Abstract: I will explain the background to my current research on a combinatorial analogue to forms naturally arising in the Chern-Weil theory of characteristic classes. These forms are called "transgression forms" and are defined as follows: Given a vector bundle over a smooth manifold, Chern-Weil theory provides a local expression for forms representing characteristic classes of the bundle. The construction is effected via a map from invariant polynomials over the general linear group to the cohomology ring of the base. The method also involves a choice of connection, but for a fixed invariant polynomial, the forms constructed by two connections differ by an exact form. A (local) form whose exterior derivative is such a difference is said to be "transgressive". Chern-Weil theory also provides a local formula for a canonical choice of such transgression forms. The talk will present ongoing work in finding a local formula over simplicial complexes for a combinatorial analogue to transgression forms.