Gradient flows in metric spaces: overview and recent advances

This DPhil short course will serve as an introduction to the theory of gradient flows with an emphasis on the recent advances in metric spaces. More precisely, we will start with an overview of gradient flows from the Euclidean theory to its generalisation to metric spaces, in particular Wasserstein spaces. This also includes a short introduction to the Optimal Transport theory, with a focus on specific concepts and tools useful subsequently. We will then analyse the time-discretisation scheme à la Jordan--Kinderlehrer-Otto (JKO), also known as minimising movement, and discuss the role of convexity in proving stability, uniqueness, and long-time behaviour for the PDE under study. Finally, we will comment on recent advances, e.g., in the study of PDEs on graphs and/or particle approximation of diffusion equations.