与哥本哈根大学的R.Nest合作的一篇论文（本报告第11项的论文3）中，我们研究了闭黎曼曲面，它们是辛流形的主要例子，并证明了C^*-代数形变量子化的存在性。在与明斯特大学的R.Nest和I.Peter的合作项目中，我们推广了这个结果，在一定的拓扑条件下，证明了任意辛流形的C^*-代数形变量子化的存在性。我们计划朝这个方向进一步调查。特别是Poisson流形的C^*-代数形变量子化，它是辛流形的推广，这是一个即将开展的项目。至于研究的初始目标，即伪微分算子的指数公式，我们当然打算继续努力，我们希望在一年左右的时间内完成这个项目。少

The aim of this project is to show an Atiyah-Singer-type index formula for pseudodiffer- ential operators on open manifolds, particularly on simply connected hyperbolic manifolds. The project is divided into two steps. The first is to isolate a class of pseudodifferential operators that have FredhoIm indices. The second is to actually prove the index theorem.In the first year of this research grant, we aimed at completing the first step. A prelim- inary study strongly indicated a difficulty in doing so, due to the lack of general spectral theory on those manifolds. This suggested us to study the geometry of hyperbolic mani- folds. Having had discussions with researchers of various fields in order to get information pertinent to our project, we managed to isolate a class of pseudodifferential operators that possibly have Fredholm indices and hoped to complete the first step. However, it was post- poned till the second year of the grant to prove that the class we isolated is the right on … More e. This goal turned out too ambitious due to technical obstacles, and unfortunately we could not comp1ete the first step of the project.While working on the objective discussed above, we worked at the same time on devel- oping basic tools needed for the proof of the index theorem. One of them is the notion of C^*-algebraic deformation quantizations of symplectic manifolds. In this direction, a significant progress has been made. In a joint paper with R.Nest of the University of Copenhagen (Paper 3 in Item 11 of this report) we studied the closed Riemannian sur- faces, which are primary examples of symplectic manifolds, and showed the existence of C^*-algebraic deformation quantizations. Generalizing this result, in a joint project with R.Nest and I.Peter of Muenster University, under a certain topological condition, we showed the existence of C^*-algebraic deformation quantization for any symplectic mani- fold. We plan further investigations into this direction. In particular, it is the up-coming project to study C^*-algebraic deformation quantization for Poisson manifolds, which are generalization of symplectic manifolds.As for the initial target of the research, that is, an index formula for pseudodifferential operators, we certainly intend to continue working on it. We will hopefully complete the project within a year or so. Less