Holomorphic curves are a central object of study in complex algebraic geometry. Such curves are meaningful even when the target has an almost complex structure. The moduli space of these curves (called pesudoholomorphic curves) is typically non-compact and not well-behaved. A nice compactification, due to Gromov, allows us to define certain invariants known as Gromov-Witten invariants.The theory of Gromov-Witten invariants can be used to deform the usual cohomology ring structure of a symplectic manifold. This has connections, on the one hand, with enumerative geometry, and on the other hand, with deformation of associative algebras and topological quantum field theory. Gromov-Witten invariants have deep connections with physics through the ideas of Mirror Symmetry. Using String Theory, physicists have made amazing predictions about the Gromov-Witten invariants of the quintic threefold. From a mathematical point of view, a large number of predictions are still open.The study of pseudoh...
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