Cluster algebras and braid varieties
January 23 to January 27, 2023
American Institute of Mathematics,
San Jose, California
and Jose Simental
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:
- 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
A list of participants.
The workshop schedule.
A report on the workshop activities.