# Mutual Stationarity and Prikry-type forcings

## Dominik Adolf

### University of California, Berkeley

Mutual stationarity is a property first introduced by Foreman and Magidor to study saturation properties of nonstationary ideals. Given a sequence $\langle\kappa_i : i < \lambda\rangle$ of regular cardinals, a sequence $\langle S_i: i < \lambda\rangle$ with $S_i \subseteq \kappa_i$ stationary for every $i$, is mutually stationary iff there are stationarily many subsets $A \subseteq \sup_{i < \lambda} \kappa_i$ s.t. $\sup(A \cap \kappa_i) \in S_i$ for all $i$ with $\kappa_i \in A$. Consider this second property of a sequence $\langle\kappa_i : i < \lambda\rangle$: there is a forcing $P$ that changes $\text{cof}(\kappa_i)$ to $\eta_i$ without changing cofinalities or cardinalites of ordinals below $\inf{\kappa_i : i < \lambda}$. We want to discuss how, and why, these properties are related.

Dr. Dominik Adolf is currently a postdoctoral fellow at the University of California, Berkeley. He holds a PhD from the University of Münster, completed under the supervision of Ralf Schindler. His research interests include set theory, inner model theory, precipitous ideals, and stationary sets.