The entropy-degree theorem for Alexandrov spaces

I will present an entropy-degree theorem for Lipschitz maps between Alexandrov spaces with curvature bounded below and negatively curved locally symmetric spaces, extending the classical volume-entropy rigidity of Besson–Courtois–Gallot to this singular setting. A central component of this result is a new degree theory for Alexandrov spaces, which I develop using the Ambrosio–Kirchheim theory of integral currents to show the equivalence of analytical and topological degrees. This framework provides a unified approach to diverse geometric problems, including volume stability for Alexandrov spaces, volume bounds for cone-manifolds, and topological obstructions for negatively curved Einstein metrics on 4-orbifolds. Furthermore, by synthesizing these methods with metric doublings, I derive quantitative volume inequalities for hyperbolic convex cores and lower bounds on the asymptotic translation lengths of end-periodic homeomorphisms. While time will not permit a full discussion of all these applications, I welcome further inquiries about these topics following the presentation.