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

 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.