The phrase "BV formalism" means many things. The first half of my talk will focus on its most basic meaning: a systematic way to organize the "integration by parts" method from undergraduate calculus into a packaging amenable to more general homological algebra (namely, into a twisted de Rham complex). Particularly useful is the Homological Perturbation Lemma.

It assures that the algorithms we teach to undergraduates terminate; it produces Feynman diagrams, the ur-example of "perturbative" physics; and, as I will explain, it also applies to nonperturbative integrals, providing a nonperturbative version of "stationary phase approximation".

The second half of my talk will generalize the earlier discussion. The "BV formalism" suggests that any system with algebraic properties similar to a twisted de Rham complex should be thought of as an "oscillating integral problem". I will explain one origin of such systems, called the "AKSZ construction". BV-type systems are amenable to homological perturbation lemma techniques. Time permitting, I will explain how I had hoped to prove Kontsevich formality, why my proof failed, and what that failure says about Poincare duality.