# Analysis and PDE seminar: First-order elliptic boundary value problems beyond self-adjoint induced boundary operators.

The Bär-Ballmann framework is a comprehensive framework to consider
elliptic boundary value problems (and also their index theory) for
first-order elliptic operators on manifolds with compact and smooth
boundary. A fundamental assumption in their work is that the induced
operator on the boundary is symmetric. Many operators satisfy this
requirement including the Hodge-Dirac operator as well as the Atiyah-
Singer Dirac operator. Recently, there has been a desire to study more
general operators with the quintessential example being the  Rarita-
Schwinger Dirac operator, which is an operator that fails to satisfy
this hypothesis.

In this talk, I will present recent work with Bär where we dispense the
symmetry assumption and consider general elliptic operators. The
ellipticity of the operator still allows us to understand the spectral
theory of the induced operator on the boundary, modulo a lower
order additive perturbation, as bi-sectorial operator. We use a mixture
of methods coming from pseudo-differential operator theory, bounded
holomorphic functional calculus, semi-group theory as well as methods
arising from the resolution of the Kato square root problem to recover
many of the results of the Bär-Ballman framework.

If time permits, I will also touch on the non-compact boundary case, and potential extensions of this to the $L^p$ setting and Lipschitz boundary.