  We study mixing times for the totally asymmetric simple exclusion process
(TASEP) on a circle of length $N$ with $k$ particles. We show that the mixing
time is of order $N^2 \min(k,N-k)^{-1/2}$, and that the cutoff phenomenon does
not occur. This confirms behavior which was separately predicted by Jara,
Lacoin and Peres, and it is more broadly believed to hold for integrable models
in the KPZ-universalty class. Our arguments rely on a connection to periodic
last passage percolation with a detailed analysis of flat geodesics, as well as
a novel random extension and time shift argument for last passage percolation.
