Optimal Transportation, Geometry and Dynamics

Explore the mathematical theory of optimal transportation and its connections to geometry and dynamical systems through this comprehensive graduate-level course from the Fields Institute. Delve into the fundamental principles of optimal transport theory, which studies the most efficient ways to move mass from one distribution to another while minimizing cost. Examine the deep geometric structures underlying optimal transport problems, including Riemannian geometry, metric spaces, and curvature conditions. Investigate the relationship between optimal transportation and dynamical systems, exploring how transport maps evolve over time and their applications to fluid dynamics and partial differential equations. Study key concepts such as the Monge-Kantorovich problem, Wasserstein distances, displacement interpolation, and the connection to gradient flows in the space of probability measures. Learn about applications to various fields including economics, image processing, machine learning, and mathematical physics. Analyze the geometric properties of optimal transport maps, including regularity theory and the role of convexity. Discover how optimal transportation provides a powerful framework for understanding geometric inequalities, concentration of measure phenomena, and the geometry of probability distributions. Through 24 detailed lectures, gain expertise in this rapidly growing field that bridges pure mathematics with practical applications across multiple disciplines.