Abstract

The study of boundary value problems on smooth manifolds with boundary is a classic and well studied subject. A significant development which highlighted the importance of the subject was the celebrated index formula of Atiyah-Patodi-Singer in the 1970s. This, in particular, enshrined the place of non-local boundary conditions in geometry, as local boundary conditions were discovered to be topologically obstructed for index formulae. However, much of the study in the subject has been restricted to compact boundary and for Dirac-type operators or their close cousins.

More recently, Bär-Bandara extended the study beyond Dirac-type operators, assuming only ellipticity of the operator, though retaining the restriction of compact boundary. This was made possible through the introduction of the H-infinity functional calculus as the central analytic instrument. An unintended consequence of the use of this "global" technique was that it suggested the possibility of understanding non-local boundary conditions for the non-compact case.

This talk will focus on recent results and advancements when the boundary is non-compact. A trace theorem is obtained on the maximal extension of the operator, ensuring the possibility to understand *all* possible boundary conditions, which is in particular significant in the study of regularity and perturbations of boundary conditions. Fredholmness is studied in contexts where the operator is invertible outside of a compact set. These results are obtained under a set of assumptions, which are shown to be satisfied under second fundamental form and scalar curvature curvature bounds  for the Spin-Dirac operator.