Given 6 dots in the plane, how many ways are there to separate them into two sets of 3 using a circle? Or using a Jordan curve? The first question can be answered by considering "higher order Voronoi diagrams"; the latter can be answered by observing these diagrams have a "plabic" structure (in the sense of Postnikov). This gives a bijection between such Jordan curves and degree 1 cluster variables in the cluster algebra of type X7, an exceptional cluster algebra about which little is known. This correspondence allows us to translate structure theorems about cluster algebras into remarkable results on the structure of Jordan curves with 3 dots on either side. As an application, we translate these results into analogous results on "separating curves in the closed surface of genus 2"; specifically, we show that the g=2 separating curve complex is a strongly connected 6-dimensional pseudomanifold. On joint work with James Beyer and Jaewon Min.

This talk will take place in East Hall 4448.