High Quality Content by WIKIPEDIA articles! In mathematics, specifically in symplectic topology and algebraic geometry, a quantum cohomology ring is an extension of the ordinary cohomology ring of a closed symplectic manifold. It comes in two versions, called small and big; in general, the latter is more complicated and contains more information than the former. In each, the choice of coefficient ring (typically a Novikov ring, described below) significantly affects its structure, as well. While the cup product of ordinary cohomology describes how subspaces of the manifold intersect each other, the quantum cup product of quantum cohomology describes how subspaces intersect in a "fuzzy", "quantum" way. More precisely, they intersect if they are connected via one or more pseudoholomorphic curves. Gromov-Witten invariants, which count these curves, appear as coefficients in expansions of the quantum cup product.

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