Index Theory on Singular Spaces

本计画探讨几何分析、拓扑学与数学物理学之汇流。首要目标是发展和关联奇异几何对象的分析和拓扑。几何学和相关物理学领域的许多问题，如广义相对论和全息术，都涉及奇异空间，并且可以通过引入带角的光滑流形来理解，这捕获了奇点的性质。通常，人们可以用一种适合于有角流形的方式来重新设计问题，然后应用一种被称为“几何微局部分析”的强大技术工具箱。“这个项目的一个方面是促进这些技术的发展和应用到一大类奇异空间（分层空间），允许分析工具被带到承担代数和拓扑问题。该项目的另一个方面是促进新的调查人员进入这一活跃的研究领域。调查员将进一步编写关于这一主题的研究专著，并举办课程和会议，传播这些技术和成果。几何分析、拓扑学和数学物理学交汇处的一个重要问题是理解分层空间的签名，例如那些作为多变量多项式的零点而出现的签名。研究者和合作者已经扩展了可以解析地理解签名的空间类，并证明了它满足光滑空间签名的一些主要性质。在这个项目中，明确的公式将建立的签名及其扭曲的类似物。该项目还旨在为奇异空间开发一个拓扑手术理论，并表明签名算子自然地从这个序列定义映射。其他项目涉及通过将通常的代数视角改变为几何视角来理解局部对称空间的基本群的表示的不变量。

This project investigates topics at the confluence of geometric analysis, topology, and mathematical physics. The overarching goal is to develop and relate the analysis and topology of singular geometric objects. Many questions in geometry and related areas of physics, such as general relativity and holography, involve singular spaces and can be profitably understood by introducing smooth manifolds with corners, which capture the nature of singularities. Often one can recast the questions in a manner adapted to the manifold with corners and subsequently apply a toolbox of strong techniques known as "geometric microlocal analysis." One aspect of this project is to contribute to the development and application of these techniques to a large class of singular spaces (stratified spaces), allowing analytic tools to be brought to bear on algebraic and topological questions. Another aspect of the project is to promote the entry of new investigators into this active area of research. The investigator will further develop a research monograph on the subject and organize courses and conferences to disseminate these techniques and results. An important example of a question at the confluence of geometric analysis, topology, and mathematical physics is to understand the signature of stratified spaces, such as those arising as the zeros of polynomials in several variables. The investigator and collaborators have extended the class of spaces on which the signature can be understood analytically and have shown that it satisfies some of the main properties of the signatures of smooth spaces. In this project explicit formulae will be established for the signature and its twisted analogues. The project also aims to develop a topological surgery theory for singular spaces and show that the signature operator naturally defines maps from this sequence. Other projects involve understanding an invariant of representations of the fundamental group of a locally symmetric space by changing the usual algebraic perspective to a geometric one.