Lajos Soukup: Independence around $\aleph_\omega$

In recent decades, several topological and set-theoretical statements have been shown to be independent of ZFC by demonstrating that they hold under GCH + $\Box_{\aleph_\omega}$, while their negations follow from GCH and a suitable version of the Chang Conjecture — specifically, the principle $(\aleph_{\omega+1}, \aleph_\omega) \twoheadrightarrow (\aleph_1, \aleph_0)$. In this talk, we will formulate several such statements and explore the possible implications between them.