  We obtain an asymptotic formula for $n\times n$ Toeplitz determinants as
$n\to \infty$, for real valued symbols with any fixed number of Fisher-Hartwig
singularities, which is uniform with respect to the location of the
singularities. As an application, we prove a conjecture by Fyodorov and Keating
regarding moments of averages of the characteristic polynomial of the Circular
Unitary Ensemble. In addition, we obtain an asymptotic formula regarding the
momentum of impenetrable bosons in one dimension with periodic boundary
conditions.
