Abstract: 

A split torus in \(S^2 \times S^2\) is a Lagrangian torus obtained as the product of circles in the factors. The goal of this talk is to give a classification up to symplectomorphisms of such tori, and illustrate that this classification is connected to billiards. Among other applications, we will answer a question about Lagrangian packings posed by Polterovich--Shelukhin. This talk is partially based on joint work with Joontae Kim. I will not suppose any preexisting knowledge in symplectic geometry.