Abstracts

12.12.2022. Greg Muller (University of Oklahoma)

The space of quasi-periodic linear recurrences

Recent work by Morier-Genoud, Ovsienko, Schwartz, and Tabachnikov gave a correspondence between superperiodic linear recurrences, subspaces of Grassmannians, and SL(k) friezes. I will describe how to extend this to a correspondence between quasiperiodic linear recurrences, points of a positroid variety, and a new class of "jugglers friezes". The key tool is an analog of Postinikov's boundary measurement map which endows the space of quasi-periodic linear recurrences with an X-type cluster structure dual to the A-type cluster structure on the corresponding positroid variety. This project is joint with Roi Docampo.

05.12.2022. Travis Mandel (University of Oklahoma)

Bracelet bases are theta bases

Cluster algebras from marked surfaces can be interpreted as skein algebras, as functions on decorated Teichmüller space, or as functions on certain moduli of SL2-local systems. These algebras and their quantizations have well-known collections of special elements called "bracelets" (due to Fock-Goncharov and Musiker-Schiffler-Williams, and due to D. Thurston in the quantum setting). On the other hand, Gross-Hacking-Keel-Kontsevich used ideas from mirror symmetry to construct canonical bases of ``theta functions'' for cluster algebras, and this was extended to the quantum setting in my work with Ben Davison. I will review these constructions and describe upcoming work with Fan Qin in which we prove that the (quantum) bracelets bases coincide with the corresponding (quantum) theta bases.

28.11.2022. Khrystyna Serhiyenko (University of Kentucky)

Leclerc’s conjecture on a cluster structure for type A Richardson varieties

Leclerc constructed a conjectural cluster structure on Richardson varieties in simply laced types using cluster categories coming from preprojective algebras. We show that in type A, his conjectural cluster structure is in fact a cluster structure. We do this by comparing Leclerc’s construction with another cluster structure due to Ingermanson, which uses the combinatorics of wiring diagrams and the Deodhar stratification. Though the two cluster structures are defined very differently, we show that the quivers coincide and clusters are related by the twist map for Richardson varieties, recently defined by Galashin–Lam. This is join work with Melissa Sherman-Bennett.

21.11.2022. Martin Kalck (University of Freiburg)

Describing Leclerc’s Frobenius categories as categories of Gorenstein projective modules

In 2016, Leclerc introduced a new class of Frobenius categories in order to obtain (partly conjectural) cluster algebra structures on coordinate rings of open Richardson varieties.

Very recently, this approach has been completed and generalized in work of Casals, Gorsky, Gorsky, Le, Shen & Simental. More precisely, for open Richardson varieties their construction corresponds to the seed introduced by Ménard. An alternative approach to obtain cluster structures for open Richardson varieties has been announced by Galashin, Lam, Sherman-Bennett & Speyer.

We explain that Leclerc's categories are equivalent to categories of Gorenstein projective modules (aka maximal Cohen-Macaulay modules) over an Iwanaga-Gorenstein ring of virtual dimension at most two. This is an analogue of Buan, Iyama, Reiten & Scott’s description of Geiss, Leclerc & Schröer’s categorification for Schuber cells in terms of Gorenstein projective modules over quotients of preprojective algebras.

Our talk will be based on https://arxiv.org/pdf/1709.04785.pdf.

14.11.2022. Lang Mou (University of Cambridge)

Categorification of cluster algebras associated to orbifold surfaces

Categorification of cluster algebras of skew-symmetrizable type is less well-understood than the skew-symmetric case. For skew-symmetrizable cluster algebras associated to unpunctured marked surfaces with certain weighted orbifold points in the sense of Felikson-Shapiro-Tumarkin, we show that there is a class of gentle algebras whose \tau-tilting theory together with a cluster character provides a categorification. This is joint work with Daniel Labardini-Fragoso.

07.11.2022. Hülya Argüz (University of Georgia)

Fock–Goncharov dual cluster varieties and Gross–Siebert mirrors

Cluster varieties come in pairs: for any X-cluster variety there is an associated Fock–Goncharov dual A-cluster variety. On the other hand, in the context of mirror symmetry, associated with any log Calabi–Yau variety is its mirror dual, which can be constructed using the enumerative geometry of rational curves in the framework of the Gross–Siebert program. I will explain how to bridge the theory of cluster varieties with the algebro-geometric framework of Gross–Siebert mirror symmetry and show that the mirror to the X-cluster variety is a degeneration of the Fock–Goncharov dual A-cluster variety. To do this, we investigate how the cluster scattering diagram of Gross–Hacking–Keel–Kontsevich compares with the canonical scattering diagram defined by Gross–Siebert to construct mirror duals in arbitrary dimensions. This is joint work with Pierrick Bousseau.

31.10.2022. Ian Le (Australian National University)

Mirror Symmetry for Truncated Cluster Varieties

Gross, Hacking and Keel gave an algebro-geometric construction of cluster varieties: take a toric variety, blow up appropriate subvarieties in the boundary, and then remove the strict transform of the boundary. We work with a modification of this construction, which we call a truncated cluster variety--roughly, this comes from performing the same procedure on the toric variety with all the codimension 2 strata removed. The resulting variety differs from the cluster variety in codimension 2. I will describe a construction of a Weinstein manifold mirror to a truncated cluster variety and explain how to prove a mirror symmetry statement via Lagrangian skeleta. We hope that this is a first step towards understanding mirror symmetry for the entire cluster variety. This is joint work with Benjamin Gammage.

24.10.2022. Merlin Christ (Universität Hamburg)

Cluster categories from Fukaya categories

Given a smooth surface with boundary and marked points on the boundary (aka a marked surface), there is an associated cluster algebra with coefficients, studied for instance by Fomin-Thurston. We discuss recent progress on the categorification of such cluster algebras with a focus on the relation to Fukaya categories.

Given such a surface S, Ivan Smith constructed a Calabi-Yau 3-fold fibred over S, whose Fukaya category recovers the 3-Calabi-Yau category associated to S. This 3-Calabi-Yau category is described by a Ginzburg dg-algebra. We will compare three different (relative) cluster categories associated to S arising from different versions of the Ginzburg algebra, and their (expected) description in terms of quotients of the Fukaya categories of the 3-fold. One of these cluster categories is the 'classical' non-relative one, a further one was introduced by Yilin Wu, the last one was introduced in my recent preprint arxiv:2209.06595. In the remaining time, we then focus on the last of these three cluster categories, which surprisingly turns out to describe another Fukaya category, namely the topological Fukaya category of the surface S itself (with a special grading).

Spring 2022

27.06.2022 Thomas Lam (University of Michigan)

Cluster structures for braid varieties

Braid varieties are affine varieties indexed by positive braids that have been studied much in this seminar series. They have connections to knot homology, to Legendrian link geometry, to the geometry of flag varieties, and also to cluster algebras.

In this talk, I will discuss a cluster structure on braid varieties based on generalized minors, Deodhar geometry, and the Louise property for quivers. This is joint work with Pavel Galashin, Melissa Sherman-Bennett, and David Speyer.

6.06.2022 Matthew Pressland (University of Glasgow):

Categorification for positroid varieties

Coordinate rings of open positroid varieties in the Grassmannian carry a rich and complex combinatorial structure arising from Postnikov diagrams (also known as alternating strand diagrams, or dimer models in the disk). A recent result in this area by Galashin and Lam, confirms the conjecture that these coordinate rings are isomorphic to cluster algebras. In this talk, I will explain how these cluster algebras can be (additively) categorised, allowing for an interpretation of much of the combinatorics of Postnikov diagrams in terms of representation theory. In particular, I will explain how face labels for Postnikov diagrams (which determine clusters of Plücker coordinates) may be understood as names of modules over an appropriate algebra. Time permitting, I will also explain how the twist automorphism (or Donaldson–Thomas transformation) corresponds to the syzygy functor on the categorification. This is based on joint work with İlke Çanakçı and Alastair King.

30.05.2022 Alfredo Nájera Chávez (UNAM Oaxaca):

Deformation theory for finite cluster complexes

The purpose of this talk is to elaborate on a geometric relationship between cluster algebras and cluster complexes. In vague words this relationship is the following: cluster algebras of finite cluster type with universal coefficients may be obtained via a torus action on a Hilbert scheme. In particular, we will discuss the deformation theory of the Stanley-Reisner ring associated to a finite cluster complex and present some applications related to the Gröbner theory of the ideal of relations among cluster and frozen variables of a cluster algebra of finite cluster type. This is based on a joint project with Nathan Ilten and Hipolito Treffinger.

23.05.2022 Orsola Capovilla-Searle (UC Davis):

Infinitely many planar exact Lagrangian fillings and symplectic Milnor fibers

We provide a new family of Legendrian links with infinitely many distinct exact orientable Lagrangian fillings up to Hamiltonian isotopy. This family of links includes the first examples of Legendrian links with infinitely many distinct planar exact Lagrangian fillings, which can be viewed as the smallest Legendrian links currently known to have infinitely many distinct exact Lagrangian fillings. As an application we find new examples of infinitely many exact Lagrangian spheres and tori in 4-dimensional Milnor fibers of isolated hypersurface singularities with positive modality.

9.05.2022 Caitlin Leverson (Bard College)

DGA Representations, Ruling Polynomials, and the Colored HOMFLY-PT Polynomial

Given a Legendrian knot $\Lambda$ in $\mathbb{R}^3$ with the standard contact structure, Rutherford showed that the ruling polynomial of $\Lambda$ appears as a specialization of the HOMFLY-PT polynomial of its topological knot type. We will extend the definition of the ruling polynomial to define the colored ruling polynomial of a Legendrian knot, analogously to how the definition of the colored HOMFLY-PT polynomial is an extension of the HOMFLY-PT polynomial, and show that the colored ruling polynomial of $\Lambda$ also appears as a specialization of the colored HOMFLY-PT polynomial of $\Lambda$'s topological knot type. We will also discuss the relationship between counts of certain representations of the Chekanov-Eliashberg algebra of $\Lambda$ to the colored ruling polynomial of $\Lambda$ and thus the colored HOMFLY-PT polynomial of $\Lambda$'s topological knot type. Little knowledge of these topics will be assumed. This is joint work with Dan Rutherford.

2.05.2022 Ralf Schiffler (University of Connecticut)

Knot theory and cluster algebras

To every knot diagram (or link diagram) K, we associate a quiver with potential (Q,W) and, hence, a cluster algebra A(Q,W) as well as a Jacobian algebra B=Jac(Q,W). The vertices of the quiver are in bijection with the segments of the knot diagram.

For every segment i of K, we construct an indecomposable B-module T(i) and let T be the direct sum of these indecomposables. Each module T(i) corresponds to an element F(i) in the cluster algebra A(Q,W), the so-called F-polynomial of the module. F(i) is a polynomial in several variables y_1,..., y_n with positive integer coefficients.

We prove that, for each segment i of K, the Alexander polynomial of K is equal to a specific specialization of F(i). Furthermore this specialization does not depend on i. For an alternating knot, this specialization is simply y_j= -t if j is even; y_j=-t^{-1} if j is odd, where we label the segments of the knot in order of appearance along the knot.

25.04.2022 Lara Bossinger (Instituto de Matemáticas UNAM)

Total positivity in the Gröbner fan and cluster algebras

18.04.2022 Linhui Shen (Michigan State University)

Clusters structures on braid varieties

Let G be a complex simple group of type ADE. Let \beta be a positive braid whose Demazure product is the longest Weyl group element. The braid variety M(\beta) generalizes many well known varieties, including positroid cells, open Richardson varieties, and double Bott-Samelson cells. We provide a concrete construction of the cluster structures on M(\beta), using the weaves of Casals and Zaslow. We show that the coordinate ring of M(\beta) is a cluster algebra, which confirms a conjecture of Leclerc as special cases. This talk is based on joint work with Roger Casals, Eugene Gorsky, Mikhail Gorsky, Ian Le, and Jose Simental.

11.04.2022 Etienne Ménard (Institut Fourier, Université Grenoble Alpes)

Cluster structures associated to open Richardson varieties: combinatorics on simply-laced types

During my PhD thesis, I worked on an algorithm to compute explicit seeds for cluster structure on a categorification of the coordinate ring of an open Richardson variety. I will first explain the motivations of this question, its context, the categorification used, then describe the algorithm, focusing on its formulation in terms of combinatorics of words representing elements of a Weyl group (among the two formulations possible). I will then end with some results linked to combinatorics of Weyl groups in type A,D or E and open questions associated.

7.03.2022 Honghao Gao (Michigan State University)

Infinitely many Lagrangian fillings

A filling is an oriented surface bounding a link. Classifications of Legendrian knots and their exact Lagrangian fillings are central questions in low-dimensional contact and symplectic topology. Lagrangian fillings can be constructed via local moves in finite steps. In this talk, I will show that most Legendrian torus links have infinitely many exact Lagrangian fillings. These fillings are constructed using Legendrian loops, and proven to be distinct using the microlocal theory of sheaves and the theory of cluster algebras. This is a joint work with Roger Casals.