Dirac operators on cobordisms: degenerations and surgery

配边的狄拉克算子：退化和手术

• 批准号: 1005745
• 负责人: Liviu Nicolaescu
• 金额: $ 10.3万
• 依托单位: University of Notre Dame
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-06-01 至 2014-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1005745&HistoricalAwards=false
• 关键词: Dirac; operators; cobordisms; degenerations; surgery

A Morse function on a manifold decomposes the manifold into infinitely many hypersurfaces (level sets). The typical level set is smooth but a few of them have singularities. A Dirac operator on the manifold induces Dirac-type operators on these level sets. The investigator intends to prove that the index of the original operator can be recovered from invariants capturing infinitesimal behavior of this family of induced operators along finitely many of these level sets. More precisely the index will be a sum of two types of invariants: soft and hard invariants. The soft invariants are described by infinitesimal spectral flows near finitely many smooth level sets, while the hard invariants are described by the Kashiwara-Wall indices of triplets of infinite dimensional lagrangian spaces canonically determined by the singular level sets.The Dirac type equations are generalizations of the famous Maxwell's equations of the electromagnetism. A solution of such an equation can be viewed as a sort of stationary electromagnetic wave on the Universe under investigation, known as the background manifold. The number of solutions of a Dirac equation is highly dependent on the shape of the manifold (or Universe). Morse theory is a technique of investigating the shape of a manifold by decomposing it into certain elementary pieces. The investigator intends to explain how to compute the number of solutions of a Dirac equation by studying the behavior of this equation on these elementary pieces of space.

流形上的一个莫尔斯函数将流形分解成无穷多个超曲面（水平集）。典型的水平集是光滑的，但其中一些具有奇异性。流形上的狄拉克算子在这些水平集上诱导出狄拉克型算子。调查人员打算证明， 算子可以从捕获该诱导算子族沿沿着许多这些水平集的无穷小行为的不变量中恢复。更确切地说，索引将是两种类型的不变量的总和：软不变量和硬不变量。软不变量由无穷多个光滑水平集附近的无穷小谱流描述，而硬不变量由奇异水平集正则确定的无穷维拉格朗日空间的三元组的Kashiwara-Wall指标描述，Dirac型方程是Dirac型方程的推广，而Dirac型方程是Dirac型方程的推广。 著名的电磁学麦克斯韦方程。 这样一个方程的解可以看作是宇宙中的一种定常电磁波 下 调查，被称为背景流形。的 狄拉克方程的解的数量高度依赖于流形（或宇宙）的形状。 莫尔斯理论是一种通过分解流形来研究流形形状的技术 一些基本的片段。调查人员打算解释如何 通过研究狄拉克方程在这些基本空间上的行为来计算狄拉克方程的解的数量。