On June 9th 2016 I gave a talk in the KGRC Research Seminar in Vienna.

**Hybrid Mice and Determinacy in the $L(\mathbb{R})$-hierarchy.**

**Abstract**: This talk will be an introduction to inner model theory the at the

level of the $L(\mathbb{R})$-hierarchy. It will

focus on results connecting inner model theory to the determinacy of

certain games.

Mice are sufficiently iterable models of set theory. Martin and Steel

showed in 1989 that the existence of finitely many Woodin cardinals

with a measurable cardinal above them implies that projective

determinacy holds. Neeman and Woodin proved a level-by-level

connection between mice and projective determinacy. They showed that

boldface $\boldsymbol\Pi^1_{n+1}$ determinacy is equivalent to the fact that the

mouse $M_n^\#(x)$ exists and is $\omega_1$-iterable for all reals $x$.

Following this, we will consider pointclasses in the $L(\mathbb{R})$-hierarchy

and show that determinacy for them implies the existence and

$\omega_1$-iterability of certain hybrid mice with finitely many

Woodin cardinals, which we call $M_k^{\Sigma, \#}$. These hybrid mice

are like ordinary mice, but equipped with an iteration strategy for a

mouse they are containing, which enables them to capture certain sets

of reals. We will discuss what it means for a mouse to capture a set

of reals and outline why hybrid mice fulfill this task. If time allows we will sketch a proof that determinacy for sets of reals in the $L(\mathbb{R})$-hierarchy implies the existence of hybrid mice.

Many thanks to Richard for the pictures!