The method of Feynman diagrams is a well-known example of \emph{algebraization} of integration. Specifically, Feynman diagrams algebraize the asymptotics of integrals of the form in the limit as along the pure imaginary axis, supposing that has only nondegenerate critical points. (In quantum field theory, is the ``action,'' and is an ``observable.'') In this talk, I will describe an analogous algebraization when --- no formal power series will appear --- and is allowed degenerate critical points. Nevertheless, some features from Feynman diagrams remain: I will explain how to algebraically ``integrate out the higher modes'' and reduce any such integral to the critical locus of ; the primary tool will be a \emph{homological} form of perturbation theory (itself almost as old as Feynman's diagrams). One of the main new features in nonperturbative integration is that the critical locus of must be
interpreted in the \emph{scheme-theoretic} sense, and in particular imaginary critical points do contribute. Perhaps this will shed light on questions like the Volume Conjecture, in which an integral over
connections is dominated by a critical point in .