On February 25, 2019 I was invited to give a talk in the Set Theory Seminar at Bar-Ilan University, Israel.

*Projective determinacy for games of length $\omega^2$ and longer*

*Abstract:* We will study infinite two player games and the large
cardinal strength corresponding to their determinacy. For games of
length $\omega$ this is well understood and there is a tight
connection between the determinacy of projective games and the
existence of canonical inner models with Woodin cardinals. For games
of arbitrary countable length, Itay Neeman proved the determinacy of
analytic games of length $\omega \cdot \theta$ for countable $\theta
> \omega$ from a sharp for $\theta$ Woodin cardinals.

We aim for a converse at successor ordinals. In joint work with Juan P. Aguilera we showed that determinacy of $\boldsymbol\Pi^1_{n+1}$ games of length $\omega^2$ implies the existence of a premouse with $\omega+n$ Woodin cardinals. This generalizes to a premouse with $\omega+\omega$ Woodin cardinals from the determinacy of games of length $\omega^2$ with $\Game^{\mathbb{R}}\boldsymbol\Pi^1_1$ payoff.

If time allows, we will also sketch how these methods can be adapted to, in combination with results of Nam Trang, obtain $\omega^\alpha+n$ Woodin cardinals for countable ordinals $\alpha$ and natural numbers $n$ from the determinacy of sufficiently long projective games.