Cominuscule homogeneous spaces, quantum cohomology, and mirror symmetry

We will present recently constructed (joint with Charles Wang) Landau-Ginzburg models for the (small) quantum cohomology of exceptional cominuscule homogeneous spaces, as well as---if time allows---Plücker coordinate cluster structures and Newton-Okounkov bodies for these mirrors. To keep the discussions at an accessible level, we will introduce the relevant concepts using familiar examples of homogeneous spaces, such as Grassmannians and quadrics, and we will compare our results to the results known in these cases.