Floer homology in symplectic geometry and in mirror symmetry

Details Floer homology in symplectic geometry and in mirror symmetry Floer homology in symplectic geometry and in mirror symmetry

2006-01-23

arxiv.org/abs/math/0601568v1

Mathematics

Symplectic Geometry

To appear in the Proceedings for ICM-2006 Madrid. This is the same version as the one submitted in December 2005 for the ICM p

Scientific paper

In this article, the authors review what the Floer homology is and what it does in symplectic geometry both in the closed string and in the open string context. In the first case, the authors will explain how the chain level Floer theory leads to the $C^0$ symplectic invariants of Hamiltonian flows and to the study of topological Hamiltonian dynamics. In the second case, the authors explain how Floer's original construction of Lagrangian intersection Floer homology is obstructed in general as soon as one leaves the category of exact Lagrangian submanifolds. They will survey construction, obstruction and promotion of the Floer complex to the $A_\infty$ category of symplectic manifolds. Some applications of this general machinery to the study of the topology of Lagrangian embeddings in relation to symplectic topology and to mirror symmetry are also reviewed.

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