The work of this proposal lies at the intersection of string theory from physics and algebraic geometry. In string theory, particles are replaced by small loops of strings moving in space-time. Conjecturally, the space-time is ten-dimensional. Over the four-dimensional space-time we are aware of, there is a six-dimensional fibration whose fiber is known as the Calabi-Yau manifold. The Gromov-Witten invariants serve as important ingredients in the study of Calabi-Yau manifolds and string theory. Since the discovery of the Gromov-Witten invariants, new insights are developing rapidly, which also provide many interesting new approaches to classical problems from algebraic geometry. The calculation of Gromov-Witten invariants is extremely difficult to perform on Calabi-Yau manifolds. This project aims to develop a new calculation method of Gromov-Witten invariants, and to apply the developments to produce new understandings in algebraic geometry.
The major focus of this project is to study the degeneration of Gromov-Witten theory. This will be carried out from the perspective of stable logarithmic maps developed jointly with Dan Abramovich, and independently by Mark Gross and Bernd Siebert. In particular, the PI proposes to prove a general gluing formula of logarithmic Gromov-Witten invariants. The theory of stable logarithmic maps also provides a useful tool for the study of rational curves on quasi-projective varieties. Another part of this project is to generalize Mori's theory in the non-proper cases, and to have a further understanding of the birational geometry of quasi-projective varieties.
