A leading theoretic candidate to unify all the forces in nature is string theory. The string-theoretic model of the universe is ten-dimensional, our usual four-dimensional space-time together with a very small six-dimensional space called a Calabi-Yau manifold. Such a small internal space will affect our space-time through certain mathematical quantities such as Gromov-Witten invariants. The computation of these invariants has been a central problem in geometry and physics. The current project envisages development of a theoretical framework as well as technical tools to compute Gromov-Witten invariants.
During the past twenty years, there has been a great deal of interaction between mathematics and physics. Various correspondences or dualities from physics have had great impact in mathematics. One of these correspondences is the Landau-Ginzburg/Calabi-Yau correspondence. A related discovery from physics is that the generating function for Gromov-Witten invariants should be a quasi-modular form, a number theoretic object. This project aims to develop a comprehensive program to establish Landau-Ginzburg/Calabi-Yau correspondence and modularity of Gromov-Witten theory in mathematics. One important application is to compute Gromov-Witten invariants of compact Calabi-Yau manifolds, a central and still difficult problem in geometry and physics. The project is interdisciplinary in nature; both physical and mathematical ideas play essential roles. The project also involves training of postdocs and students.