Schedule for: 25w5439 - Representation Theory, Symplectic Geometry, and Cluster Algebras

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

The informal gathering will take place in the BIRS lounge in the Professional Development Centre, where you checked in. The front desk or other participants should be able to provide specific room and directions.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this series of talks, I will explain the additive categorification of cluster algebra structures on the Grassmannian and more general positroid varieties. For the Grassmannian itself, the cluster structure is due to Scott, generalising a special case by Fomin and Zelevinsky, and the categorification is by Jensen, King and Su, building on an “almost-categorification” by Geiß, Leclerc and Schröer. Scott’s description of the cluster structure uses combinatorics introduced by Postnikov in his study of total positivity for the Grassmannian, and I will explain how the same combinatorics encodes several non-commutative algebras. This leads to a re-interpretation and generalisation of Jensen–King–Su’s construction, and additive categorifications of the cluster structure on the positroid variety associated to any connected positroid. The ultimate aim of the series is to explain a proof, based on the homological algebra of these categorifications, that the two most canonical cluster structures on a positroid variety quasi-coincide in the sense of Fraser, confirming an expectation of Muller and Speyer.

The object of these talks is to introduce braid varieties and explore their geometry. The focus will be the construction of cluster structures on their rings of regular functions. The technique that we will employ to construct cluster seeds is that of weaves. This present a rather general framework which builds cluster structures for positroid and Richardson varieties, double Bruhat cells and double Bott-Samelson varieties, and other. Weaves also connect to known associated combinatorics, such as reduced plabic graphs, 3D plabic graphs or k-Grassmannian permutations. The first lecture will focus on defining braid varieties, presenting examples and first properties. The second lecture is centered around weave combinatorics and cluster algebras. The third lecture shall describe applications and generalizations of results in and around braid varieties, and some speculations towards potential results. Throughout these talks, relations to symplectic topology will be emphasized, letting the geometry and the algebra guide each other hand in hand.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In previous work with Ralf Schiffler, we associated a module T(i) to every segment i of a link diagram K and showed that there is a poset isomorphism between the submodules of T(i) and the Kauffman states of K relative to i. In this talk, I will demonstrate that these posets form distributive lattices and that the subposet of join irreducible Kauffman states is isomorphic to the poset of the coefficient quiver of T(i). Finally, I will describe a method to identify the join irreducible Kauffman states.

For a cluster algebra A, its deep locus is the complement to the union of cluster tori in Spec(A). We put forth a conjecture that, for a reasonably wide class of cluster algebras, characterizes their deep locus as the points with non-trivial stabilizer under a natural group action on Spec(A). We verify the conjecture for cluster algebras of finite cluster type, as well as for the maximal positroid strata in Gr(3,n). All these cluster algebras admit an explicit realization as cluster algebras associated to type A braid varieties, and we will see a relationship between the geometry of the deep locus and that of the corresponding link.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The object of these talks is to introduce braid varieties and explore their geometry. The focus will be the construction of cluster structures on their rings of regular functions. The technique that we will employ to construct cluster seeds is that of weaves. This present a rather general framework which builds cluster structures for positroid and Richardson varieties, double Bruhat cells and double Bott-Samelson varieties, and other. Weaves also connect to known associated combinatorics, such as reduced plabic graphs, 3D plabic graphs or k-Grassmannian permutations. The first lecture will focus on defining braid varieties, presenting examples and first properties. The second lecture is centered around weave combinatorics and cluster algebras. The third lecture shall describe applications and generalizations of results in and around braid varieties, and some speculations towards potential results. Throughout these talks, relations to symplectic topology will be emphasized, letting the geometry and the algebra guide each other hand in hand.

In this series of talks, I will explain the additive categorification of cluster algebra structures on the Grassmannian and more general positroid varieties. For the Grassmannian itself, the cluster structure is due to Scott, generalising a special case by Fomin and Zelevinsky, and the categorification is by Jensen, King and Su, building on an “almost-categorification” by Geiß, Leclerc and Schröer. Scott’s description of the cluster structure uses combinatorics introduced by Postnikov in his study of total positivity for the Grassmannian, and I will explain how the same combinatorics encodes several non-commutative algebras. This leads to a re-interpretation and generalisation of Jensen–King–Su’s construction, and additive categorifications of the cluster structure on the positroid variety associated to any connected positroid. The ultimate aim of the series is to explain a proof, based on the homological algebra of these categorifications, that the two most canonical cluster structures on a positroid variety quasi-coincide in the sense of Fraser, confirming an expectation of Muller and Speyer.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Generalized associahedra are a well-studied family of polytopes associated to a finite-type cluster algebra and choice of starting cluster. We show that the generalized associahedra constructed by Padrol, Palu, Pilaud, and Plamondon, building on ideas from Arkani-Hamed, Bai, He, and Yan, can be naturally viewed as moment polytopes for an open patch of the quotient of the cluster A-variety with universal coefficients by its maximal natural torus action. We prove our result by showing that their construction can be understood on the basis of the way that moment polytopes behave under symplectic reduction. This talk is based on joint work with Misha Gekhtman, arXiv:2402.03437.

A positroid gives rise to a positroid variety with a cluster structure. Pressland showed that this cluster structure may be categorified by finding a dimer model corresponding to the positroid, and taking the Gorenstein-projective module category of its boundary algebra. Boundary algebras are not well understood when the positroid is not uniform. We (joint with Jon Boretsky) give the first explicit description of the boundary algebra of a connected positroid as a quiver with relations. Our description sidesteps the need to choose a dimer model by working directly with positroid combinatorics: the quiver is obtained from the facets of positroid base polytope and a minimal set of relations is obtained by using in addition the decorated permutation of the positroid.

First introduced by Fuchs and Chekanov-Pushkar, normal rulings are decompositions of front diagrams of Legendrian knots. In this talk, we will introduce normal rulings and, if time permits, some connections between normal rulings and other topics.

Lagrangian fillings of Legendrian links equipped with rank 1 local systems form a moduli space, which are defined using either augmentations of the Legendrian contact homology or microlocal rank 1 sheaves. We give a sufficient condition on when the holonomy of rank 1 local systems for a given Lagrangian filling can be extended globally to the whole moduli space of all Lagrangian fillings and provide a symplectic geometric interpretation on these regular functions. In the case of braid varieties, we recover the cluster variables which are often constructed through algebraic computations. This is joint work with Roger Casals.

In this talk, I will construct some subcategories GP B of the Grassmannian cluster category CM C. I will explain the connections between GP B and Grassman necklaces, and construct a cluster character on GP B, which generalises flow polynomials defined on plabic graphs. I will also discuss the mutations of those cluster characters. This talk is based on the joint work with A King and B T Jensen, Categorification and mirror symmetry for Grassmannians, arXiv: 2404.14572.

The g-vector (aka index) of an object in a cluster category is the leading exponent of its cluster character. This talk will describe various things we know about the set of g-vectors (a priori a monoid) In the case of the Grassmannian cluster category, including that it is a rational polyhedral cone. This is joint work with B.T. Jensen and X. Su.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

The object of these talks is to introduce braid varieties and explore their geometry. The focus will be the construction of cluster structures on their rings of regular functions. The technique that we will employ to construct cluster seeds is that of weaves. This present a rather general framework which builds cluster structures for positroid and Richardson varieties, double Bruhat cells and double Bott-Samelson varieties, and other. Weaves also connect to known associated combinatorics, such as reduced plabic graphs, 3D plabic graphs or k-Grassmannian permutations. The first lecture will focus on defining braid varieties, presenting examples and first properties. The second lecture is centered around weave combinatorics and cluster algebras. The third lecture shall describe applications and generalizations of results in and around braid varieties, and some speculations towards potential results. Throughout these talks, relations to symplectic topology will be emphasized, letting the geometry and the algebra guide each other hand in hand.

In this series of talks, I will explain the additive categorification of cluster algebra structures on the Grassmannian and more general positroid varieties. For the Grassmannian itself, the cluster structure is due to Scott, generalising a special case by Fomin and Zelevinsky, and the categorification is by Jensen, King and Su, building on an “almost-categorification” by Geiß, Leclerc and Schröer. Scott’s description of the cluster structure uses combinatorics introduced by Postnikov in his study of total positivity for the Grassmannian, and I will explain how the same combinatorics encodes several non-commutative algebras. This leads to a re-interpretation and generalisation of Jensen–King–Su’s construction, and additive categorifications of the cluster structure on the positroid variety associated to any connected positroid. The ultimate aim of the series is to explain a proof, based on the homological algebra of these categorifications, that the two most canonical cluster structures on a positroid variety quasi-coincide in the sense of Fraser, confirming an expectation of Muller and Speyer.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Given a marked surface, there is a well-known associated cluster algebra whose coefficients are the boundary arcs of the surface. One can obtain a corresponding additive categorification via Wu's extriangulated Higgs category. We will explain how this Higgs category can be identified with the topological Fukaya category of the surface with 1-periodic coefficients. This category can be understood from multiple perspectives: In representation theoretic terms, it is the orbit category of the derived category of a gentle algebra by the shift functor [1]. Symplectically, the category can be constructed as the cosingularity category of a partially wrapped Fukaya category of a 3-fold fibered over the surface. More abstractly, it arises via the categorical gluing of simple categories over a triangulation of the surface. We will also indicate how the above picture naturally extends to the setting of higher rank cluster algebras of surfaces in the sense of higher Teichmüller theory. Based on arxiv:2209.06595, arxiv:2501.13666, and work in preparation.

Legendrian links can arise as the boundary of exact Lagrangian surfaces in the standard symplectic 4-ball. Such surfaces are called fillings of the link. In the last decade, our understanding of the moduli space of fillings for various families of Legendrians has greatly improved thanks to tools from sheaf theory, Floer theory and cluster algebras. From a Floer perspective, fillings can be distinguished using augmentations of the Chekanov-Eliashberg differential graded algebra of the link. Henry and Rutherford showed that the augmentation variety of a Legendrian with a fixed front admits a decomposition into pieces of the form $\mathbb{C}^n\times (\mathbb{C}^*)^m$ constructed using Morse Complex sequences. Morse Complex sequences are a discrete cousin of generating families which in turn are related to normal graded rulings and to augmentations. We focus on Legendrians given by the $(-1)$ closure of $\beta\Delta$ where $\beta$ is a positive braid whose Demazure product is $\Delta$. For such a Legendrian, the moduli space of fillings (and the augmentation variety) is a braid variety $X(\beta)$ which admits a cluster structure. Casals-Gorsky-Gorsky-Simental showed braid varieties admit a decomposition into pieces of the form $\mathbb{C}^n\times (\mathbb{C}^*)^m$ constructed using algebraic weaves. In joint work with Johan Asplund, James Hughes, Caitlin Leverson, Wenyuan Li, and Angela Wu we show how to compare both decompositions and that they in fact agree.

The classification of Lagrangian submanifolds has been an influential problem in symplectic topology for a long time. A more recent trend is to look at this through the lens of homological algebra, trying to classify objects of the Fukaya category instead. I will describe a program aimed at proving that the Fukaya category of G/P is generated by Lagrangian tori, where cluster algebras play a crucial role. I see this effort as an open-string version of Schubert calculus, continuing a story that started in the 19th century with Hilbert’s 15th problem, and spurred the development of classical and quantum intersection theory.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will discuss relations, both proven and conjectural, between the three types of algebras in the title. The main focus will be on the relation between Legendrian skein algebras and Hall algebras of Fukaya categories of surfaces. Partly based on arXiv:1908.10358.

Since their introduction, quivers with potential have been used in the context of additive categorification of cluster algebras. Recently, Casals and Gao applied this notion in the study of Lagrangian fillings of Legendrian links. They associated quivers with potential with certain configurations of curves in surfaces, proved their non-degeneracy for conjugate surfaces of plabic fences, and applied this to study the relation between cluster seeds and embedded exact Lagrangian fillings for the corresponding links. I will sketch some of these constructions and report on work in progress towards establishing similar results for Demazure weaves, with potential applications to fillings and to the categorification of cluster structures on braid varieties in mind.

After establishing a cluster structure on the sheaf moduli of (e.g. braid positive) Legendrian knots, a natural next step is to understand automorphisms that respect this cluster structure. In this talk, I will describe how to interpret a broad class of these cluster automorphisms as Legendrian loops, i.e. isotopies that fix the Legendrian setwise. I will then give some explicit computations of cluster automorphisms using Legendrian front diagrams, and describe applications of this approach to constructing and distinguishing exact Lagrangian fillings.

The relationship between cluster algebras and symplectic geometry has led to breakthrough results in understanding the exact Lagrangian fillability of Legendrian submanifolds in R^3. Conversely, it has also led to applications to cluster theory, giving a geometric interpretation of cluster variables and mutations. Some of this story can be extended to Legendrian surfaces -- which are Legendrian submanifolds in R^5. In joint work with James Hughes, we observe that for a certain class of Legendrian surfaces, called "doubles", one can associate a notion of complexity related to the mutation distance between seeds in certain cluster algebras. This talk will review notions of Legendrian surfaces, doubles, the ideas behind this result, and some hopes about how we hope to push this further.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In symplectic geometry, tropical varieties arise as minimal degenerate limits of amoebae of Lagrangian submanifolds of (C*)^n. Here the amoeba of a subset of (C*)^n is its image under the log norm projection to R^n. In this talk, we introduce and study tropical Lagrangian coamoebae, analogous objects associated with the argument projection to T^n. Their formal definition is rooted in mirror symmetry: a tropical Lagrangian coamoeba is a certain geometric representation of a free resolution of a mirror coherent sheaf. A succinct characterization of this construction, and a separate source of motivation, is that in a precise sense it generalizes the spectral theory of dimer models. Recall that the dimer model on a bipartite graph in T^2 may be solved in terms of its spectral data, a family of curves with line bundles in (C*)^2. This relationship turns out to admit a robust generalization, where the spectral data is allowed to be any coherent sheaf on an algebraic torus of any dimension, and where the bipartite graph is generalized by a tropical Lagrangian coamoeba. From another perspective, tropical Lagrangian coamoebae also provide a symplectic counterpart to the theory of brane brick models, a class of generalized dimer models studied in the physics literature.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.