Stationary reflection and the failure of SCH at $\aleph_{\omega_1}$

Stationary reflection is a compactness type principle for infinite objects. Roughly speaking, it says that if a set is “positive”, then it has “positive” initial segments. A recurring question in set theory is what are the implications between such principles and cardinal arithmetic. Here we focus on the singular cardinal hypothesis (SCH), which is parallel of CH for singular cardinals.  In contrast to stationary reflection, the failure of SCH presents an instance of incompactness. Obtaining stationary reflection together with the not SCH has been a long standing project. In the last several years, it was shown that this can happen for a singular cardinal $\kappa$ high up, and also for $\aleph_\omega$. Ben-Neria, Hayut and Unger asked if it can also hold for $\aleph_\omega_1$. In this talk we give a positive answer. This is joint work with Tom Benhamou.