Around cellular BV theories with Atiyah-Segal gluing

We will present an explicit example of a Lagrangian topological field theory living on manifolds endowed with cellular decompositions, which is a gauge theory (in the sense that it satisfies the BV master equation), is consistent with cellular aggregations (represented by BV pushforwards) and satisfies a Segal-like gluing formula with respect to gluing/cutting of manifolds. This example corresponds to the "BF-theory". Time permitting, we will explain another example, working very similarly - the 1-dimensional Chern-Simons theory. This is based in part on joint work with A. S. Cattaneo and N. Reshetikhin.