On January 22, 2019 I gave a talk at the (Arctic Set Theory Workshop 4) in Kilpisjärvi, Finland.

*Homogeneous Spaces and Wadge Theory*

*Abstract:* In his PhD thesis Wadge characterized the notion of
continuous reducibility on the Baire space ${}^\omega\omega$ in form
of a game and analyzed it in a systematic way. He defined a refinement
of the Borel hierarchy, called the Wadge hierarchy, showed that it is
well-founded, and (assuming determinacy for Borel sets) proved that
every Borel pointclass appears in this classification. Later Louveau
found a description of all levels in the Borel Wadge hierarchy using
Boolean operations on sets. Fons van Engelen used this description to
analyze Borel homogeneous spaces and show that every homogeneous Borel
space is in fact strongly homogeneous.

In this talk, we will show how to generalize these results under the Axiom of Determinacy. In particular, we will outline that under AD every homogeneous space is in fact strongly homogeneous.

This is joint work with Raphaël Carroy and Andrea Medini.

Slides can be found here.