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

We prove that all monomials of κ-classes and ψ-classes are independent in $R^k(\ocM_{g,n})/R^k(\partial\ocM_{g,n})$ for all k≤[g/3]. We also give a simple argument for κ_l≠0 in $R^l(\ocM_g)$ for l≤g−2.

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

For ordinary flops, the correspondence defined by the graph closure is shown to give equivalence of Chow motives and to preserve the Poincaré pairing. In the case of simple ordinary flops, this correspondence preserves the big quantum cohomology ring after an analytic continuation over the extended Kähler moduli space. For Mukai flops, it is shown that the birational map for the local models is deformation equivalent to isomorphisms. This implies that the birational map induces isomorphisms on the full quantum rings and all the quantum corrections attached to the extremal ray vanish.

We verify the Invariance Conjectures of tautological equations in genus two. In particular, a uniform derivation of all known genus two equations is given.