The reinterpretation of Maxwell's equations in terms of differential forms which has occured these last years has proved quite beneficial to
computational electromagnetism.  We now understand edge elements, for instance, as a natural brand of finite elements for differential forms of
degree one (electric and magnetic field).  Well-established methods of  "generalized finite differences" flavor, such as those implemented in the
MAFIA codes, can now be understood as a natural discretization of the differential-geometric version of Maxwell's equations, thanks to the
Finite Integration Technique and to the view of network constitutive laws as a discretized version of the Hodge operator.

Since the marriage between finite elements and boundary-integral methods is also a natural thing to do in electromagnetism, one feels the need to 
a reinterpretation in geometric language of standard useful notions such as "single layer" and "double layer" integral representations of fields, aiming at a unified body of concepts. 

This talk will take a few steps in this direction.  We classify various brands of single or double layers of charges or currents in a comprehensive way (they appear as differential forms of order 0 to 2, straight or twisted, living on the surface) and sketch a potential theory for such objects.  We then look for generalizations of the "Dirichlet-to-Neumann map", that connects the trace of a harmonic function with the values of its normal derivative at the boundary.  It appears that there is such a map for each kind of differential form.  They go by pairs, due to a deep duality oif topological nature.  Moreover, these pairs of operators live together in a tightly intertwined structure.  The exploration of this structure sheds a new light on some standard things such as the duality between magnetic charges and amperian currents, and suggests a rational classification of "all possible integral methods".  We may also understand how degrees of freedom for the different kinds of charge- and current-layers should be assigned, and how such discretizations fit with those inside conductive regions.