**February 23**

Shoshana Friedman,
CUNY

**Large cardinals and HOD**

In his work on the HOD conjecture, Woodin isolates the concept of of an inner model $N$ being a *weak extender model for $\delta$ is supercompact*: $N$ is an inner model of ZFC, and for every $\gamma>\delta$, there is a $\delta$-complete normal fine measure $U$ on $P_\delta(\gamma)$ such that $N\cap P_\delta(\gamma)\in U$ and $U\cap N\in N$. In the case of HOD, when it is a weak extender model for $\delta$ is supercompact that implies that $\delta$ is HOD-supercompact. In this talk I will separate the concepts of supercompactness, supercompactness in HOD and being HOD-supercompact and how these concepts will be affected by the assumption of the HOD hypothesis.

This is joint work with Arthur Apter and Gunter Fuchs.