Boise extravaganza in set theory, Riverside, June 2014. (Riverside slides)
*Abstract: Considerable work has been done to study descriptive set theory in spaces of size larger than the continuum. In order to generalize the notion of Polish space or standard Borel space to this setting, it is necessary to find a replacement for the notion of complete metrizability. In this talk we will study a class of spaces satisfying a variant of the Choquet property, where the Choquet game is replaced by an analogous game of length $\kappa$. We will show for example that there is a surjectively universal space in this class. We then give a Kuratowski-like result that under appropriate hypotheses, any two such spaces are isomorphic by a $\kappa$-Borel function. This is joint work with Philipp Schlicht.