In the 1980's, we saw much activity involving the application of classicaltechniques to the study of holomorphic curves on Calabi-Yau threefolds.In particular, it was conjectured by Clemens that for each degree d,there are finitely many smooth rational curves of degree don a generic hypersurface of degree 5 in projective 4 space. This conjecture is stillunsettled.
In the 1990's, these techniques came to be replaced by new techniques inspired by string theory, especially mirror symmetry, Gromov-Witten theory, and quantum cohomology. The physics-inspiredtechniques led to a complete solution of reformulations and generalizations of classicalproblems, involving the definition and computation of the "virtual number" of holomorphiccurves, a rational number. It is highly desirable to define an integer invariant which playsthe role of the "number" of curves.
In this talk, I give an overview of the exciting development of this area, together with new directions in mathematics inspired by the study of branes in string theory and M-theory, together with classical techniques of excess intersection theory.