  In this paper, we prove a longstanding conjecture on the asymptotic behavior
of a pair of oscillatory matrix integrals: the Harish-Chandra/Itzykson-Zuber
(HCIZ) integral, and the Brezin-Gross-Witten (BGW) integral. The main result
gives a complete asymptotic expansion of these integrals for small complex
parameters. The coefficients of these asymptotic expansions are generating
functions for monotone Hurwitz numbers sorted by genus.
