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 $\VV^q(X) \subset H^1(X,\mathbb{C}^{*})$
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.