Dirac operators on cobordisms: degenerations and surgery

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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.