  First, we review the Dirac operator folklore about basic analytic and
geometrical properties of operators of Dirac type on compact manifolds with
smooth boundary and on closed partitioned manifolds and show how these
properties depend on the construction of a canonical invertible double and are
related to the concept of the Calderon projection. Then we summarize a recent
construction of a canonical invertible double for general first order elliptic
differential operators over smooth compact manifolds with boundary. We derive a
natural formula for the Calderon projection which yields a generalization of
the famous Cobordism Theorem. We provide a list of assumptions to obtain a
continuous variation of the Calderon projection under smooth variation of the
coefficients. That yields various new spectral flow theorems. Finally, we
sketch a research program for confining, respectively closing, the last re-
maining gaps between the geometric Dirac operator type situation and the
general linear elliptic case.

Differential Geometry;Functional Analysis