Collection Convex analysis in geodesic spaces
In this course, we will go through the development of a young area of convex analysis in geodesic spaces, especially the CAT(k) spaces. This space can be thought of as the nondifferentiable counterpart of Riemannian manifolds. Riemannian manifolds of certain curvatures, metric trees, cubical complexes, and the space of phylogenetics are typical examples of a CAT(k) space. The theory flourished with an attempt to generalize fixed point theory for nonexpansive maps outside of a linear space, followed by the study of the proximal algorithm of (geodesic) convex functions that enriches the study of abstract gradient flows. Further development of convex subdifferentials and monotone vector fields are then addressed, and a few interesting applications will be presented.