Archive on FD Seminar
https://www.fd-seminar.xyz/talks/
Recent content in Archive on FD SeminarHugo -- gohugo.ioen-usThu, 30 Jul 2020 13:00:00 +0000TBA
https://www.fd-seminar.xyz/talks/2020-07.30/
Thu, 30 Jul 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-07.30/Finite generation of cohomology for Drinfeld doubles of finite group schemes
https://www.fd-seminar.xyz/talks/2020-07-03/
Thu, 23 Jul 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-07-03/Recall that the Drinfeld double of a finite group scheme \(G\) is a finite-dimensional Hopf algebra which integrates, in a certain way, both the group ring of \(G\) and the algebra of global functions on \(G\). I will discuss our recent proof of the fact that Drinfeld doubles of arbitrary finite group schemes have finitely generated cohomology. (One should view this result in light of Etingof and Ostrik’s conjecture, which proposes that any finite tensor categories has finitely generated cohomology.Postnikov diagrams and orbifolds
https://www.fd-seminar.xyz/talks/2020-07-16/
Thu, 16 Jul 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-07-16/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 as defined by 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.Bounded extension algebras and Han's conjecture
https://www.fd-seminar.xyz/talks/2020-07-09/
Thu, 09 Jul 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-07-09/Given a finite dimensional algebra \(A\) over a field \(k\), Han’s conjecture relates two homological invariants associated to \(A\): its global dimension and its Hochschild homology. In the commutative case – non necessarily finite dimensional but finitely generated – the finiteness of the global dimension is equivalent to the fact that \(A\) is geometrically regular, see for example [3, 9.3.13]. More precisely, Han’s conjecture states that for \(A\) finite dimensional, \(A\) is smooth if and only if \(HH_n(A)=0\) for \(n\gg0\).Derived equivalences for skew-gentle algebras
https://www.fd-seminar.xyz/talks/2020-07-02/
Thu, 02 Jul 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-07-02/Opper, Plamondon and Schroll described a geometric model for the derived category of gentle algebras. In this talk I will explain how to use this model to get one for the derived category of skew-gentle algebras. The main tool is the use of the skew-group algebra structure of the skew-gentle algebras. This is a collaboration with Thomas Brüstle.A geometric model for the syzygies over certain 2-Calabi-Yau tilted algebras
https://www.fd-seminar.xyz/talks/2020-06-25/
Thu, 25 Jun 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-06-25/This is a joint work in progress with Khrystyna Serhiyenko.
The syzygy functor maps a module to the kernel of its projective cover. Thus a syzygy is a submodule of a projective. We want to study the stable category of syzygies over a 2-Calabi–Yau tilted algebra. For these algebras, this category is equivalent to the stable category of Cohen-Macauley modules, as well as to the singularity category of the algebra. It is a triangulated 3-Calabi–Yau category whose shift is given by the syzygy functor.Simple objects in torsion-free classes over preprojective algebras of Dynkin type
https://www.fd-seminar.xyz/talks/2020-06-18/
Thu, 18 Jun 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-06-18/In this talk, I propose to study exact-categorical structures of torsion(-free) classes of module categories. For functorially finite torsion-free class, indecomposable projective and injective objects are easily described by \(\tau^-\)-tilting modules, and in particular, the numbers of them coincide. However, there can be more simple objects in torsion-free class, which I propose to study. I explain that the number of simple objects controls the validity of the Jordan–Hölder type theorem in a torsion-free class.The quiver of n-hereditary algebras
https://www.fd-seminar.xyz/talks/2020-06-11/
Thu, 11 Jun 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-06-11/In 2004, Iyama introduced a generalisation of some of the central ideas of Auslander-Reiten theory to a “higher dimensional” setting. This has inspired a lot of interesting research over the years. One important concept is that of \(n\)-hereditary algebras, which enjoy some of the key properties of hereditary algebras in the context of higher AR-theory. Their defining characteristic is rather strong, so they form a very small subset of the finite-dimensional algebras of global dimension \(n\).Schemes of modules over gentle algebras and laminations of surfaces
https://www.fd-seminar.xyz/talks/2020-06-04/
Thu, 04 Jun 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-06-04/I will speak about some geometric aspects of the representation theory of gentle algebras. Some results regarding the irreducible components of the affine schemes of modules over gentle algebras will be presented. In the case of gentle algebras arising from triangulations of unpunctured surfaces, a bijection between the set of laminations on the surface and the set of generically \(\tau\)-reduced irreducible components (formerly called “strongly reduced” by Geiss–Leclerc–Schröer) will be described.Simple-mindedness: negativity and positivity
https://www.fd-seminar.xyz/talks/2020-05-28/
Thu, 28 May 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-05-28/Given an acyclic quiver Q and an integer \(w\), the orbit category \[ C_w(Q) = \mathcal{D}^b(\mathbf{k} Q)/\Sigma^{-w} \mathbb{S} \] is a \(w\)-Calabi-Yau triangulated category, where \(\Sigma\) is the shift functor on \(\mathcal{D}^b(\mathbf{k} Q)\) and \(\mathbb{S}\) is the Serre functor. When \(w \geq 2\), these orbit categories are called (higher) cluster categories and the key generators are (higher) cluster-tilting objects; they are in bijection with silting objects in the fundamental domain. Amongst other connections, the combinatorics of (higher) cluster-tilting objects serve as a categorical model of those of (higher) noncrossing partitions.Calabi–Yau properties of Postnikov diagrams
https://www.fd-seminar.xyz/talks/2020-05-21/
Thu, 21 May 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-05-21/A Postnikov diagram is a collection of strands in the disk, satisfying combinatorial conditions on their crossings. The diagram determines many other mathematical objects, including a cluster algebra, which Galashin and Lam have recently shown to be isomorphic to the homogeneous coordinate ring of a certain subvariety of the Grassmannian, called a positroid variety. In this talk, I will explain how to categorify this cluster algebra, using a second (non-commutative) algebra attached to the Postnikov diagram.