Title: Conceptual approaches to Fukaya categories and mirror symmetry
Abstract: Symplectic geometry has seen an explosion of powerful tools coming from Floer theory and pseudoholomorphic curve theory, with a central role played by a categorical invariant known as the Fukaya category. The Fukaya category is also a key puzzle piece in mirror symmetry, a series of remarkable conjectural correspondences - first put forward by string theorists - between the symplectic geometry of one space and the algebraic geometry of a ‘mirror’ space. Nevertheless, computations have until recently remained difficult and ad hoc, largely due to the global analytic nature of pseudoholomorphic curves. I will give an introduction to this area through pictures, emphasizing recent structural developments that have enabled new systematic and conceptual calculations and verifications of mirror symmetry.