Optimal Transport: Gradient Flows in Rd

This lecture covers the concept of optimal transport and its application in gradient flows in Rd, focusing on the analysis of minimizing movements and the convergence of piecewise constant solutions. The instructor explains the theory behind convex functions, geodesic spaces, and the implicit Euler scheme, emphasizing the importance of Lipschitz and Picard-Lindelöf theorems in guaranteeing convergence. The lecture concludes with a discussion on constructing discrete solutions and the implications of weak convergence in Hilbert spaces.