The local topological monodromy conjecture

Let $f$ be a polynomial with coefficients in a field of characteristic $0$, and let $Z_\mathrm{top}(s)$ be its local topological zeta function. If $\alpha$ is a pole of $Z_\mathrm{top}(s)$, then $\exp(2\pi i\alpha)$ is a nearby eigenvalue of monodromy. The local topological monodromy conjecture. If $\alpha$ is a pole of $Z_\mathrm{top}(s)$, then $\exp(2\pi i\alpha)$ is a nearby eigenvalue of monodromy. The conjecture is the topological counterpart of the local $p$-adic and motivic monodromy conjectures; the surrounding discussion explains its relationship with poles arising from resolutions and with $p$-adic zeta functions.