Dirac operators on cobordisms: degenerations and surgery

流形上的莫尔斯函数将流形分解成无穷多个超曲面（水平集）。典型的水平集是平滑的，但其中一些有奇点。流形上的狄拉克算子推导出这些水平集上的狄拉克型算子。研究者打算证明原始算子的指标可以从不变量中恢复，这些不变量捕获了这类诱导算子沿着有限多个水平集的无穷小行为。更准确地说，索引将是两种不变量的总和：软不变量和硬不变量。软不变量用有限多个光滑水平集附近的无限小谱流来描述，而硬不变量用无限维拉格朗日空间三元组的Kashiwara-Wall指标来描述，这些三元组通常由奇异水平集决定。狄拉克方程是著名的电磁学麦克斯韦方程的推广。这样一个方程的解可以被看作是宇宙中一种固定的电磁波，被称为背景流形。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将流形分解成若干基本块来研究流形形状的技术。研究者打算通过研究狄拉克方程在这些基本空间块上的行为来解释如何计算狄拉克方程的解的数目。

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.