I am invited to give a talk on January 13, 2025 at the Workshop on Set Theory at Mathematisches Forschungsinstitut Oberwolfach, Germany.
Generic absoluteness for Chang-type models and LSA
Universally Baire sets play an important role in studying canonical models with large cardinals. Inspired by work of Sargsyan and Trang, we introduce a new technique for establishing generic absoluteness results for models containing the universally Baire sets. 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 the universe. Inspired by core model induction, we introduce the definable powerset $\mathcal{A}^\infty$ of the universally Baire sets 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.
In a second step, we study extensions of the model $L(\mathcal{A}^\infty)$ with additional countable sequences of reals and argue that, if there are two supercompact cardinals, after collapsing the powerset of the first supercompact, there are LSA pointclasses cofinally in the universally Baire sets.
This is joint work with Lukas Koschat and Grigor Sargsyan.