The BV pushforward is a simple construction that adapts the idea of integrating
the ultraviolet fields in Wilson renormalization to the Batalin–Vilkovisky setting. If one
applies it to AKSZ theories, that are topological, one can see interesting relations with
physical theories defined in the infra red. After a brief review of basic facts of BV, I will discuss
examples of this construction in two dimensions. I will also discuss its application to the
equivariant extension of Yang-Mills theory in four dimensions.