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.