In September 2018 I am invited to give a talk in the Thematic Session in Set Theory and Topology at the joint meeting of the Italian Mathematical Union, the Italian Society of Industrial and Applied Mathematics and the Polish Mathematical Society (UMI-SIMAI-PTM) in Wrocław.

*Large Cardinals in the Stable Core*

*Abstract:* The Stable Core $\mathbb{S}$, introduced by Sy Friedman in
2012, is a proper class model of the form $(L[S],S)$ for a simply
definable predicate $S$. He showed that $V$ is generic over the
Stable Core (for $\mathbb{S}$-definable dense classes) and that the
Stable Core can be properly contained in HOD. These remarkable
results motivate the study of the Stable Core itself. In the light of
other canonical inner models the questions whether the Stable Core
satisfies GCH or whether large cardinals is $V$ imply their existence
in the Stable Core naturally arise. We answer these questions and
show that GCH can fail at all regular cardinals in the Stable
Core. Moreover, we show that measurable cardinals in general need not
be downward absolute to the Stable Core, but in the special case
where $V = L[\mu]$ is the canonical inner model for one measurable
cardinal, the Stable Core is in fact equal to $L[\mu]$.

This is joint work with Sy Friedman and Victoria Gitman.