# From Lusternik-Schnirelmann theory to Conley conjecture

 Princeton/IAS Symplectic Geometry Seminar Topic: From Lusternik-Schnirelmann theory to Conley conjecture Speaker: Başak Gürel Affiliation: University of Central Florida Date: Tuesday, October 18 Time/Room: 3:00pm - 4:00pm/Fine 224, Princeton University

In this talk I will discuss a recent result showing that whenever a closed symplectic manifold admits a Hamiltonian diffeomorphism with finitely many simple periodic orbits, the manifold has a spherical homology class of degree two with positive symplectic area and positive integral of the first Chern class. This theorem encompasses all known cases of the Conley conjecture (symplectic CY and negative monotone manifolds) and also some new ones (e.g., weakly exact symplectic manifolds with non-vanishing first Chern class). The proof hinges on a general Lusternik-Schnirelmann type result that, under some natural additional conditions, the sequence of mean spectral invariants for the iterations of a Hamiltonian diffeomorphism never stabilizes. Based on joint work with Viktor Ginzburg.