Mathematical Sciences: Spectral Geometry and Index Theory on Foliated Manifolds

该项目继续对叶理理论进行数学研究。主要的重点将是代数拓扑学(次级特征类)和谱几何方法的应用。我们将研究分片空间上横截Dirac算子的指数和谱、相应的Cheeger-Chern-Simons类、ETA不变量以及由合适的模空间参数化的横截Dirac算子族。数学物理的应用，特别是叶状空间的规范理论和全球手征反常，将进一步进行探索。该项目的总体目标是通过热方程方法和奇异空间上的谱渐近理论，建立黎曼叶层基本截面上横切椭圆算子的指数定理。主要的要求是，该定理及其推广将以与叶理的横切几何和相关Zeta函数的光谱性质相关的数据的形式明确给出。几何分析中叶的研究是微分算子域的自然推广。可以证明这些运算符以特别简单的方式作用于以层方式分解域的集合。在这些层次上，可以使用类似于研究微分方程式的一些基本工具来分析许多算子。

This project continues mathematical research on foliation theory. The main emphasis will be the application of the methods of algebraic topology (secondary characteristic classes) and spectral geometry. Work will be done investigating the index and spectrum of transversal Dirac operators on a foliated space, associated Cheeger-Chern-Simons classes, eta invariant and families of transversal Dirac operators parametrized by suitable moduli spaces. Applications to mathematical physics, in particular to gauge theory on foliated spaces, and global chiral anomalies will be further pursued. The overall objective of the project is to establish via heat equation methods and the theory of spectral asymptotics on singular spaces, an index theorem for transversally elliptic operators on the basic section of a Riemannian Foliation. The main requirement is that the theorem and its extensions would be explicitly given in terms of data relative to the transversal geometry of the foliation and of the spectral properties of the associated zeta-function. The study of foliations in geometric analysis is a natural generalization of domains for differential operators. These operators can be shown to act in particularly simple ways on sets which decompose domains in a layer fashion. On the layers it is possible to analyze many operators using analogues of some of the fundamental tools applied to the study of differential equations.