The first talk will be aimed at a general geometry & number theory audience. It will give an introduction to symplectic mapping glass groups, focusing on Stein manifolds (i.e. complex submanifolds of affine space), with an overview of some known (and unknown) properties. No knowledge of symplectic topology will be assumed. After the break, we will present further details on some recent results proved using tools from mirror symmetry (joint with Hacking, and with Smith); our exact focus will be fine-tuned depending on the audience's interest. For this part, familiarity with basic properties of derived categories of coherent sheaves (on smooth varieties) would be helpful.