Homotopy rigidity of Lagrangian embeddings

Understanding Lagrangian embeddings inside a symplectic manifold is one of the central questions of symplectic topology. Arnold’s nearby Lagrangian conjecture, which classifies closed exact Lagrangians in a cotangent bundle, is one of the major conjectures in this direction. Although it remains open in general, significant progress has been made on homotopy theoretic approximations of this conjecture and some of its variants. In this talk, I will survey these developments and describe some work-in-progress on using techniques from Floer homotopy theory to obtain homotopy rigidity results for Lagrangians in one such variant of the nearby Lagrangian conjecture.