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Here are the abstracts for the selected research talks.
Esther Banaian: A geometric model for semilinear locally gentle algebras
Semilinear gentle algebras are path algebras over a division ring, where the underlying quiver with relations is subject to restriction similar to those for a “classical” gentle algebra. They are a type of semilinear clannish algebras studied by Bennett-Tennenhaus and Crawley-Boevey. Classically, one can model gentle algebras using geometric means. In this talk, we explain how we can extend these models to the semilinear case. Along the way we will also discuss how semilinear gentle algebras are nodal, and demonstrate how the Zembyk decomposition for nodal algebras can be interpreted geometrically. This is based on joint work (arXiv 2402.04947) with Raphael Bennett-Tennenhaus, Karin Jacobsen, and Kayla Wright.Merlin Christ: Cluster categories of surfaces and topological Fukaya categories
Given a marked surface without punctures the associated cluster category is closely related to the topological Fukaya category of the surface. In this talk, we will survey the different ingredients needed to make this into a precise relation. These include the Higgs category, the generalized cluster category of the relative Ginzburg algebra and also constructible sheaves of enhanced triangulated categories.Tal Gottesman: Fractionally Calabi Yau posets: corroborating a conjecture by Chapoton
In 2023, Chapoton published a fascinating conjecture linking combinatorial formulas, the representation theory of partially ordered sets and symplectic geometry. In 2018, Rognerud proved that Tamari lattices illustrate the first two aspects of this conjecture. In this talk I will present the lattices of order ideals of products of two chains as a first illustration all three aspects of the conjecture. In this example both the fractionally Calabi-Yau property and the link to symplectic geometry derive from the study of one family of representations defined by antichains of the lattice that have remarkable properties.Vitor Gulisz: A functorial approach to n-abelian categories
We show how to reformulate the axioms of an n-abelian category in terms of its functor categories. Since these are abelian, such a reformulation allows the use of classical homological algebra to understand higher homological algebra phenomena. As an application, we present generalizations of the axioms “every monomorphism is a kernel” and “every epimorphism is a cokernel” of an abelian category to n-abelian categories. Moreover, by restricting our results to rings, we are able to describe when the category of finitely generated projective modules over a ring is n-abelian. This description leads to a correspondence for n-abelian categories with additive generators, which extends the Higher Auslander Correspondence.Maximilian Kaipel: A lattice of categories of an algebra
For a finite-dimensional algebra, it is well known that the partially ordered sets of functorially finite torsion classes, 2-term silting complexes and support τ-tilting modules are isomorphic. By definition, the picture group encodes the structure of these posets. To study this group, a categorical analogue, the τ-cluster morphism category was introduced, which in many cases defines a K(π,1) space for the picture group.We focus on a geometric construction of the τ-cluster morphism category from the g-vector fan of an algebra. In particular, we choose to relax certain constraints in the construction to obtain a lattice of categories containing the τ-cluster morphism category. In the talk, I will explain how the surrounding categories allow us to study the τ-cluster morphism category.
David Nkansah: The Nakayama functor is a wannabe Serre functor
Classic Auslander-Reiten theory is a useful tool used to describe the category of modules over an Artinian ring. Nakayama functors have a significant role in this process. In this presentation, we will construct Nakayama functors and Auslander-Reiten translates on proper abelian subcategories by using approximation theory. These abelian subcategories, which Jørgensen defined in 2022, are generalisations of hearts of t-structures. As a result, under some standing assumptions, we will prove that our proper abelian subcategories are dualizing k-varieties and that they have enough projectives if and only if they have enough injectives. We will also provide a new proof of the existence of Auslander-Reiten sequences in the module category of a finite dimensional algebra.This talk is based on the following arXiv preprint: arXiv:2312.07323.
Here is the list of speakers for the introductory lectures of the mini-courses.
You can access the recordings of the talks from the first day at https://lmv.math.cnrs.fr/ecole-dete-charms-exposes/ using the password CHARMS2024.