$J$-holomorphic curves revolutionized the learn of symplectic geometry whilst Gromov first brought them in 1985. via quantum cohomology, those curves at the moment are associated with a few of the most fun new principles in mathematical physics. This booklet provides the 1st coherent and whole account of the idea of $J$-holomorphic curves, the main points of that are almost immediately scattered in a variety of examine papers. the 1st 1/2 the booklet is an expository account of the sphere, explaining the most technical elements. McDuff and Salamon supply whole proofs of Gromov's compactness theorem for spheres and of the life of the Gromov-Witten invariants. the second one half the booklet makes a speciality of the definition of quantum cohomology. The authors identify that this multiplication exists, and provides a brand new facts of the Ruan-Tian consequence that's associative on acceptable manifolds. They then describe the Givental-Kim calculation of the quantum cohomology of flag manifolds, resulting in quantum Chern sessions and Witten's calculation for Grassmannians, which pertains to the Verlinde algebra. The Dubrovin connection, Gromov-Witten strength on quantum cohomology, and curve counting formulation also are mentioned. The publication closes with an overview of connections to Floer conception.
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