  In this paper, we study a positive characteristic analogue of the centers of
log canonicity of a pair $(R, \Delta)$. We call these analogues centers of
$F$-purity. We prove positive characteristic analogues of subadjunction-like
results, prove new stronger subadjunction-like results, and in some cases, lift
these new results to characteristic zero. Using a generalization of centers of
$F$-purity which we call uniformly $F$-compatible ideals, we give a
characterization of the test ideal (which unifies several previous
characterizations). Finally, in the case that $\Delta = 0$, we show that
uniformly $F$-compatible ideals coincide with the annihilators of the
$\mathcal{F}(E_R(k))$-submodules of $E_R(k)$ as defined by Smith and Lyubeznik.
