Index Theory on Singular Spaces

主要研究奇异空间上指标论的各个方面。例如，PI和他的合作者将把他们最近对群作用的轨道空间分辨率的分析应用于一类等变算子的指标理论。一个类似的解析，但是是任意伪流形，将在一个项目中使用，以理解签名算子的性质，并扩展已知的空间类，以满足拓扑中的诺维科夫猜想。一个相关的项目旨在将几何微局部分析的许多方法扩展到分层空间的设置。PI还将与他的合作者一起研究非紧致或奇异空间的谱几何，研究非紧致曲面的等谱性，并扩展拓扑Reidemeister扭转可以解析表达的空间类别。最后，PI还参与了各种努力，以扩大意识到这些结果并在这一领域工作的数学家的数量，包括制作讲义和调查文章以及组织会议和研讨会。奇异空间在数学和物理中自然出现，即使人们对研究光滑现象感兴趣。例如，宇宙学和粒子物理学的最新模型需要理解具有角和边等奇点的空间。然而，几何分析中使用的许多技术并不适用于奇异空间。这一建议的一个方面是促进新技术的发展和应用，以解决指数理论和光谱几何中出现的问题。这些问题是分析与其他领域的交叉，如拓扑、代数几何和数学物理，是问题和应用的源泉。

The principal investigator proposes to study various aspects of index theory on singular spaces. The PI and his collaborators will, for example, apply their recent analysis of the resolution of the orbit space of a group action to the index theory of a family of equivariant operators. A similar resolution, but of an arbitrary pseudomanifold, will be used in a project to understand the properties of the signature operator and extend the class of spaces known to satisfy the Novikov conjecture from topology. A related project aims more generally to extend many of the methods of geometric microlocal analysis to the setting of stratified spaces. The PI will also work with his collaborators to understand the spectral geometry of non-compact or singular spaces, with projects to understand isospectrality for non-compact surfaces and extend the class of spaces where the topological Reidemeister torsion can be expressed analytically. Finally, the PI is involved in various efforts to extend the number of mathematicians aware of these results and working in this field, including the production of lecture notes and survey articles as well as the organization of conferences and seminars.Singular spaces arise naturally in mathematics and physics, even when one is interested in studying smooth phenomena. For instance, recent models in cosmology and particle physics require an understanding of spaces with singularities such as corners and edges. However, many of the techniques used in geometric analysis do not apply to singular spaces. One aspect of this proposal is to contribute to the development and application of new techniques to problems arising in index theory and spectral geometry. These problems are in the intersection of analysis with other fields, such as topology, algebraic geometry, and mathematical physics, a confluence which is a source of both problems and applications.