Reductions of non-lc-ideals and non-$F$-pure ideals assuming weak ordinarity

Abstract: It has been known for several decades that there are close connections between certain classes of singularities in the Minimal Model Program over $\mathbb{C}$ and so-called $F$-singularities which are defined in positive characteristic via Frobenius. There is a more refined conjecture which relates multiplier ideal filtrations and test ideal filtrations. The triviality of certain parts of this filtration may be used to define some of the singularities mentioned previously. It is known that this conjecture is equivalent to the so-called weak ordinarity conjecture from arithmetic geometry (which roughly speaking asserts that if $X$ is a smooth projective varietiy of dimension $d$ defined over $\Spec \mathbb{Z}$ then its reductions mod $p$ admit bijective Frobenius action on $H^{d-1}(X_p,\mathcal{O}_{X_p})$ for inifinitely many $p$) by work of Srinivas, Mustata, Bhatt, Schwede, Takagi. I will survey these results and time permitting talk about a similar conjectural relation between maximal non-lc ideal filtrations and non-$F$-pure ideal filtrations which in some cases is also equivalent to weak ordinarity.