On September 23, 2015, I gave a talk in the Minisymposium *Set Theory* at the DMV-Jahrestagung in Hamburg.

*Mice with finitely many Woodin cardinals from optimal determinacy hypotheses*

*Abstract:* Mice are countable sufficiently iterable models of set theory. Itay Neeman has shown that the existence of such mice with finitely many Woodin cardinals implies that projective determinacy holds. In fact he proved that the existence and $\omega_1$-iterability of $M^{\sharp}_n(x)$ for all reals $x$ implies that boldface $\Pi^1_{n+1}$-determinacy holds.

We prove the converse of this result, that means boldface $\Pi^1_{n+1}$-determinacy implies that $M^{\sharp}_n(x)$ exists and is $\omega_1$-iterable for all reals $x$. This level-wise connection between mice and projective determinacy is an old so far unpublished result by W. Hugh Woodin. As a consequence we can obtain the determinacy transfer theorem for all levels $n$. These results connect the areas of inner model theory and descriptive set theory, so we will give an overview of the relevant topics in both fields and briefly sketch a proof of the result mentioned above. The first goal is to show how to derive a model of set theory with Woodin cardinals from a determinacy hypothesis. The second goal is to prove that there is such a model which is iterable. For this part the odd and even levels of the projective hierarchy are treated differently.

This is joint work with Ralf Schindler and W. Hugh Woodin