  We introduce and study a new Radon-like transform that averages projected
differential p-forms in R^n over affine (n-k)-planes. We then prove an explicit
inversion formula for our transform on the space of rapidly-decaying smooth
p-forms. Our transform differs from the one in Gelfand-Graev-Shapiro. Moreover,
if it can be extended to a somewhat larger space of p-forms, our inversion
formula will allow the synthesis of any rapidly-decaying smooth p-form on R^n
as a (continuous) superposition of pullbacks from p-forms on k-dimensional
subspaces. In turn, such synthesis implies an explicit formula (which we
derive) for reconstructing compactly supported currents in R^n (e.g., compact
oriented k-dimensional subvarieties) from their oriented projections onto
k-planes.
