  We consider the Hodge Laplacian on manifolds with incomplete edge
singularities, with infinite dimensional von Neumann spaces and intricate
elliptic boundary value theory. We single out a class of its algebraic
self-adjoint extensions. Our microlocal heat kernel construction for algebraic
boundary conditions is guided by the method of signaling solutions by Mooers,
though crucial arguments in the conical case obviously do not carry over to the
setup of edges. We then establish the heat kernel asymptotics for the algebraic
extensions of the Hodge operator on edges, and elaborate on the exotic
phenomena in the heat trace asymptotics which appear in case of a
non-Friedrichs extension.
