Andrey Lazarev (University of Lancaster)
Chern-Simons type invariants of a smooth manifold $X$ are correlation functions of a certain quantum field theory associated to $X$. There are several ways to make mathematical sense of it; one of them is due to Costello. In his interpretation an invariant is a solution of the quantum master equation (QME), up to a suitable notion of homotopy. Such a solution also goes by the name `quantum L-infinity algebra'. When one removes the adjective `quantum' the theory becomes homotopy invariant and is reduced, effectively, to rational homotopy theory.
In this talk I explain the homological underpinnings of this theory, formulate an alternative non-analytic approach and discuss some open problems. This is joint work with C. Braun.