Mathematical Sciences: Spectral Geometry and Index Theory on Foliated Manifolds

本项目继续进行叶理的数学研究 理论 主要重点将是方法的应用 代数拓扑（第二特征类）和 光谱几何学 将对指数进行调查， 叶空间上横截Dirac算子谱， 相关Cheeger-Chern-Simons类，eta不变量和 族的横向狄拉克算子参数化适当的 模空间 数学物理的应用，在 特别是规范理论上的叶状空间，和全球 将进一步研究手性异常。 总体目标 该项目的目的是通过热方程方法和 奇异空间的谱渐近理论，一个指标 横截椭圆算子在基本截面上的一个定理 一个黎曼叶丛 主要要求是， 定理及其扩展将明确地给出， 数据相对于横向几何的叶理和 相关ζ函数的光谱特性。 在几何分析中对叶理的研究是一种自然的 微分算子域的推广 这些操作者可以用特别简单的方式来表现 在以层的方式分解域的集合上。 上 层，可以使用类似物来分析许多操作符 一些基本的工具应用于研究 微分方程

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.