Index Theory on Singular Spaces

本项目研究几何分析、拓扑学和数学物理的融合。总体目标是发展和联系奇异几何对象的分析和拓扑。几何和物理相关领域的许多问题，如广义相对论和全息术，都涉及到奇异空间，并且可以通过引入带角的光滑流形来理解，这些流形捕捉到了奇点的本质。通常，人们可以以一种适合于带有角的歧管的方式重新设计问题，并随后应用称为“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在大量奇异空间（分层空间）中的发展和应用，使分析工具能够承担代数和拓扑问题。该项目的另一个方面是促进新的研究人员进入这一活跃的研究领域。研究人员将进一步就这一主题编写研究专著，并组织课程和会议来传播这些技术和结果。几何分析、拓扑学和数学物理融合的一个重要问题是理解分层空间的特征，例如在几个变量中多项式的零点。研究者和合作者扩展了签名可以解析理解的空间类，并证明它满足光滑空间签名的一些主要性质。在这个项目中，将为签名及其扭曲类似物建立显式公式。该项目还旨在发展奇异空间的拓扑外科理论，并表明签名算子自然地从这个序列中定义映射。其他项目包括通过将通常的代数视角转换为几何视角来理解局部对称空间基本群的不变性表示。

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.