Linear Request Problem: Rigorous Results

This lecture explores the linear request problem, focusing on the setup where a smooth map on a closed manifold preserves a smooth volume form. It delves into infinitesimal perturbations, transport plans, Wasserstein metric, and deterministic transport plans. The instructor presents the concept of infinitesimal transport plans, differentiability, and continuity equations. The main result, a theorem from 2018, is discussed, showing the existence of a smooth family preserving volume forms. The lecture concludes by addressing optimality, regularity, and the importance of defining perturbations for a good linear response and request.