The goals of the research project 'Counting curves on higher dimensional complex projective manifolds' are1) to explicitly calculate the expected number of curves (one-complex-dimensional submanifolds) on higher dimensional manifolds in settings where one expects finitely many such curves, 2) to determine when the expected number is actually realized by (rigid) curves. One aim is to simplify recent advances by Givental, Yau, et al. carrying out 1) in the case of Calabi-Yau threefolds. Another is to study the general deformation of a special situation in which the curves, although expected to be finite, actually move in families.
This project lies within the area of research called complex projective geometry, that is, the study of geometric properties of solutions of systems of polynomial equations in several variables. The research is related to fundamental questions in quantum physics and differential equations.
