Recent developments on Gromov-Witten theory of Hilbert schemes of points on certain surfaces

In This paper, we survey recent progress on the theory of Gromov- Witten invariants on Hilbert schemes of points mainly on elliptic surfaces and simply connected minimal surface of general type. In particular, we focus on the aspects of computational progress that has been done in the cases of lower genus. Then, we discuss some important conjectures that have been proposed and gather all available information and progress toward answering them.
Comment: To rewrite the review paper again