I was invited to give an **invited talk** at the first Gdansk Logic Conference, University of Gdansk, Poland, May 5-7, 2023.

*Determinacy and generic absoluteness for the definable powerset of
the universally Baire sets*

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.