Principal Investigator: Yong-Geun Oh
This project aims at providing a description of Fukaya categories of toric manifolds and of Calabi-Yau threefolds, and establishing the mirror correspondence between the Fano toric A-model and the Landau-Ginzburg B-model based on the general Lagrangian Floer theory established by the PI jointly with Fukaya, Ohta and Ono. It also proposes to solve the simpleness problem of the area preserving homeomorphism group of the two sphere (and the disc) by investigating the extension problem of Calabi homomorphisms and Entov-Polterovich's quasi-morphisms to the Hamiltonian homeomorphism group. In addition, the PI proposes to apply the machinery of Floer theory to symplectic topology of toric manifolds and construct new quasi-morphisms on the Hamiltonian diffeomorphism group and Entov-Polterovich's symplectic quasi-states on toric manifolds. The PI anticipates that the proposed research will not only provide solution to the homological mirror symmetry of Fano toric A-model and Landau-Ginzburg B-model but also lead to deeper understanding of open-closed Floer theory and its applications to dynamical systems and symplectic topology.
The Hamiltonian formalism played a fundamental role not only for solving problems in classical mechanics but also for transforming the classical mechanics into quantum mechanics. It also plays an important role in deriving many basic physical equations in high energy physics ranging from quantum field theory to modern string theory. The natural space where the Hamiltonian formalism can be exercised is the symplectic manifold (or more generally the Poisson manifold). One of the most distinguished geometric objects of study in symplectic manifold is the Lagrangian submanifold; For example, `the state of a particle with zero momentum in space' forms a Lagrangian submanifold in `all possible states of a particle in space' which forms a symplectic manifold. Understanding the interplay between geometry of Lagrangian submanifolds and dynamics of Hamiltonian flows is the core theme of symplectic topology. The PI's proposed research aims at extending the Hamiltonian dynamics to the level of continuous dynamics and solving various problems arising from Hamiltonian dynamics and mirror symmetry. It also aims at easing the access of graduate students and researchers from other related fields into the study of Floer theory and mirror symmetry by writing a graduate level textbook on Floer homology and its applications to symplectic geometry.
