Preservation Theorems and Forcing Axioms for Topological Properties

An active line of research in set-theoretic topology concerns how robust certain topological properties are under forcing extensions. More precisely, we are interested in questions of the form: when does a specific forcing or a forcing of such-and-such type preserve some topological property P? In this talk we will discuss how strongly proper forcings can be used to provide some new and general preservation theorems. Then we will see how to leverage this insight to obtain a forcing axiom for proper forcings that preserve a given Lindelöf space. The first part of the talk concerns joint work with Jared Holshouser, and the second part concerns work just of the speaker.