Title: On the Lagrangian version of the Conley Conjecture

Abstract: In 1984 Charles Conley conjectured that every Hamiltonian diffeomorphism of the standard symplectic 2n-torus has infinitely many periodic points. This conjecture was recently proved by Hingston in 2004, and extended to general closed symplectically aspherical manifolds by Ginzburg in 2006. If we consider a 1-periodic Lagrangian function on the tangent bundle of a closed manifold, a Lagrangian reformulation of the Conley conjecture would assert the existence of infinitely many periodic points of the time-1 map of the Lagrangian flow. In 1997, Long established this conjecture for fiberwise quadratic Lagrangians on the tangent bundle of an n-torus. In this talk we will outline a generalization of this proof for a broader class of Lagrangians defined on the tangent bundle of a general closed manifold.