researchseminars.org

Sira Gratz (University of Glasgow)
2021-02-24 13:15 - 14:15 UTC | https://stable.researchseminars.org/talk/AarHomAlg/1/
Title: Grassmannians, Cluster Algebras and Hypersurface Singularities (https://stable.researchseminars.org/talk/AarHomAlg/1/)
by Sira Gratz (University of Glasgow) as part of Aarhus Homological Algebra Seminar

Abstract
Grassmannians are objects of great combinatorial and geometric beauty, which arise in myriad contexts. Their coordinate rings serve as a classic example of cluster algebras, as introduced by Fomin and Zelevinsky at the start of the millennium, and their combinatorics is intimately related to algebraic and geometric concepts such as to representations of algebras and hypersurface singularities. At the core lies the idea of generating an object from a so-called "cluster" via the concept of "mutation".

In this talk, we offer an overview of Grassmannian combinatorics in a cluster theoretic framework, and ultimately take them to the limit to explore the a priori simple question: What happens if we allow infinite clusters? In particular, we introduce the notion of a cluster algebra of infinite rank (based on joint work with Grabowski), and of a Grassmannian category of infinite rank (based on joint work with August, Cheung, Faber and Schroll).

Rosanna Laking (University of Verona)
2021-03-10 13:15 - 14:15 | https://stable.researchseminars.org/talk/AarHomAlg/2/
Title: Mutation and minimal inclusions of torsion classes (https://stable.researchseminars.org/talk/AarHomAlg/2/)
by Rosanna Laking (University of Verona) as part of Aarhus Homological Algebra Seminar

Abstract
Torsion pairs are fundamental tools in the study of abelian categories, which contain important information related to derived categories and their t-structures. In this talk we will consider the lattice of torsion classes in the category of finite-dimensional modules over a finite-dimensional algebra, with a particular focus on the minimal inclusions of torsion classes.

It was shown by Adachi, Iyama and Reiten that minimal inclusions of functorially finite torsion classes correspond to irreducible mutations of associated two-term silting complexes in the category of perfect complexes. In this talk we will explain how minimal inclusions of arbitrary torsion classes correspond to irreducible mutations of associated two-term cosilting complexes in the unbounded derived category.

This talk will be based on joint work with Lidia Angeleri Hügel, Jan Stovicek and Jorge Vitória.

Karin Baur (University of Leeds)
2021-03-17 13:15 - 14:15 | https://stable.researchseminars.org/talk/AarHomAlg/3/
Title: Orbifold diagrams and skew group categories (https://stable.researchseminars.org/talk/AarHomAlg/3/)
by Karin Baur (University of Leeds) as part of Aarhus Homological Algebra Seminar

Abstract
Alternating strand diagrams (as introduced by Postnikov) on the disk have been used in the study of the coordinate ring of the Grassmannian. In particular, they give rise to clusters of the Grassmannian cluster algebras (Scott) or to cluster-tilting objects of the Grassmannian cluster categories of Jensen-King-Su (Baur-King-Marsh). On the other hand, orbifolds have also been related to cluster structures as Paquette-Schiffler (or Chekhov-Shapiro for a geometric approach). Here we introduce orbifold diagrams as quotients of symmetric Postnikov diagrams and show how to associate quivers with potentials to them. This is joint work with Andrea Pasquali (Stuttgart) and Diego Velasco (Cali).

Ilke Canakci (Vrije Universiteit Amsterdam)
2021-04-21 12:15 - 13:30 | https://stable.researchseminars.org/talk/AarHomAlg/4/
Title: Infinite friezes (https://stable.researchseminars.org/talk/AarHomAlg/4/)
by Ilke Canakci (Vrije Universiteit Amsterdam) as part of Aarhus Homological Algebra Seminar

Abstract
Frieze patterns, introduced by Coxeter, are infinite arrays of numbers where neighbouring numbers satisfy a local arithmetic rule. Under a certain finiteness assumption, they are in one-to-one correspondence with triangulations of polygons [Conway–Coxeter] and they come from triangulations of annuli in an infinite setting [Baur–Parsons–Tschabold]. In this talk, we will discuss a relationship between pairs of infinite friezes associated with a triangulation of the annulus and explore how one determines the other in an essentially unique way. We will also consider module categories associated with triangulated annuli where infinite friezes may be recovered using a homological formula. This is joint work with Karin Baur, Karin Jacobsen, Maitreyee Kulkarni, and Gordana Todorov.

Jorge Vitoria (University of Cagliari)
2021-04-26 11:00 - 12:00 | https://stable.researchseminars.org/talk/AarHomAlg/5/
Title: Hearts for commutative noetherian rings: derived equivalences and torsion pairs (https://stable.researchseminars.org/talk/AarHomAlg/5/)
by Jorge Vitoria (University of Cagliari) as part of Aarhus Homological Algebra Seminar

Abstract
The structure of the category of modules over a commutative noetherian ring R and of its derived category is largely controlled by the prime spectrum of R. In this talk we discuss how this control extends to the structure of hearts of t-structures in the derived category. We will focus in particular on hearts arising from hereditary torsion pairs in Mod(R). These turn out to be Grothendieck categories which are derived equivalent to R and such that part of the lattice of torsion pairs can be studied using the prime spectrum of R. This talk is based on joint work with Sergio Pavon.

Susan J. Sierra (University of Edinburgh)
2021-05-05 12:15 - 13:15 | https://stable.researchseminars.org/talk/AarHomAlg/6/
Title: The Poisson spectrum of the symmetric algebra of the Virasoro algebra (https://stable.researchseminars.org/talk/AarHomAlg/6/)
by Susan J. Sierra (University of Edinburgh) as part of Aarhus Homological Algebra Seminar

Abstract
Let W be the Witt algebra of polynomial vector fields on the punctured
complex plane, and let Vir be the Virasoro algebra, the unique nontrivial central
extension of W. We discuss work in progress with Alexey Petukhov to analyse
Poisson ideals of the symmetric algebra of Vir. We focus on understanding
maximal Poisson ideals, which can be given as the Poisson cores of maximal
ideals of Sym(Vir) and of Sym(W). We give a complete classification of maximal
ideals of Sym(W) which have nontrivial Poisson cores. We then lift this
classification to Sym(Vir), and use it to show that if $\lambda \neq 0$, then $(z-
\lambda)$ is a maximal Poisson ideal of Sym(Vir).

Man-Wai Cheung (Harvard University)
2021-05-10 11:00 - 12:00 | https://stable.researchseminars.org/talk/AarHomAlg/7/
Title: Tropical disks counting, stability conditions in symplectic geometry and quiver representations (https://stable.researchseminars.org/talk/AarHomAlg/7/)
by Man-Wai Cheung (Harvard University) as part of Aarhus Homological Algebra Seminar

Abstract
Bridgeland developed stability scattering diagrams relating scattering diagrams with quiver representations. Scattering diagrams were developed as a machinery in mirror symmetry. Together with Travis Mandel, we associate tropical disks counting with
quiver representations by using the stability scattering diagrams.
Next, together with Yu-Wei Fan and Yu-Shen Lin, we look at the stable objects for the $A_2$ quiver. It is known that the derived Fukaya-Seidel category of the rational elliptic surface is the derived category of the $A_2$ quiver. We made use of the relation and corresponded the special Lagrangian with the stable objects in the derived category of coherent sheaves.
