The book "Statistical Optimal Transport" provides an introduction to the field of optimal transport, its applications in statistics, and machine learning. It covers foundational concepts, such as the optimal transport problem, Wasserstein distances, and Brenier's theorem, as well as more advanced topics like entropic optimal transport, Wasserstein gradient flows, and metric geometry of the Wasserstein space. The book is structured into several chapters, each focusing on different aspects of optimal transport, including its statistical estimation, computational aspects, and geometric properties. Key topics include: 1. **Optimal Transport**: Introduces the optimal transport problem, including the Monge and Kantorovich formulations, and discusses the existence and uniqueness of solutions. 2. **Wasserstein Distances**: Explains the definition and properties of Wasserstein distances, including their role in metrizing weak convergence and their relationship to other distances like total variation. 3. **Optimal Transport in One Dimension**: Provides a simpler case study where optimal transport is characterized in terms of cumulative distribution functions. 4. **Brenier's Theorem**: Discusses the existence of monotone transport maps in higher dimensions, generalizing the univariate fact that uniform random variables can be used to sample from any distribution. 5. **Entropic Optimal Transport**: Introduces the entropic relaxation of the optimal transport problem and its applications in machine learning. 6. **Wasserstein Gradient Flows**: Explores the dynamics of optimal transport as gradient flows in the Wasserstein space. 7. **Metric Geometry of the Wasserstein Space**: Covers curvature, geodesics, and other geometric aspects of the Wasserstein space. 8. **Wasserstein Barycenters**: Discusses the computation of barycenters in the Wasserstein space. The book aims to provide a concise yet comprehensive introduction to the field, suitable for students and researchers in statistics, machine learning, and applied mathematics. It also includes exercises to help readers apply the concepts learned.The book "Statistical Optimal Transport" provides an introduction to the field of optimal transport, its applications in statistics, and machine learning. It covers foundational concepts, such as the optimal transport problem, Wasserstein distances, and Brenier's theorem, as well as more advanced topics like entropic optimal transport, Wasserstein gradient flows, and metric geometry of the Wasserstein space. The book is structured into several chapters, each focusing on different aspects of optimal transport, including its statistical estimation, computational aspects, and geometric properties. Key topics include: 1. **Optimal Transport**: Introduces the optimal transport problem, including the Monge and Kantorovich formulations, and discusses the existence and uniqueness of solutions. 2. **Wasserstein Distances**: Explains the definition and properties of Wasserstein distances, including their role in metrizing weak convergence and their relationship to other distances like total variation. 3. **Optimal Transport in One Dimension**: Provides a simpler case study where optimal transport is characterized in terms of cumulative distribution functions. 4. **Brenier's Theorem**: Discusses the existence of monotone transport maps in higher dimensions, generalizing the univariate fact that uniform random variables can be used to sample from any distribution. 5. **Entropic Optimal Transport**: Introduces the entropic relaxation of the optimal transport problem and its applications in machine learning. 6. **Wasserstein Gradient Flows**: Explores the dynamics of optimal transport as gradient flows in the Wasserstein space. 7. **Metric Geometry of the Wasserstein Space**: Covers curvature, geodesics, and other geometric aspects of the Wasserstein space. 8. **Wasserstein Barycenters**: Discusses the computation of barycenters in the Wasserstein space. The book aims to provide a concise yet comprehensive introduction to the field, suitable for students and researchers in statistics, machine learning, and applied mathematics. It also includes exercises to help readers apply the concepts learned.