Abstract: Given a foliation on a variety in positive characteristic, one can define an associated infinitesimal quotient of the variety. This construction is the source of many surprising examples: while it does not change the topology of the variety, it may alter drastically its singularities and its cohomology. In this talk, I will present an efficient way of understanding the MMP singularities of such infinitesimal quotients. Then I will apply this method to exhibit pathological MMP singularities (such as non-Cohen--Macaulay terminal singularities, or locally stable families with non-S2 special fibers).