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Bethany Marsh - Categorification of the Grassmannian cluster structure (slides) (recording)
The homogeneous coordinate ring of the Grassmannian has a beautiful cluster algebra structure, discovered by J. Scott. This structure is described by the combinatorics of certain diagrams in a disk which were introduced by A. Postnikov. The aim of this talk is to give an introduction to this cluster algebra structure and the categorification developed by B. T. Jensen, A. D. King and X. Su using a Frobenius category of maximal Cohen-Macaulay modules. I will also discuss the relationship with dimer models developed in joint work with K. Baur and A. D. King.
Marta Mazzocco - Quantum uniformisation and CY algebras (slides) (recording)
In this talk, I will discuss a special class of quantum del Pezzo surfaces. In particular I will introduce the generalised Sklyanin-Painlevé algebra and characterise its PBW/PHS/Koszul properties. This algebra contains as limiting cases the generalised Sklyanin algebra, Etingof-Ginzburg and Etingof-Oblomkov-Rains quantum del Pezzo and the quantum monodromy manifolds of the Painlevé equations.
Vanessa Miemietz - Simple transitive 2-representations of Soergel bimodules (slides) (recording)
I will explain how to reduce the classification of ‘simple’ 2-representations of the 2-category of Soergel bimodules in many (most) cases to the known problem of the same classification for certain fusion categories.
Matthew Pressland - Cluster categories from Postnikov diagrams (slides) (recording)
Many rings of interest in geometry can be equipped with the additional combinatorial structure of a cluster algebra, which one would like to understand representation-theoretically by means of a cluster category. A result of Jensen, King and Su provides such a category for the cluster algebra structure on the coordinate ring of the Grassmannian, and Baur, King and Marsh show how this category is related to Postnikov diagrams, certain collections of oriented paths in a disc. In this talk I will explain how to reverse this logic, and use Postnikov diagrams to produce cluster categories. As an application, this allows us to categorify the cluster algebra structures on positroid subvarieties in the Grassmannian.
Katerina Hristova - 2-categories with one cell and their representations (slides) (recording)
We look at weakly fiat 2-categories with one object and one cell, apart from possibly a cell consisting only of the identity one morphism of the unique object. We explain two interesting examples of such categories - one coming from symmetric bimodules of a finite dimensional basic unital algebra, and the other constructed from the category of A-modules, where A has the additional property of being a Hopf algebra. We look at the relation between these categories and classify their simple transitive 2-representations. Joint work with Vanessa Miemietz.
Johanne Haugland - Subcategories of n-exangulated categories (slides) (recording)
The notion of extriangulated categories was introduced by Nakaoka and Palu as a simultaneous generalisation of exact and triangulated categories. Many concepts and results concerning exact and triangulated structures have been unified and extended using this framework. Herschend, Liu and Nakaoka defined n-exangulated categories, which is a higher dimensional analogue of extriangulated categories. In this talk, we give an introduction to such categories and discuss how we can understand their subcategories in terms of subgroups of the associated Grothendieck group.
Jordan McMahon - Categorifying maximal collections of non-k-intertwining subsets (slides) (recording)
Maximal collections of non-crossing subsets are an easy to understand abstraction of the triangulations of a convex polygon. They have interesting combinatorics in their own right, closely connected to the Grassmannian. They may be categorified through Grassmannian cluster algebras and cluster categories. Maximal collections of non-k-intertwining subsets are a natural generalisation of these combinatorics.
In the first part of this presentation we will briefly discuss (using pictures) how Grassmannian cluster algebras are related to current research trends including Topological Data Analysis, Pseudocircle arrangements and Morsifications. Then we discuss joint work with N. Williams on a new categorification of maximal collections of non-k-intertwining subsets using higher precluster-tilting subcategories.
Uran Meha - Coherent presentations of plactic monoids (slides) (recording)
Plactic monoids are certain monoids that codify the representation theory of symmetrizable Kac-Moody algebras. In classical types, these monoids admit finite convergent presentations, called column presentations. Convergence is a property of a presentation formalized in terms of rewriting theory, a computational theory that has recently found application in categorifications of quantum groups. Here we explain results of recent work by the speaker on type C (and type A), where these convergent presentations are extended to coherent ones by the use of rewriting theory and certain new graph theoretical tools called C-trees. We note the appearance of certain intrinsic parameters of types A and C in these coherent presentations.
Kaveh Mousavand - A categorification of biclosed sets of strings (slides)
For any gentle algebra of finite representation type, one can consider the closure space on the set of strings. Palu, Pilaud, and Plamondon proved that the collection of all biclosed sets of strings forms a lattice, and moreover, that this lattice is congruence-uniform. Many interesting examples of finite congruence-uniform lattices may be represented as the lattice of torsion classes of an associative algebra. We introduce a generalization, the lattice of torsion shadows, and we prove that the lattice of biclosed sets of strings is isomorphic to a lattice of torsion shadows.
If time permits, we also introduce the analogous notion of wide shadows, and prove that the shard intersection order of the lattice of biclosed sets is isomorphic to a lattice of wide shadows.
Yadira Valdivieso - Skew-gentle algebras and orbifolds (slides) (recording)
Skew-gentle algebras, a generelisation of gentle algebras, naturally appear in many different contexts such as in the framework of cluster algebras where they arise as Jacobian algebras of certain triangulations of surfaces with punctures. In this talk, we will give a geometric model of the bounded derived category of a skew-gentle algebra in the terms of graded curves in a generelised orbifold dissection with orbifold points of order two with boundary and punctures. We show that the geometric model of a skew-gentle algebras is closed related to the model of the underlying gentle algebra defined in joint work with Opper-Plamondon-Schroll and which by work of Haiden-Katzarkov-Kontsevich and Lekili-Polishchuk is closely linked with the partially wrapped Fukaya category of a surface with stops. This is a report on joint work with Sibylle Schroll and Daniel Labardini-Fragoso.
Ana Garcia Elsener - Monomial Jacobian algebras (slides) (recording)
A celebrated result by Keller–Reiten says that 2-Calabi–Yau tilted algebras are Gorenstein and stably 3-Calabi–Yau. We show that the converse holds in the monomial case: a 1-Gorenstein monomial algebra and stably 3-Calabi–Yau has to be 2-Calabi–Yau tilted, moreover it is Jacobian.
Bernhard Keller - Quantum Cartan matrices categorified (slides) (recording)
Quantum Cartan matrices are of importance for the representation theory of quantum affine algebras. We show how to categorify them using bigraded 2-dimensional Ginzburg algebras. These also appear in beautiful recent work by Ikeda-Qiu on "quantized" Bridgeland stability conditions.