The descriptive theory of represented spaces

On the surface, the emergent descriptive theory of represented spaces is the extension of both classical [36] and effective [52] descriptive set theory from its usual setting of Polish spaces to the much larger class of represented spaces. There is far more to it, though: By extending the setting, suddenly connections to areas such as category theory (with a certain topos-theoretic flavour) and computability-theoretic degree theory appear at a fundamental level. Initial results do inspire some hope that this new theory can help to solve existing open problems, elucidate further the core aspects of known theorems and provide both new questions and theorems in its own right. The present text is meant both as a survey of existing work relevant to (or part of) the descriptive theory of represented spaces, and to outline some exciting directions of further research.