How to construct symplectic invariants by counting J-holomorphic curves

I will begin by explaining the fundamental recipe in categorical symplectic geometry, which allows one to define symplectic invariants by counting rigid J-holomorphic curves. I will mention a variety of examples: quantum cohomology, Floer cohomology, the Fukaya category, and the focus of my research, the symplectic $(A_\infty,2)$-category. Finally, I will describe a new use of this recipe, to equip the Fukaya category of an elliptic K3 surface with a monoidal structure (this is work in progress with Mohammed Abouzaid and Yunpeng Niu). I will do my best to make this talk broadly accessible to topologists.