Abstract: Given a normal integral scheme of finite type over a mixed characteristic complete DVR or a perfect field of characteristic p, one can define the notion of a test ideal. This sheaf is used to characterize a class of mild singularities known as splinter singularities, which are analogous to rational singularities in characteristic 0. In equal characteristics, it is a well-known result that test and multiplier ideals are stable under small perturbations. In this talk, I will explain how to extend this stability result to the mixed characteristic setting and discuss some of its applications. Time permitting, I will also outline the key ideas and tools from p-adic geometry that underlie the proof. This is based on joint work in progress with Bhargav Bhatt and Linquan Ma.

Abstract: A recent manuscript written by Claude under the direction of Levent Alpöge constructs a 1-parameter family of compact complex threefolds diffeomorphic to S^6. I will describe the main geometric ideas behind the construction and outline the proof. If time permits, I will remark on possible future research directions.