Why are people interested in defining GW invariant in algebraic geometry category Originally it is in symplectic geometry. Is it just curosity or any other special reason? Thank you for clarifying.
 A: Part of it is that, for the special case of homogeneous spaces and genus 0, it can be shown that GW invariants count the solutions to certain enumerative problems (how many rational curves of degree $d$ are there that intersect general translates of given cohomology classes) and some rather old problems in algebraic geometry were solved this way, for instance, Kontsevich's formula for (stable) rational plane curves of degree $d$ passing through $3d-1$ points in general position.
More generally, they are invariants that let us distinguish different varieties of the same dimension, by generalizing the cohomology ring.
A: The first complete definition of GWI in algebraic geometry is more or less contemporary to the first complete definition in symplectic geometry. In algebraic geometry you can, e.g., use virtual localization techniques which (as far as I know) have no counterpart on the symplectic side.
More generally, using both symplectic and algebraic techniques on the same problem is almost always a good idea since the strengths of one approach tend to be complementary to those of the other.
