## Logic Seminar

The Ultrapower Axiom is a structural principle for ultrafilters expected to hold in the canonical inner model for a supercompact cardinal if it exists. The existence of such a model should also imply the HOD Conjecture, which asserts that under large cardinal hypotheses, HOD correctly computes successors of singular cardinals. The proof that the HOD Conjecture follows from UA motivates the study of definability from ultrafilters under large cardinal assumptions. Combined with techniques from set theoretic geology, this investigation leads to some new structural properties of large cardinals related to both UA and the HOD Conjecture. For example, while it is consistent that there is an inner model $M$ that admits $2^{2^\kappa}$ many elementary embeddings $j : V\to M$ where $\kappa$ is the least measurable cardinal, we show that if there is an extendible cardinal $\delta$ and a proper class of strongly compact cardinals, then any inner model $M$ admits at most one elementary embedding $j : V\to M$ with critical point greater than $\delta$.