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

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

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 [LLW].

Let X denote an equivariant embedding of a connected reductive group X over an algebraically closed field X . Let X denote a Borel subgroup of X and let X denote a X -orbit closure in X . When the characteristic of X is positive and X is projective we prove that X is globally X -regular. As a consequence, X is normal and Cohen-Macaulay for arbitrary X and arbitrary characteristics. Moreover, in characteristic zero it follows that X has rational singularities. This extends earlier results by the second author and M. Brion.