I am invited to give a talk at the 17th International Luminy Workshop in Set Theory in Luminy, France, October 9 - 13, 2023.

*Generic absoluteness for the definable powerset of the universally Baire sets*

Universally Baire sets play an important role in studying canonical models with large cardinals. But to reach higher large cardinals more complicated objects, for example, canonical subsets of universally Baire sets come into play. Inspired by core model induction, we introduce the definable powerset $\mathcal{A}^\infty$ of the universally Baire sets $\Gamma^\infty$ and show that, after collapsing a large cardinal, $L(\mathcal{A}^\infty)$ is a model of determinacy and its theory cannot be changed by forcing. Our main technical tool is an iteration that realizes the universally Baire sets as the sets of reals in a derived model of some iterate of $V$, from a supercompact cardinal $\kappa$ and a proper class of Woodin cardinals. This is joint work with Grigor Sargsyan.