The aim of Part II is to explore the technique of invariance of tautological equations in the realm of Gromov-Witten theory. The main result is a proof of Invariance Theorem (Invariance Conjecture 1 in [14]), via the techniques from Gromov-Witten theory. It establishes some general inductive structure of the tautological rings, and provides a new tool to the study of this area.Research partially supported by NSF and AMS Centennial Fellowship.

We show that the generating functions of Gromov-Witten invariants with ancestors are invariant under a simple flop, for all genera, after an analytic continuation in the extended Kähler moduli space. This is a sequel to [14].

ABSTRACT. 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. CONTENTS

We prove that all monomials of κ-classes and ψ-classes are independent in. We also give a simple argument for κ l = 0 in R l (M g ) for l g − 2.

ABSTRACT. We discuss selected topics on the topology of moduli spaces of curves and maps, emphasizing their relation with Gromov-Witten theory and integrable systems. CONTENTS