Floer Theory, Symplectic Geometry and Mirror Symmetry

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

DMS-0203593 Yong-Geun Oh Floer homology in symplectic geometry was introduced by Floer in an attempt to prove the Arnold conjecture. Various results by Chekanov, Oh, Polterovich and Seidel have proved that the Floer homology is a general powerful tool to investigate symplectic topology. In this project, Oh proposes to further investigate structure and applications of the Floer theory for deeper understanding of the Hamiltonian diffeomorphism group and Lagrangian submanifolds or more generally symplectic topology. The Floer theory of Lagrangian submanifolds, via Fukaya's A-infinity category, also provides a geometric framework for Kontsevich's homological mirror symmetry proposal on Calabi-Yau manifolds. Oh's recent work with Fukaya, Ohta and Ono provides several key steps towards a rigorous construction of Fukaya's category by developing an obstruction theory for defining the Lagrangian intersection Floer homology. Complete construction will involve study of singular Lagrangian submanifolds, Lagrangian surgery and their relations to the Floer homology. Oh proposes to investigate these new aspects of the Floer theory in relation the mirror symmetry on the Calabi-Yau manifolds. The Hamiltonian formalism plays important roles not only for solving problems in classical mechanics but also for quantizing the classical mechanics into quantum mechanics. The Poisson bracket is the crucial geometric structure that plays a key role in the classical mechanics and the quantum mechanics through the quantization process. When one considers mechanics in a constrained system, i.e., mechanics on a curved space, description of the corresponding phase space and the geometric structure corresponding to the Poisson bracket requires the notion of the symplectic structure and symplectic manifolds. Symplectic geometry and topology is the study of symplectic manifolds. In symplectic geometry, there are two most important objects of study. One is the study of Hamiltonian systems, a special type of differential equation, and their periodic orbits. This is dynamical in nature. The other is the study of geometry and topology of Lagrangian submanifolds. This is geometric in nature. Understanding the intersection theory of Lagrangian submanifolds is the core of symplectic topology. Floer homology introduced by Floer in the end of 80's is a general machinery to study this intersection theory. The Floer theory also provides a geometric framework for the mirror symmetry phenomenon that was discovered by physicists in string theory. Oh's proposed research aims at, on the one hand, deeper understanding of symplectic topology, and also aims at understanding inter-relations between the symplectic and the complex geometry via the study of mirror symmetry.