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 establish a generic vanishing theorem for surfaces in characteristic p that lift to p and use it for classification of surfaces of general type with Euler characteristic p and large Albanese dimension.

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

1. INDEX THEORY, ELLIPTIC CURVES AND LOOP GROUPS One can look at elliptic genus from several different points of view; from index theory, from representation theory of Kac-Moody affine Lie algebras or from the theory of elliptic functions and modular forms. Each of them shows us some quite different interesting features of ellitic genus. On the other hand we can also combine the forces of these three different mathematical fields to derive many interesting results in topology such as rigidity, divisibility and vanishing of topological invariants..

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].