Archive on FD Seminar
https://www.fd-seminar.xyz/talks/
Recent content in Archive on FD SeminarHugo -- gohugo.ioen-usThu, 01 Apr 2021 14:00:00 +0000TBA
https://www.fd-seminar.xyz/talks/2021-04-01/
Thu, 01 Apr 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-04-01/TBA
https://www.fd-seminar.xyz/talks/2021-03-25/
Thu, 25 Mar 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-03-25/TBA
https://www.fd-seminar.xyz/talks/2021-03-18/
Thu, 18 Mar 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-03-18/TBA
https://www.fd-seminar.xyz/talks/2021-03-11/
Thu, 11 Mar 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-03-11/TBA
https://www.fd-seminar.xyz/talks/2021-03-04/
Thu, 04 Mar 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-03-04/TBA
https://www.fd-seminar.xyz/talks/2021-02-25/
Thu, 25 Feb 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-02-25/TBA
https://www.fd-seminar.xyz/talks/2021-02-18/
Thu, 18 Feb 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-02-18/TBA
https://www.fd-seminar.xyz/talks/2021-02-11/
Thu, 11 Feb 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-02-11/Two vertex geometrically irreducible algebras - the final step
https://www.fd-seminar.xyz/talks/2021-02-04/
Thu, 04 Feb 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-02-04/It has been proved by Schröer and the author that are there are basically two possible classes of geometrically irreducible algebras with exactly two vertices in their Gabriel quiver. The first class is connected with the famous Birkhoff problem and one can interpret the points of the corresponding varieties as homomorphisms between modules over truncated polynomial algebras. It has been confirmed that the algebras appearing in this case are actually geometrically irreducible.Quasihereditary algebras with exact Borel subalgebras
https://www.fd-seminar.xyz/talks/2021-01-28/
Thu, 28 Jan 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-01-28/Exact Borel subalgebras of quasihereditary algebras emulate the role of classic Borel subalgebras of complex semisimple Lie algebras. Not every quasihereditary algebra $A$ has an exact Borel subalgebra. However, a theorem by Koenig, Külshammer and Ovsienko establishes that there always exists a quasihereditary algebra Morita equivalent to $A$ that has a (regular) exact Borel subalgebra. Despite that, an explicit characterisation of such “special” Morita representatives is not directly obtainable from Koenig, Külshammer and Ovsienko’s work.Stability and tilts on triangulated categories
https://www.fd-seminar.xyz/talks/2021-01-21/
Thu, 21 Jan 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-01-21/In representation theory, a theorem of Chindris' states that stability for modules is preserved under a tilting equivalence if weights are chosen in a suitable way. In algebraic geometry, a fundamental result on elliptic curves states that the Fourier-Mukai transform from a Poincaré line bundle preserves stability of coherent sheaves. In this talk, I will explain how these two results are connected, and how understanding this connection helps us prove new results on Bridgeland stability and polynomial stability on elliptic surfaces.Partial Serre duality and cocompact objects
https://www.fd-seminar.xyz/talks/2021-01-14/
Thu, 14 Jan 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-01-14/Let \(\mathsf{T}\) be a triangulated category with coproducts and let \(\mathsf{X}\) be a set of compact objects in \(\mathsf{T}\). It is known that \(\mathsf{X}\) generates a certain t-structure, and in particular describes explicitly a left adjoint to the inclusion of the coaisle. Unfortunately, it does not make much sense to consider the naive dual of this setup; cocompact objects rarely appear in categories which occur naturally. Motivated by this we introduced a weaker version of cocompactness called 0-cocompactness, and showed that such objects cogenerate t-structures.The category of local representations of a finite group
https://www.fd-seminar.xyz/talks/2021-01-07/
Thu, 07 Jan 2021 14:00:00 +0000https://www.fd-seminar.xyz/talks/2021-01-07/We consider modular representations of a finite group and focus for each prime ideal of the cohomology ring on the stable category of representations supported at that prime. This category is tensor triangulated, but compact and dualising objects do not coincide. For instance, the tensor unit is not compact. This is in contrast to the global category of representations and leads to an interesting completion of the category of compact objects.Torsion pairs and mutation
https://www.fd-seminar.xyz/talks/2020-12-17/
Thu, 17 Dec 2020 14:00:00 +0000https://www.fd-seminar.xyz/talks/2020-12-17/Every torsion pair in the category \(\operatorname{mod}A\) of finitely generated modules over a noetherian ring \(A\) corresponds bijectively to a torsion pair in the large module category \(\operatorname{Mod}A\) which is determined by a large (that is, not necessarily compact) two-term cosilting complex in the unbounded derived category \(D(\operatorname{Mod}A)\). Motivated by this observation, we investigate a notion of mutation for large cosilting objects in triangulated categories. In this context, mutation is not always possible: it is controlled by properties of certain torsion pairs in the heart of the t-structure induced by the cosilting object.Finite triangulated categories and DG-enhancements
https://www.fd-seminar.xyz/talks/2020-12-10/
Thu, 10 Dec 2020 14:00:00 +0000https://www.fd-seminar.xyz/talks/2020-12-10/We say that a triangulated category over a field is finite if it is idempotent complete, its \(\operatorname{hom}\)-sets are finite-dimensional, and there are finitely many indecomposables. Such categories arise in different contexts and have been widely studied by Xiao and Zhu (2005), Amiot (2007), Krause (2012), Keller (2018), Hanihara (2020), and others.
Most triangulated categories arise as the derived category of something, often called enhancement, such as a DG-algebra. That enhancement is essentially unique in several cases of interest, as established by Franke (1996), Schwede (2007), Lunts and Orlov (2010), Canonaco and Stellari (2018), Antieau (2018), etc.Exact structures and degeneration of Hall algebras
https://www.fd-seminar.xyz/talks/2020-12-03/
Thu, 03 Dec 2020 14:00:00 +0000https://www.fd-seminar.xyz/talks/2020-12-03/Hall algebras and various related structures play a prominent role in the modern representation theory. I will explain the interplay between different exact structures on an additive category and degenerations of the associated Hall algebras. For the categories of representations of Dynkin quivers, this recovers degenerations of the negative part of the corresponding quantum group. I will sketch the proofs of our results in the general case based on Auslander-Reiten theory.An algebraic interpretation of the higher Stasheff–Tamari orders
https://www.fd-seminar.xyz/talks/2020-11-26-2/
Thu, 26 Nov 2020 14:45:00 +0000https://www.fd-seminar.xyz/talks/2020-11-26-2/Read the full research snapshot hereRefined Coxeter Polynomials
https://www.fd-seminar.xyz/talks/2020-11-26-1/
Thu, 26 Nov 2020 14:00:00 +0000https://www.fd-seminar.xyz/talks/2020-11-26-1/Read the full research snapshot hereMonomorphism categories for generalised species
https://www.fd-seminar.xyz/talks/2020-11-19-2/
Thu, 19 Nov 2020 14:45:00 +0000https://www.fd-seminar.xyz/talks/2020-11-19-2/Read the full research snapshot hereMorita theorem for hereditary cluster categories
https://www.fd-seminar.xyz/talks/2020-11-19-1/
Thu, 19 Nov 2020 14:00:00 +0000https://www.fd-seminar.xyz/talks/2020-11-19-1/Read the full research snapshot hereRecollements and injective generation of the derived category
https://www.fd-seminar.xyz/talks/2020-11-12-2/
Thu, 12 Nov 2020 14:45:00 +0000https://www.fd-seminar.xyz/talks/2020-11-12-2/Read the full research snapshot hereDerived equivalences for skew-gentle algebras
https://www.fd-seminar.xyz/talks/2020-11-12-1/
Thu, 12 Nov 2020 14:00:00 +0000https://www.fd-seminar.xyz/talks/2020-11-12-1/Read the full research snapshot here.Linear quasi-categories as templicial modules
https://www.fd-seminar.xyz/talks/2020-11-05/
Thu, 05 Nov 2020 14:00:00 +0000https://www.fd-seminar.xyz/talks/2020-11-05/This is joint work with my supervisor Wendy Lowen. After laying out the basics of quasi-categories as defined by Joyal, we introduce a notion of linear quasi-categories over a unital commutative ring. We make use of certain colax monoidal functors, which we call templicial modules, as a variant of simplicial modules respecting the monoidal structure. It turns out that templicial modules with a Frobenius monoidal structure are equivalent to (homologically) non-negatively graded dg-categories.Finite-dimensional DG algebras and their properties
https://www.fd-seminar.xyz/talks/2020-10-29/
Thu, 29 Oct 2020 14:00:00 +0000https://www.fd-seminar.xyz/talks/2020-10-29/The talk will focus on finite-dimensional DG algebras and categories of perfect complexes over such DG algebras. These categories can be considered as proper derived noncommutative schemes. We are going to discuss basic properties of these noncommutative schemes and to establish a connection between such categories and DG categories with (semi)exceptional collections.Categorification of representation theory with an application to Soergel bimodules
https://www.fd-seminar.xyz/talks/2020-10-22/
Thu, 22 Oct 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-10-22/We explain how to categorify various basic results from the representation theory of finite-dimensional algebras, which are useful for classifying simple modules, to the 2-representation theory of fiat 2-categories. We then apply these in order to obtain a classification of simple 2-representations of the 2-category of Soergel bimodules.Preprojective algebras and fractional Calabi-Yau algebras
https://www.fd-seminar.xyz/talks/2020-10-15/
Thu, 15 Oct 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-10-15/Given a quiver we consider two algebras: its path algebra and its preprojective algebra. If the quiver is Dynkin (ADE) then both have nice properties: the path algebra is fractionally Calabi-Yau and the preprojective algebra has a Nakayama automorphism of finite order. I will explain what these words mean and how these properties are related, using 2-dimensional category theory. This gives a useful criterion to check if a d-representation finite algebra is fractionally Calabi-Yau.Homological mirror symmetry for invertible polynomials in two variables
https://www.fd-seminar.xyz/talks/2020-10-08/
Thu, 08 Oct 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-10-08/The starting point for homological mirror symmetry for invertible polynomials is an n x n invertible matrix with non-negative integer entries. To such a matrix, as well as to its transpose, one can associate polynomials. These polynomials are called invertible if they are weighted homogeneous, and both define isolated singularities at the origin. Homological mirror symmetry for invertible polynomials is a series of conjectures which posits the equivalence of the different flavours of Fukaya category associated to the Lefschetz fibration defined by one polynomial with various flavours of graded matrix factorisations defined by the transpose polynomial.tau-Tilting Finite Algebras With Non-Empty Left Or Right Parts Are Representation-Finite
https://www.fd-seminar.xyz/talks/2020-10-01/
Thu, 01 Oct 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-10-01/\(\tau\)-tilting theory was introduced by Adachi, Iyama and Reiten as a far-reaching generalization of classical tilting theory for finite dimensional associative algebras. One of the main classes of objects in the theory is that of \(\tau\)-rigid modules: a module \(M\) over an algebra \(\Lambda\) is \(\tau\)-rigid if \(\operatorname{Hom}_{\Lambda}(M,\tau M)=0\), where \(\tau M\) denotes the Auslander-Reiten translation of \(M\); such a module \(M\) is called \(\tau\)-tilting if the number \(|M|\) of non-isomorphic indecomposable summands of \(M\) equals the number of isomorphism classes of simple \(\Lambda\)-modules.Leavitt path algebras, B-infty-algebras and Keller’s conjecture for singular Hochschild cohomology
https://www.fd-seminar.xyz/talks/2020-09-24/
Thu, 24 Sep 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-09-24/I will first recall the relation between Leavitt path algebras and the singularity categories of radical-square-zero algebras. Using Leavitt path algebras, we confirm Keller’s conjecture for any radical-square-zero algebra: there is an isomorphism in the homotopy category of $B_\infty$-algebras between the Hochschild cochain complex of the dg singularity category and the singular Hochschild cochain complex of the algebra. Moreover, we prove that Keller’s conjecture is invariant under one-point (co)extensions and singular equivalences with levels.Grassmanian categories of infinite rank
https://www.fd-seminar.xyz/talks/2020-09-17/
Thu, 17 Sep 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-09-17/In this talk, I’ll describe our work towards providing an infinite rank version of the Grassmanian cluster categories introduced by Jensen, King and Su. We develop a new combinatorial tool for determining when two k-subsets of the integers are “non-crossing”, or equivalently when two Plücker coordinates of a Grassmanian cluster algebra of infinite rank are “compatible”. We use this tool to show that there is a structure preserving bijection between these Plücker coordinates and the generically free modules of rank 1 in our Grassmanian category of infinite rank, mirroring a result of Jensen, King and Su in the finite case.Hopf algebras of discrete representation type
https://www.fd-seminar.xyz/talks/2020-09-10/
Thu, 10 Sep 2020 14:00:00 +0000https://www.fd-seminar.xyz/talks/2020-09-10/Hopf algebra is an important topic in geometric representation theory. A basic algebra is of finite representation type if there are only finitely many non-isomorphic indecomposable representations. Basic Hopf algebras of finite representation type have been classified by Liu and Li in 2004. As algebras, they are just copies of Nakayama algebras. A pointed coalgebra is of discrete representation type, if there are only finitely many non-isomorphic indecomposable representations for each dimension vector.Relative Calabi-Yau completions and higher preprojective algebras
https://www.fd-seminar.xyz/talks/2020-09-03/
Thu, 03 Sep 2020 13:00:00 +0000https://www.fd-seminar.xyz/talks/2020-09-03/In this talk, we will give an introduction to (relative) Calabi-Yau structures following Brav-Dyckerhoff and (relative) Calabi-Yau completions following Yeung. We will then illustrate the relevance of these constructions to higher Auslander-Reiten theory on the example of higher preprojective algebras in the sense of Iyama-Oppermann. This is a report on part of Yilin Wu’s ongoing Ph. D. thesis.GKZ systems and perverse schobers
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/The Riemann-Hilbert correspondence yields an equivalence between the triangulated category of (regular holonomic) D-modules and that of constructible sheaves. Under this equivalence the abelian category of D-modules corresponds to the abelian category of perverse sheaves. In some contexts this abelian category has a concrete combinatorial description; in particular Kapranov and Schechtman showed that this is the case for perverse sheaves on real hyperplane arrangements. Such perverse sheaves arise from (some) GKZ systems of differential equations.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.