Applications of holomorphic curves in symplectic topology

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This thesis will investigate applications of closed and open holomorphic curve invariants in symplectic topology. The key tools will be invariants related to quantum and symplectic cohomology and to Floer theory, and its algebraic structures. Basic questions to be addressed include properties of symplectomorphism groups, the action of symplectomorphisms on Lagrangian submanifolds, and connections to algebraic and tropical geometry and to mirror symmetry.