Calabi–Yau properties of Postnikov diagrams

A Postnikov diagram is a collection of strands in the disk, satisfying combinatorial conditions on their crossings. The diagram determines many other mathematical objects, including a cluster algebra, which Galashin and Lam have recently shown to be isomorphic to the homogeneous coordinate ring of a certain subvariety of the Grassmannian, called a positroid variety. In this talk, I will explain how to categorify this cluster algebra, using a second (non-commutative) algebra attached to the Postnikov diagram. The approach depends on studying Calabi–Yau symmetries of this algebra, and relates to work of Broomhead and others concerning dimer models on closed surfaces.