Background: Participants should understand graduate-level differential and algebraic topology. Some exposure to symplectic and contact geometry would also be helpful. Recommendations for background readings for the series can be found here.
The plan for the lecture series is
- Introduction to Legendrian links and the microlocal theory of sheaves: This first lecture will introduce the main objects of study. Legendrian links on the geometric side and sheaves and singular support on the algebraic side. The main goals of the lecture series will be stated in detail, with motivation and examples being provided. The central objective will be to introduce and study the derived dg-category of sheaves in R2 with singular support on a specified Legendrian link. All the necessary definitions and ingredients will be introduced from the ground up. The main results will illustrate how to use this dg-category to study Legendrian links, e.g. proving Legendrian invariance, distinguishing infinitely many Lagrangian fillings and the construction and use of cluster structures on its moduli of objects.
- Lagrangian fillings of Legendrian links: This second lecture will introduce the geometric ingredients: Legendrian links and their Lagrangian fillings. Their definitions and several constructions will be presented in detail. In particular, both singularities of wavefronts and Lagrangian handle attachments will be discussed at length.
- Singular support of sheaves and Legendrian fronts: In this lecture we will introduce the algebraic ingredients: the definition of the derived dg- category of sheaves with specified singular support. Several examples and motivation will be provided, explaining in particular the need for working with derived dg-categories and the fact that singular support is always coisotropic.
- Legendrian Invariance via sheaf kernels of Hamiltonian isotopies: This lecture will prove the invariance of the derived dg-category at hand under contact isotopies of the ideal contact boundary of the cotangent bundle. The cases of Reidemeister moves for Legendrian links will be discussed in detail.
- Explicit computations of sheaf categories: The purpose of this lecture will be to explicitly compute these sheaf categories for the class of Legendrian links arising as the (−1)-closures of positive braids. The lecture will include detailed descriptions of their objects and their morphisms in terms of Lie-theoretic concepts. The lecture will also present examples outside of this class and explain some of the challenges that arise for more general situations.
- Legendrian weaves and compressible systems: This lecture will introduce an important class of Lagrangian fillings, given by weaves. Weaves and their Lusztig cycles are the starting point for constructing Lagrangian fillings and L-compressible systems. In turn, these are the main input in the connection between cluster algebras and contact and symplectic topology. The lecture will also discuss how sheaf categories can be computed combinatorially in the context of weaves via flag moduli.
- Cluster theory and contact topology: The purpose of this lecture is to introduce cluster algebras and explain how the study of cluster algebras can be applied to prove results in contact topology (and vice versa). The lecture will ddiscuss the necessary ingredients to prove that the ring of functions on the moduli of Lagrangian fillings is a cluster algebra. It will be explained how the moduli of Lagrangian fillings is presented as the moduli D-stack of pseudoperfect objects of the category of sheaves and how Lagrangian surgery on L-compressible systems leads to mutations for cluster algebras.
- Cluster structures on moduli of Lagrangian fillings: This lecture will give the proof that the ring of functions on the moduli of Lagrangian fillings is a cluster algebra.
- Applications to symplectic topology and cluster algebras: This lecture will explain how the connection between cluster algebras and symplectic topology can be used to prove results in both areas. On the one hand, we will use cluster algebras to distinguish infinitely many Lagrangian fillings, compute the cohomology and Hodge structure of augmentation varieties for (−1)-closures and establish a conjectural classification of Lagrangian fillings. On the other hand, we will explain how to prove Leclerc’s conjecture that Richardson varieties admit cluster structures (and in any Lie type) and how to prove that the Muller-Speyer twist is a cluster automorphism via symplectic topology and Legendrian weaves.
- Progress and conjectures on the classification of Lagrangian fillings: This last lecture will present open problems in the area and (suggested) strategies to approach them. In particular, there will be a detailed discussion on the classification of Lagrangian fillings for Legendrian links (up to Hamiltonian isotopy), precise conjectures on the existence and properties of cluster structures on moduli of Lagrangian fillings.
