Persistence modules in symplectic topology
This talk is about how persistent homology theory is used in symplectic geometry. I will start from the main algebraic object - persistence module, and explain how various filtered homology theories in symplectic geometry can be formulated in this language. Then I will focus on its application on symplectic embeddings. In a joint work with V. Stojisavljevic, we defined symplectic Banach-Mazur distance in the space of Liouville domains in the cotangent bundle of a base manifold. Also, we obtained conclusions on this pseudometric space from a coarse geometric viewpoint when the base manifold is a closed surface. Finally, I will emphasize how this technique can be used to study closed geodesics.