Cluster algebras and braid varieties

January 23 to January 27, 2023

at the

American Institute of Mathematics, San Jose, California

organized by

Roger Casals, Mikhail Gorsky, Melissa Sherman-Bennett, and Jose Simental

Original Announcement

This workshop will be devoted to the study of cluster algebras and their relation to symplectic geometry, bringing together people with expertise in these two fields. In recent years deep and intriguing connections between these two areas have been discovered, including the existence of cluster structures on moduli spaces of Lagrangian fillings, the cluster compatibility of Poisson structures on wild character varieties, the cluster algebras associated to plane curve singularities, and the cluster nature of spectral networks. This bridge between cluster algebras and symplectic geometry is proving fruitful in both directions and a central aim of the workshop is to crystalize those connections and bring forward further applications.

For instance, cluster algebras have already been used to prove the existence of infinitely many Lagrangian fillings for certain Legendrian links, and now provide a conjectural classification of these symplectic geometric objects. Conversely, sheaf and Floer-theoretic methods in symplectic geometry have led to new combinatorial descriptions of cluster structures in a wide range of cases, including positroid varieties, and have additionally provided insight on the structure of certain classes of cluster varieties and their cohomology rings. In particular, the workshop will feature the study of braid varieties and Legendrian weaves, which lie at the intersection of many facets of symplectic geometry where cluster algebras have appeared. From the viewpoint of cluster algebras, braid varieties provide a new and unifying perspective on Richardson varieties. Weaves for braid varieties provide a diagrammatic calculus to access infinitely many cluster seeds, e.g. allowing mutations at non-square faces of plabic graphs, and also lead to geometric constructions bridging known tools, such as plabic graphs, alternating strand diagrams, conjugate surfaces and spectral networks. Some of the topics for the workshop are:

Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.