Uniform asymptotics of Toeplitz determinants with Fisher-Hartwig singularities

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.