# C^\infty closing lemma for three-dimensional Reeb flows via embedded contact homology

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Kei Irie , Kyoto University, Japan
IAS - West Building Lecture Hall

I will explain an application of embedded contact homology (ECH) to Reeb dynamics. Specifically, I prove so-called $C^\infty$ closing lemma for Reeb flows on closed contact three-manifolds. The key ingredient of the proof is a result by Cristofaro-Gardiner, Hutchings and Ramos, which claims that the asymptotics of ECH spectral invariants recover the volume of a contact manifold. Applications to closed geodesics on Riemannian two-manifolds and Hamiltonian diffeomorphisms of symplectic two-manifolds (joint work with M.Asaoka) will be also presented.