the mirror symmetry conjecture (inspired by physics) has spurred a lot of development in symplectic geometry. in the last few years, a wave of modern homotopy theory has also entered the symplectic landscape, and begun to present new questions about the structure of symplectic manifolds. in this talk, we’ll explain a basic invariant in symplectic geometry (the fukaya category) and, as time allows, give a survey of new inroads being opened through lagrangian cobordisms, derived geometry, and deformation theory.