The purpose of this short article is to prove a product formula relating the log Gromov-Witten invariants of V×W with those of V and W in the case the log structure on V is trivial.

This an expository article on Givental's axiomatic Gromov--Witten theory and some of its applications.

The celebrated Mirror Theorem states that the genus zero part of the A model (quantum cohomology, rational curves counting) of the Fermat quintic threefold is equivalent to the B model (complex deformation, variation of Hodge structure) of its mirror dual orbifold. In this article, we establish a mirror-dual statement. Namely, the B model of the Fermat quintic threefold is shown to be equivalent to the A model of its mirror, and hence establishes the mirror symmetry as a true duality.

A new tautological equation of $\bar{M}_{3,1}$ in codimension 3 is derived and proved, using the invariance condition explained in earlier works.

A reconstruction theorem for genus 0 gravitational quantum cohomology and quantum K-theory is proved. A new linear equivalence in the Picard group of the moduli space of genus 0 stable maps relating the pull-backs of line bundles from the target via different markings is used for the reconstruction result. Examples of calculations in quantum cohomology and quantum K-theory are given.