The Atiyah-Patodi-Singer Index Theorem

Boundary value problems and layer potentials on manifolds with cylindrical ends

“…There are several other potential applications of our results and methods to index theory and to spectral theory on non-compact manifolds, not necessarily with cylindrical ends. See [28,37,38]. When extended to Dirac operators, our results, we hope, will be useful to study Hamiltonians whose potentials have "flat directions," which is important for some questions in string theory.…”

“…As an application, we prove the well-posedness of the non-homogeneous Dirichlet problem on manifolds with boundary and cylindrical ends. We also prove the existence of the Dirichletto-Neumann map, which we show to be a pseudodifferential operator in the calculus of pseudodifferential operators that are "almost translation invariant at infinity," a calculus that is closely related to Melrose's b-calculus [29,28], which we study in this paper. The proof of the convergence of the layer potentials and of the existence of the Dirichlet-to-Neumann map are based on a good understanding of resolvents of elliptic operators that are translation invariant at infinity.…”

Spectral invariants of operators of Dirac type on partitioned manifolds

“…This depends on the background and tastes of the beholder. Geometers and analysts (as opposed to topologists) are likely to find the heat kernel method appealing, because K-theory, Bott periodicity and cobordism theory are avoided, not only for geometric operators which are expressible in terms of twisted Dirac operators, but also largely for more general elliptic pseudo-differential operators, as Melrose has done in [Me93]. Moreover, the heat method gives the index of a "geometric" elliptic differential operator naturally as the integral of a characteristic form (a polynomial of curvature forms) which is expressed solely in terms of the geometry of the operator itself (e.g., curvatures of metric tensors and connections).…”

Operator algebras related to the Bochner–Martinelli integral

“…for instance Chapter 3 in [Mel93]. Of course, having found a solution for any one n, it works for smaller values, simply by dropping some of the matrices.…”