The book "Topological Methods in Hydrodynamics" by Vladimir I. Arnold and Boris A. Khesin is a comprehensive treatise on the application of topological and geometric methods to the study of fluid dynamics. The authors explore the fundamental aspects of hydrodynamics, emphasizing the deep connections between fluid dynamics and various branches of mathematics, such as complex analysis, topology, stability theory, and bifurcation theory. The book is structured into six main parts: 1. **Group and Hamiltonian Structures of Fluid Dynamics**: This section introduces the symmetry groups and Hamiltonian structures relevant to fluid dynamics, including the Euler equations and their Hamiltonian formulation. It also discusses the group setting of ideal magnetohydrodynamics and finite-dimensional approximations of the Euler equation. 2. **Topology of Steady Fluid Flows**: Here, the authors classify three-dimensional steady flows and present variational principles for steady solutions. They explore stability criteria, linear and exponential stretching of particles, and the features of higher-dimensional steady flows. 3. **Topological Properties of Magnetic and Vorticity Fields**: This part delves into the minimal energy and helicity of frozen-in fields, topological obstructions to energy relaxation, and the Sakharov–Zeldovich minimization problem. It also covers asymptotic linking and crossing numbers, energy of knots, and generalized helicities. 4. **Differential Geometry of Diffeomorphism Groups**: This section examines the curvature properties of diffeomorphism groups, the unreliability of long-term weather predictions, and the exterior geometry of volume-preserving diffeomorphisms. It discusses conjugate points and the infinite diameter of Hamiltonian diffeomorphism groups. 5. **Kinematic Fast Dynamo Problems**: This part explores kinematic dynamos, discrete dynamos, and main antidynamo theorems. It includes dissipative dynamo models and the topological entropy of dynamical systems. 6. **Dynamical Systems with Hydrodynamical Background**: This final section applies hydrodynamic concepts to various dynamical systems, including the Korteweg–de Vries equation, equations of gas dynamics, Kähler geometry, and Sobolev’s equation. The book is rich in mathematical detail and includes numerous references and an index, making it a valuable resource for researchers and students in the fields of fluid dynamics and applied mathematics.The book "Topological Methods in Hydrodynamics" by Vladimir I. Arnold and Boris A. Khesin is a comprehensive treatise on the application of topological and geometric methods to the study of fluid dynamics. The authors explore the fundamental aspects of hydrodynamics, emphasizing the deep connections between fluid dynamics and various branches of mathematics, such as complex analysis, topology, stability theory, and bifurcation theory. The book is structured into six main parts: 1. **Group and Hamiltonian Structures of Fluid Dynamics**: This section introduces the symmetry groups and Hamiltonian structures relevant to fluid dynamics, including the Euler equations and their Hamiltonian formulation. It also discusses the group setting of ideal magnetohydrodynamics and finite-dimensional approximations of the Euler equation. 2. **Topology of Steady Fluid Flows**: Here, the authors classify three-dimensional steady flows and present variational principles for steady solutions. They explore stability criteria, linear and exponential stretching of particles, and the features of higher-dimensional steady flows. 3. **Topological Properties of Magnetic and Vorticity Fields**: This part delves into the minimal energy and helicity of frozen-in fields, topological obstructions to energy relaxation, and the Sakharov–Zeldovich minimization problem. It also covers asymptotic linking and crossing numbers, energy of knots, and generalized helicities. 4. **Differential Geometry of Diffeomorphism Groups**: This section examines the curvature properties of diffeomorphism groups, the unreliability of long-term weather predictions, and the exterior geometry of volume-preserving diffeomorphisms. It discusses conjugate points and the infinite diameter of Hamiltonian diffeomorphism groups. 5. **Kinematic Fast Dynamo Problems**: This part explores kinematic dynamos, discrete dynamos, and main antidynamo theorems. It includes dissipative dynamo models and the topological entropy of dynamical systems. 6. **Dynamical Systems with Hydrodynamical Background**: This final section applies hydrodynamic concepts to various dynamical systems, including the Korteweg–de Vries equation, equations of gas dynamics, Kähler geometry, and Sobolev’s equation. The book is rich in mathematical detail and includes numerous references and an index, making it a valuable resource for researchers and students in the fields of fluid dynamics and applied mathematics.