Abstract: It has been known for several decades that there are close connections between certain classes of singularities in the Mini mal Model Program over $\\mathbb{C}$ and so-called $F$-singularities which are defined in positive characteristic via Frobenius. There is a more ref ined conjecture which relates multiplier ideal filtrations and test ideal filtrations. The triviality of certain parts of this filtration may be use d to define some of the singularities mentioned previously. It is known th at this conjecture is equivalent to the so-called weak ordinarity conjectu re from arithmetic geometry (which roughly speaking asserts that if $X$ is a smooth projective varietiy of dimension $d$ defined over $\\Spec \\math bb{Z}$ 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 Srini vas\, 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.

LOCATION:PH H3 33 https://plan.epfl.ch/?room=PHH333 STATUS:CONFIRMED END:VEVENT END:VCALENDAR