Index theory of unbounded Fredholm operators

Abstract: Index theory for norm continuous families of bounded Fredholm operators was developed in the classical work of Atiyah; its analog for self-adjoint operators was developed in the work of Atiyah and Singer. The index theory of elliptic differential operators on closed manifolds is based on these classical results: one can pass from operators of positive order to operators of zeroth order, and such a transformation is continuous.

However, in other situations one needs to deal with weaker topologies on the space of unbounded operators. For example, for elliptic boundary value problems on compact manifolds with boundary, the graphs of corresponding unbounded operators depend continuously on parameter. The topology determined by passing from a closed operator to its graph is called the graph topology. The homotopy type of relevant spaces of unbounded Fredholm operators was determined by M. Joachim in 2003.

My talk is devoted to an index theory of graph continuous families of unbounded Fredholm operators in a Hilbert space. I will show how this theory is related to the classical index theory of bounded Fredholm operators. The talk is based on my recent preprints arXiv:2110.14359 and arXiv:2202.03337.

geometric topologynumber theoryoperator algebrasrepresentation theory

Audience: researchers in the topic

( slides | video )

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Noncommutative Geometry in NYC

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