Abstract: The Bieri--Neumann--Strebel--Renz invariants $\Sigma^q(X)$ of a connected, finite-type CW-complex $X$ are the vanishing loci for the Novikov--Sikorav homology of $X$ in degrees up to $q$. These invariants live in the unit sphere inside $H^1(X,\mathbb{R})$; this sphere can be thought of as parametrizing all free abelian covers of $X$, while the $\Sigma$-invariants keep track of the geometric finiteness properties of those covers. On the other hand, the characteristic varieties $V^q(X) \subset H^1(X,\mathbb{C}^{*})$ are the non-vanishing loci in degree $q$ for homology with coefficients in rank $1$ local systems. After explaining these notions and providing motivation, I will describe a rather surprising connection between these objects, to wit: each BNSR invariant $\Sigma^q(X)$ is contained in the complement of the tropicalization of $V^{\le q}(X)$. I will conclude with some examples and applications pertaining to complex geometry, group theory, and low-dimensional topology.