Open Gromov-Witten Theory, Mirror Symmetry, and Toric Geometry

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Mathematics and physics are traditionally closely related. In recent years many new mathematical subjects have been created due to the influence of attempts to study the nature of subatomic particles. One important example of such new mathematics is the rapidly-developing subject of Gromov-Witten theory. One reason for the importance of Gromov-Witten theory is its deep connections with many other subjects, such as string theory, integrable systems of partial differential equations, counting of geometric objects, and parameter spaces for geometric objects. Another example is mirror symmetry, which is a duality arising from physics and is interpreted in mathematics as the equivalence of two seemingly completely unrelated geometric objects. Such a highly nontrivial correspondence is extremely interesting in its own right, while also important for theoretical physics. This research project will explore several key questions in the deep relations between Gromov-Witten theory and mirror symmetry. The research will further advance knowledge about mirror symmetry, enumeration of holomorphic disks, and the geometry of parameter spaces of surfaces, and it will also further promote the existing interactions between algebraic geometry, symplectic geometry, combinatorics, mathematical physics, and string theory. This research project aims to study several aspects of open Gromov-Witten theory and mirror symmetry: calculations of open Gromov-Witten invariants of compact Calabi-Yau manifolds; understanding the relations between mirror maps defined using period integrals and open Gromov-Witten invariants; proving mirror symmetry statements for open Gromow-Witten invariants of symplectic toric orbifolds; and studying the behaviors of open Gromov-Witten invariants under birational transformations. The PI plans to study open Gromov-Witten invariants by relating them to other quantities, such as closed Gromov-Witten invariants, and to derive relations among open Gromov-Witten invariants. Techniques such as virtual localization, mirror theorems, analysis of Kuranishi spaces, tropical geometry, Seidel representations, and quantum differential equations will be used in the research.