Special Events Seminar

In this talk, I will tell a story that starts with the classification of cluster algebras of finite type by Fomin and Zelevinsky. The combinatorial essence of cluster algebras of finite type is a fan (in fact, the normal fan of a polytope--a generalized associahedron). The fan records the d-vectors or g-vectors of clusters, and in infinite type, there is an analogous infinite fan of d-vectors or g-vectors. These infinite fans are not complete, however, and it became clear from work in rank 2 and in the surfaces case that the space outside the g-vector/d-vector fan is vital to understanding the cluster algebra. Recently, Gross, Hacking, Keel, and Kontsevich constructed cluster scattering diagrams and used them to prove many of Fomin and Zelevinsky's structural conjectures on cluster algebras. They showed that the cluster scattering diagram (a collection of codimension-1 cones) "cuts out" the g-vector fan. More recently, I showed that the cluster scattering diagram cuts out a complete fan containing the g-vector fan as a subfan. This fan is the most general "generalized associahedron fan", the combinatorial essence of a general cluster algebra.

The cluster scattering diagram arises from a non-constructive (in practice) existence theorem. Thus combinatorial models for cluster scattering diagrams and fans are crucially needed. After reviewing some of the backstory, I will describe work in progress to construct cluster scattering diagrams and fans in affine type (with Stella) and in the surfaces case (with Muller and Viel). Speaker(s): Nathan Reading (North Carolina State University)