  The timescales of many physical, chemical, and biological processes are
determined by first passage times (FPTs) of diffusion. The overwhelming
majority of FPT research studies the time it takes a single diffusive searcher
to find a target. However, the more relevant quantity in many systems is the
time it takes the fastest searcher to find a target from a large group of
searchers. This fastest FPT depends on extremely rare events and has a
drastically faster timescale than the FPT of a given single searcher. In this
work, we prove a simple explicit formula for every moment of the fastest FPT.
The formula is remarkably universal, as it holds for $d$-dimensional diffusion
processes (i) with general space-dependent diffusivities and force fields, (ii)
on Riemannian manifolds, (iii) in the presence of reflecting obstacles, and
(iv) with partially absorbing targets. Our results rigorously confirm,
generalize, correct, and unify various conjectures and heuristics about the
fastest FPT.
