**May 28**

Gabriel Goldberg,
University of Berkeley

**The HOD conjecture and the structure of elementary embeddings**

Woodin's HOD conjecture asserts that in the context of very large cardinals, the inner model HOD closely approximates the universe of sets in the same way GĂ¶del's constructible universe does assuming 0# does not exist. The subject of these two talks is the relationship between Woodin's conjecture and certain constraints on the structure of elementary embeddings of the universe of sets. For example, in the second talk, we will prove that any two elementary embeddings of the universe of sets into the same inner model agree on HOD, while if a local version of this theorem held, then the HOD conjecture would follow.