## Wenliang Zhang

F-singularities are classes of singularities defined via the Frobenius endomorphism in characteristic p. One prominent method to measure these singularities is to introduce ideals that define the loci, such as the test ideal (a characteristic p analog of the multiplier ideal). However, these ideals may behave quite differently from their characteristic 0 counterparts. For instance, the test ideal may fail to satisfy the generic restriction theorem that holds for the multiplier ideal. This prompts a natural question, how do we study F-singularities in families? In this talk, I will discuss recent joint work with Zsolt Patakfalvi and Karl Schwede on the behavior of F-singularities in families. In particular, I will discuss a relative version of the ideals mentioned in the previous paragraph and some restriction theorems for them. If time permits, some global geometric consequences will also be discussed.

## Vu Thanh

Let $k$ be an arbitrary field. Let \$S = (a_1< ...