Index Theory on Singular Spaces

奇异空间索引论

• 批准号: 1104533
• 负责人: Pierre Albin
• 金额: $ 14.78万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2015-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1104533&HistoricalAwards=false
• 关键词: Index; Theory; Singular; Spaces

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.

主要研究者建议研究奇异空间的指数理论的各个方面。例如，PI和他的合作者将把他们最近对群作用的轨道空间的分解的分析应用于一族等变算子的指数理论。一个类似的解决方案，但任意的伪流形，将用于一个项目，以了解签名算子的性质，并扩展已知的空间类，以满足诺维科夫猜想从拓扑。一个相关的项目旨在更广泛地扩展几何微局部分析的许多方法，以设置分层空间。PI还将与他的合作者合作，了解非紧或奇异空间的谱几何，开展了解非紧表面的等谱性的项目，并扩展可以解析表达拓扑Reidemeister挠率的空间类别。最后，PI参与了各种努力，以扩大数学家的数量知道这些结果，并在这一领域的工作，包括生产的讲义和调查的文章，以及组织会议和研讨会。奇异空间自然出现在数学和物理，即使当一个人有兴趣研究光滑现象。例如，宇宙学和粒子物理学中的最新模型需要理解具有奇异性的空间，如角和边。然而，几何分析中使用的许多技术并不适用于奇异空间。这项建议的一个方面是有助于发展和应用新技术的问题所产生的指数理论和光谱几何。这些问题是在交叉的分析与其他领域，如拓扑，代数几何，数学物理，合流这是一个来源的问题和应用。