This book, authored by Dusa McDuff and Dietmar Salamon, is a comprehensive treatment of J-holomorphic curves and their applications in symplectic topology. The content is divided into several chapters, each focusing on different aspects of the subject: 1. **Introduction**: Provides an overview of symplectic manifolds, moduli spaces, evaluation maps, and Gromov-Witten invariants. 2. **J-holomorphic Curves**: Discusses almost complex structures, the nonlinear Cauchy-Riemann equations, unique continuation, critical points, and the adjunction inequality. 3. **Moduli Spaces and Transversality**: Explores moduli spaces of simple curves, transversality, regularity criteria, and implicit function theorems. 4. **Compactness**: Covers energy, bubbling phenomena, mean value and isoperimetric inequalities, singularities, and convergence. 5. **Stable Maps**: Introduces stable maps, Gromov convergence, compactness, uniqueness of limits, and the Gromov topology. 6. **Moduli Spaces of Stable Maps**: Focuses on simple stable maps, transversality, semipositivity, pseudocycles, and Gromov-Witten pseudocycles. 7. **Gromov-Witten Invariants**: Discusses counting pseudoholomorphic spheres, variations in definitions, counting pseudoholomorphic graphs, and rational curves in projective spaces. 8. **Hamiltonian Perturbations**: Examines trivial bundles, locally Hamiltonian fibrations, pseudoholomorphic sections, and counting pseudoholomorphic sections. 9. **Applications in Symplectic Topology**: Highlights periodic orbits of Hamiltonian systems, obstructions to Lagrangian embeddings, the nonsqueezing theorem, and symplectic 4-manifolds. 10. **Gluing**: Introduces the gluing theorem, connected sums of J-holomorphic curves, weighted norms, cutoff functions, and the construction and properties of the gluing map. 11. **Quantum Cohomology**: Covers the small quantum cohomology ring, the Gromov-Witten potential, examples, the Seidel representation, and Frobenius manifolds. 12. **Floer Homology**: Discusses Floer's cochain complex, ring structure, Poincaré duality, spectral invariants, the Seidel representation, and Donaldson's quantum category. The book also includes appendices on Fredholm theory, elliptic regularity, the Riemann-Roch theorem, stable curves of genus zero, singularities and intersections, and a bibliography with a list of symbols and an index.This book, authored by Dusa McDuff and Dietmar Salamon, is a comprehensive treatment of J-holomorphic curves and their applications in symplectic topology. The content is divided into several chapters, each focusing on different aspects of the subject: 1. **Introduction**: Provides an overview of symplectic manifolds, moduli spaces, evaluation maps, and Gromov-Witten invariants. 2. **J-holomorphic Curves**: Discusses almost complex structures, the nonlinear Cauchy-Riemann equations, unique continuation, critical points, and the adjunction inequality. 3. **Moduli Spaces and Transversality**: Explores moduli spaces of simple curves, transversality, regularity criteria, and implicit function theorems. 4. **Compactness**: Covers energy, bubbling phenomena, mean value and isoperimetric inequalities, singularities, and convergence. 5. **Stable Maps**: Introduces stable maps, Gromov convergence, compactness, uniqueness of limits, and the Gromov topology. 6. **Moduli Spaces of Stable Maps**: Focuses on simple stable maps, transversality, semipositivity, pseudocycles, and Gromov-Witten pseudocycles. 7. **Gromov-Witten Invariants**: Discusses counting pseudoholomorphic spheres, variations in definitions, counting pseudoholomorphic graphs, and rational curves in projective spaces. 8. **Hamiltonian Perturbations**: Examines trivial bundles, locally Hamiltonian fibrations, pseudoholomorphic sections, and counting pseudoholomorphic sections. 9. **Applications in Symplectic Topology**: Highlights periodic orbits of Hamiltonian systems, obstructions to Lagrangian embeddings, the nonsqueezing theorem, and symplectic 4-manifolds. 10. **Gluing**: Introduces the gluing theorem, connected sums of J-holomorphic curves, weighted norms, cutoff functions, and the construction and properties of the gluing map. 11. **Quantum Cohomology**: Covers the small quantum cohomology ring, the Gromov-Witten potential, examples, the Seidel representation, and Frobenius manifolds. 12. **Floer Homology**: Discusses Floer's cochain complex, ring structure, Poincaré duality, spectral invariants, the Seidel representation, and Donaldson's quantum category. The book also includes appendices on Fredholm theory, elliptic regularity, the Riemann-Roch theorem, stable curves of genus zero, singularities and intersections, and a bibliography with a list of symbols and an index.