Polynomial structures in higher genus enumerative geometry

It is important to calculate the enumerative invariants from various moduli theories in mirror symmetry. The polynomial structure is often appeared in those quantum theories, including the Calabi-Yau type and the Fano type theories. Such conjectural structure is also called the finite generation conjecture in the literature. For each genus, it is conjectured that the computation of infinite many enumerative invariants can be converted to a finite computation problem. The original motivation of studying such structures will also be mentioned. This talk is based on the joint work with Janda-Ruan, Chang-Li-Li, Bousseau-Fan-Wu and Zhang respectively.