**November 18**

**Dima Sinapova**,
Rutgers University

**Prikry sequences and square properties at $\aleph_\omega$**

It is well known that if an inaccessible cardinal $\kappa$ is singularized to countable cofinality while preserving cardinals, then $\square_{\kappa_\omega}$ holds in the outer model. Moreover, this remains true even when relaxing the cardinal preservation assumption a bit. In this talk we focus on when Prikry forcing adds weaker forms of square in a more general setting. We prove abstract theorems about when Prikry forcing with interleaved collapses to bring down the singularized cardinal to $\aleph_\omega$ will add a weak square sequence. This can be viewed as a partial positive result to a question of Woodin about whether the failure of SCH at $\aleph_\omega$ implies weak square.