Cluster algebras and braid varieties

Original Announcement

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:

• Develop the dictionary between cluster seeds and Legendrian weaves. For instance, given a cluster algebra for the braid variety of a positive braid Legendrian link, can each cluster seed be obtained from a weave? How many exact Lagrangian fillings can inhabit the same cluster seed? When can we recover a filling from a cluster seed?
• Study algebraic symmetries of cluster algebras, such as the cluster modular group, in terms of symplectic isotopies of Legendrian knots. For instance, understand the many interesting group actions appearing in the irregular Riemann-Hilbert correspondence, e.g. by the Weyl group and braid groups, in terms of continuous symmetries of alternating strand diagrams, and study salient cluster automorphisms, such as the Donaldson-Thomas transformation, in terms of Hamiltonian isotopies.
• Establish the necessary techniques in symplectic geometry to construct cluster structures on Richardson varieties beyond Dynkin type A. Utilize them to gain understanding of their cohomology rings, as have been successfully done for positroid varieties in type A.
• Identify the relationship between cluster structures and partially wrapped Fukaya categories. For instance, compute morphisms in these symplectic categories in terms of spectral networks (i.e. pseudo-holomorphic strips) or directly in terms of plabic graphs. In particular, apply techniques in the study of compactifications and toric degenerations of cluster varieties to the study of Lagrangian surfaces in Weinstein 4-manifolds.

Material from the workshop

The workshop schedule.

A report on the workshop activities.

A list of open problems.

Papers arising from the workshop: