  The Whitney forms on a simplex $T$ admit high-order generalizations that have
received a great deal of attention in numerical analysis. Less well-known are
the shadow forms of Brasselet, Goresky, and MacPherson. These forms generalize
the Whitney forms, but have rational coefficients, allowing singularities near
the faces of $T$. Motivated by numerical problems that exhibit these kinds of
singularities, we introduce degrees of freedom for the shadow $k$-forms that
are well-suited for finite element implementations. In particular, we show that
the degrees of freedom for the shadow forms are given by integration over the
$k$-dimensional faces of the blow-up $\tilde T$ of the simplex $T$.
Consequently, we obtain an isomorphism between the cohomology of the complex of
shadow forms and the cellular cohomology of $\tilde T$, which vanishes except
in degree zero. Additionally, we discover a surprising probabilistic
interpretation of shadow forms in terms of Poisson processes. This perspective
simplifies several proofs and gives a way of computing bases for the shadow
forms using a straightforward combinatorial calculation.
