I will give an invited talk in the Section in Logic at the annual meeting GDMV of the German Mathematical Society (DMV) joint with GDM taking place March 5th to 9th in Paderborn, Germany.

*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.

In this talk, we will discuss the basics behind these results and show the first steps towards generalizing them to the projective hierarchy, assuming projective determinacy (PD). In particular, we will outline that under PD every homogeneous projective space is in fact strongly homogeneous.

This is joint work with Raphaël Carroy and Andrea Medini.