The Strong Monodromy Conjecture predicts a mysterious relationship between the non-Archimedean and Archimedean worlds. It suggests that, for a polynomial f, the operation of integrating its p-adic norm is governed by the integrals of its Archimedean norm.

Hyperplane arrangements provide a fertile testing ground to investigate these connections. In 2009, Budur-Mustațǎ-Teitler proved the weak Monodromy Conjecture for hyperplane arrangements and introduced the n/d conjecture, demonstrating that the latter implies the strong version. Although verified for many cases, a full resolution of the n/d conjecture has remained elusive.

In this talk, I will present a proof of the n/d conjecture, which relies on a new approach that emerges from a confluence of recent Hodge-theoretic approaches to singularities, and the recent breakthrough on the unitary dual of Lie groups. This is based on the joint work with Dougal Davis.