Optimal Transport: Introduction and Kantorovich Problem

This lecture introduces the course on optimal transport, discussing the schedule, references, and format. The instructor emphasizes the importance of interactive learning and provides technical details about the course structure, including exercise classes and exams. The lecture covers the historical background of the optimal transport problem, starting with Monge's work in the 18th century. It explains the concepts of push forward measures, transport maps, and couplings of measures. The Kantorovich problem is presented as a minimization task involving cost functions and transport plans. The lecture concludes with a discussion on the Brenier theorem, which establishes the existence of unique optimal transport maps in specific scenarios.