%{search_type} search results

• Introduction $J$-holomorphic curves Moduli spaces and transversality Compactness Stable maps Moduli spaces of stable maps Gromov-Witten invariants Hamiltonian perturbations Applications in symplectic topology Gluing Quantum cohomology Floer cohomology Fredholm theory Elliptic regularity The Riemann-Roch theorem Stable curves of genus zero Singularities and intersections (written with Laurent Lazzarini) Bibliography List of symbols Index.
• (source: Nielsen Book Data)

The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. Its mathematical applications include many key results in symplectic topology. It was also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov-Witten invariants and quantum cohomology-two important ingredients of the mirror symmetry conjecture. This book establishes the fundamental theorems of the subject in full and rigorous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associativity of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: There are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of $J$-holomorphic curves in four dimensional manifolds. The book is suitable for graduate students and researchers interested in symplectic geometry and its applications, especially in the theory of Gromov-Witten invariants.
(source: Nielsen Book Data)

• Introduction--
• 1. Closed holomorphic curves in symplectic 4-manifolds--
• 2. Intersections, ruled surfaces and contact boundaries--
• 3. Asymptotics of punctured holomorphic curves--
• 4. Intersection theory for punctured holomorphic curves--
• 5. Symplectic fillings of planar contact 3-manifolds-- Appendix A. Properties of pseudoholomorphic curves-- Appendix B. Local positivity of intersections-- Appendix C. A quick survey of Siefring's intersection theory-- References-- Index.
• (source: Nielsen Book Data)

Intersection theory has played a prominent role in the study of closed symplectic 4-manifolds since Gromov's famous 1985 paper on pseudoholomorphic curves, leading to myriad beautiful rigidity results that are either inaccessible or not true in higher dimensions. Siefring's recent extension of the theory to punctured holomorphic curves allowed similarly important results for contact 3-manifolds and their symplectic fillings. Based on a series of lectures for graduate students in topology, this book begins with an overview of the closed case, and then proceeds to explain the essentials of Siefring's intersection theory and how to use it, and gives some sample applications in low-dimensional symplectic and contact topology. The appendices provide valuable information for researchers, including a concise reference guide on Siefring's theory and a self-contained proof of a weak version of the Micallef-White theorem.
(source: Nielsen Book Data)

The theory of $J$-holomorphic curves has been of great importance since its introduction by Gromov in 1985. In mathematics, its applications include many key results in symplectic topology. It was also one of the main inspirations for the creation of Floer homology. In mathematical physics, it provides a natural context in which to define Gromov-Witten invariants and quantum cohomology, two important ingredients of the mirror symmetry conjecture. The main goal of this book is to establish the fundamental theorems of the subject in full and rigourous detail. In particular, the book contains complete proofs of Gromov's compactness theorem for spheres, of the gluing theorem for spheres, and of the associatively of quantum multiplication in the semipositive case. The book can also serve as an introduction to current work in symplectic topology: there are two long chapters on applications, one concentrating on classical results in symplectic topology and the other concerned with quantum cohomology. The last chapter sketches some recent developments in Floer theory. The five appendices of the book provide necessary background related to the classical theory of linear elliptic operators, Fredholm theory, Sobolev spaces, as well as a discussion of the moduli space of genus zero stable curves and a proof of the positivity of intersections of $J$-holomorphic curves in four-dimensional manifolds. The second edition clarifies various arguments, corrects several mistakes in the first edition, includes some additional results in Chapter 10 and Appendices C and D, and updates the references to recent developments.
(source: Nielsen Book Data)