Alexandrov spaces naturally appear in the collapsing theory of Riemannian manifolds, which are metric spaces having the notion of lower curvature bound. In the study of Alexandrov spaces, Perelman's stability theorem is very important.
As a byproduct of the stability theorem, we know that Alexandrov spaces contained in a suitable moduli space have finitely many topological types. In 2019, Yamaguchi and the speaker obtained Lipschitz homotopy finiteness of Alexandrov spaces.
Lipschitz homotopy means homotopy which is Lipschitz continuous. Recently, Fujioka, Yamaguchi and myself refine the previous result. I will talk about these contents.