From 13th to 17th June 2016 the Young Set Theory Workshop will be held in Kopenhagen. For more information see the webpage of the YSTW 2016. The title of my talk will be the following:

**A Journey Through the World of Mice and Games – Projective and Beyond.**

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

level of the projective hierarchy and 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.