Seminar
No talks have been announced for this week.
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Seminar Archive
Friday, 01 July 2022
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13:15, Zoom
Haruhisa Enomoto (Osaka): From the lattice of torsion classes to wide and ICE-closed subcategories
Abstract: For a module category, we can consider the posets of various subcategories. In this talk, I explain that we can compute the posets of wide and ICE-closed (Image, Cokernel, Extension-closed) subcategories from the lattice of torsion classes in a purely combinatorial way. Also I give a simple expression of the poset of wide subcategories from the lattice of torsion classes, and give an application to the combinatorics of Reading's shard intersection order on Coxeter groups.
Friday, 24 June 2022
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13:15, Room T2-149
Vanessa Miemietz (Norwich): Uniqueness of bocses corresponding to a quasi-hereditary algebra
Abstract: Koenig-Kuelshammer-Ovsienko proved that an algebra is quasi-hereditary if and only if it is Morita equivalent to the right algebra of a normal directed bocs. I will review their construction and explain my joint work with Julian Kuelshammer, which proves that the basic bocs associated to a Morita equivalence class of quasi-hereditary algebras is unique up to isomorphism.
Friday, 17 June 2022
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13:15, Room T2-149
Georges Neaime (Bielefeld): Towards the linearity of complex braid groups
Abstract: Complex braid groups are a generalisation of Artin–Tits groups. They are attached to complex reflection groups, which are themselves a generalisation of finite Coxeter groups. It is an ongoing challenge to extend the theory of Artin–Tits groups to all complex braid groups. A part of this theory extension was established by Broué, Malle, and Rouquier in their seminal work. An important feature of spherical Artin groups is that they are linear groups, i.e., they admit a faithful linear representation of finite dimension. For the usual braid group, this property was shown to hold independently by Bigelow and Krammer. We seek to extend the theory of linearity to the context of complex braid groups, with a focus on the infinite families. Indeed, we will describe a definition of BMW and Brauer algebras, from which we can construct suitable linear representations and conjecture their faithfulness. We present a number of theorems and conjectures related to the structure of the aforementioned algebras, as well as properties of the relevant representations. We will finally propose a research programme for the sequel.
Friday, 20 May 2022
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13:15, Room T2-149
Estanislao Herscovich (Grenoble): Vertex algebras and 2-monoidal categories
Abstract: In an attempt to make the theory of vertex algebras more natural, R. Borcherds proposed a new foundation based on two tensor products. In this talk I will explain how these ideas sit well within the framework of 2-monoidal categories. More precisely, I will present a certain 2-monoidal categories of functors and show how vertex algebras can be regarded as commutative algebras with respect to one of the tensor products.
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14:30, Room T2-149
Thorsten Heidersdorf (Bonn): Categorical quotient constructions and representations of the general linear supergroup GL(m|n)
Abstract: Finite dimensional representations of complex algebraic supergroups are nowadays reasonably well understood as abelian categories (in terms of blocks, extensions etc) but their monoidal structure is largely unknown. I will describe some quotient constructions to approximate this monoidal structure in the case of GL(m|n).
Friday, 13 May 2022
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13:15, Room T2-149
Isaac Bird (Prague): Duality and definable categories in triangulated categories
Abstract: Over a ring, a submodule is pure if and only if its dual is a summand, while a module is pure-injective if and only if it is a direct summand of a dual module. I will discuss how these statements can be generalised to triangulated categories through the framework of duality triples, before turning attention to duality and definable categories. This is done through the use of duality pairs, which I will introduce. Several applications and examples will be given. This talk is based on joint work with Jordan Williamson.
Friday, 22 April 2022
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13:15, Room T2-149
Achim Krause (Münster): Integral endotrivial modules
Abstract: Endotrivial modules are invertible objects in stable module categories, i.e. modules "invertible modulo projectives". These have been studied intensively over p-groups with coefficients in a finite field, and more recently also for more general finite groups. In joint work with Jesper Grodal, we generalize stable module categories to arbitrary coefficients, and study the resulting notion of endotrivial modules over arbitrary rings, with special focus on integer coefficients. The relationship between the integer coefficient case and the finite field coefficient case leads to surprising character-theoretic integrality conditions.
Saturday, 09 April 2022
Friday, 04 February 2022
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13:15, Room T2-205
Jun Maillard (Saint-Etienne): A Cartan-Eilenberg formula for Mackey 2-functors
Abstract: The Cartan-Eilenberg stable elements formula expresses the (mod p) cohomology of a group G as a limit of the cohomology of its p-subgroup. This can be generalized by replacing the (mod p) cohomology functor by any global cohomological Mackey functor. A large class of 'categories indexed by finite groups', such as the categories of modules and their stable and derived versions, can be endowed with a structure of Mackey 2-functors, a 2-dimensional counterpart of Mackey functors. I present an analogous Cartan Eilenberg formula for these Mackey 2-functors, and how it can be reformulated as a structure of (higher) sheaf.
Friday, 28 January 2022
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13:15, Zoom
Charley Cummings (Bristol): The left-right symmetry of the finitistic dimension
Abstract: The finitistic dimension is a numerical invariant associated to a finite dimensional algebra that is a measure of the complexity of its representation theory. This dimension can be defined in terms of left or right modules. In general, the left and right finitistic dimensions of an algebra are not equal. However, it is unknown if the finiteness of these two dimensions is connected. In this talk, we consider the connection between the finiteness of the left and right finitistic dimensions of an algebra and relate this to the longstanding, open finitistic dimension conjecture.
Friday, 21 January 2022
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13:15, Room T2-205
Wassilij Gnedin (Bochum): Lifting derived equivalences and Serre duality for gentle Gorenstein algebras
Abstract: Recent work by Iyengar and Krause established Serre duality results for a notion of Gorenstein algebras which include symmetric orders as well as gentle algebras. My talk is concerned with the derived representation theory of these algebras.
In the first part, I will discuss whether the tilting property of a perfect complex over a Gorenstein algebra is local. This turns out to be true for symmetric orders in a broad sense that allows to consider other rings than localizations at prime ideals. After base change to a complete local ring R, the derived equivalence problem of a symmetric R-order can be reduced to a finite-dimensional setup.
The second part deals with an explicit description of the dualising bimodule of certain infinite-dimensional gentle algebras, which allows for a homological point of view on their string and band complexes. Combined with Auslander-Reiten theory of orders, this description yields an approach with modest use of combinatorics to recover results on the singularity category of a gentle Gorenstein algebra.
This talk is a report on joint work with S. Iyengar and H. Krause.
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14:30, Room T2-205
Didrik Fosse (Trondheim): A combinatorial rule for tilting mutation
Abstract: Tilting mutation is a way of producing new tilting complexes from old ones by replacing one indecomposable summand. In this talk, we give a purely combinatorial procedure for performing tilting mutation of suitable algebras. For the path algebra of a (sufficiently nice) quiver with relations, this procedure allows us to perform tilting mutation of the algebra by only modifying the quiver with relations.
As an application, we show how the mutation rules can be used to show that two algebras are derived equivalent.
Friday, 14 January 2022
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13:15, Zoom
Shiquan Ruan (Xiamen): Nakayama algebras and Fuchsian singularities
Abstract: In this talk, we will classify all the Nakayama algebras having Fuchsian type, that is, derived equivalent to the extended canonical algebras, or equivalently, derived equivalent to a kind of stable category of vector bundles over weighted projective lines. This is achieved by constructing certain tilting complexes in the bounded derived category of coherent sheaves and also in the stable category of vector bundles for weighted projective lines, and the strategy of one-point extension and the perpendicular approach will be used. As a byproduct, we reprove the classification result of Nakayama algebras of piecewise hereditary type due to Happel—Seidel.
This is joint work with Helmut Lenzing and Hagen Meltzer.
Wednesday, 08 December 2021
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10:15, Room V4-112
Hernán Giraldo (Medellin): Shapes of Auslander-Reiten triangles in the stable category of modules over repetitive algebras
Abstract: Let k be an algebraically closed field, let A be a finite dimensional k-algebra, and let  be the repetitive algebra of A. For the stable category stmod(Â) of finitely generated left Â-modules, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in stmod(Â). We use the fact (and prove) that every Auslander-Reiten triangle in stmod(Â) is induced from an Auslander-Reiten sequence of finitely generated left Â-modules. Finally, we will talk about applications of this last result.
Friday, 03 December 2021
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13:15, Room T2-149
Tiago Cruz (Stuttgart): Relative dominant dimension and quality of split quasi-hereditary covers
Abstract: The dominant dimension of a finite-dimensional algebra A is a homological invariant measuring the connection between module categories A-mod and B-mod, where B is the endomorphism algebra of a faithful projective-injective A-module.
In this talk, we will discuss generalisations of dominant dimension and how they can be used to measure the quality of split quasi-hereditary covers. The Schur algebra S(d, d) together with its faithful projective-injective module is a classic example of a split quasi-hereditary cover of the group algebra of the symmetric group on d letters.
Friday, 26 November 2021
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13:15, Room T2-149
Tobias Barthel (Bonn): Stratifying integral representations of finite groups
Abstract: Classifying all integral representations of finite groups up to isomorphism is essentially impossible. In this talk, we will introduce an integral version of the stable module category for a finite group G and then explain how to use it to give a `generic' classification of integral G-representations. Our results globalize the modular case established by Benson, Iyengar, and Krause and relies on the notion of stratification in tensor triangular geometry developed in joint work with Heard and Sanders. Time permitting, I will discuss some further directions.
Friday, 19 November 2021
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13:15, Room T2-149
Adam-Christiaan van Roosmalen (Hasselt): Auslander's formula and correspondence for exact categories
Abstract: Following Auslander's philosophy, one can study a small abelian category A by studying the category of finitely presented functors on A. Auslander's formula provides a way to move back: one can recover A as the quotient of the category of finitely presented functors on A by the Serre subcategory of all effaceable functors.
One impediment to formulating a version of Auslander's formula for an exact category E is that the category of finitely presented functors on E is independent of the chosen exact structure: it only depends on the underlying additive category. To address this, we introduce the category of admissibly presented functors on an exact category. In this talk, I will focus on this category of admissibly presented functors, and use it to formulate a version of Auslander's formula and correspondence for exact categories.
This talk is based on joint work with Ruben Henrard and Sondre Kvamme.
Saturday, 13 November 2021
Friday, 12 November 2021
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13:15, Room T2-149
Scott Balchin (Bonn): The smashing spectrum of a tt category
Abstract: In joint work with Greg Stevenson, we prove that the frame of smashing tensor ideals of a big tt-category is always spatial. As such, by Stone duality, we are afforded a space: the smashing spectrum. In this talk, I will report on the construction of this new invariant via lattice theoretic techniques, and its relation to the Balmer spectrum. In particular, we will see that there is a surjective comparison map which detects the failure of the telescope conjecture.
Friday, 29 October 2021
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13:15, Room T2-149
Håvard Terland (Trondheim): Identifying components of mutation quivers
Abstract: Tau-tilting theory completes tilting theory from the perspective of mutation. Letting points be support-tau tilting pairs and arrows indicate (left) mutation, one then obtains a so-called mutation quiver whose underlying graph is regular.
The goal of this talk will be to introduce tau-tilting theory and discuss recent efforts to better understand the connected components of (the underlying graphs of) mutation quivers of support tau-tilting pairs.
Friday, 22 October 2021
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13:15, Zoom
Kaveh Mousavand (Kingston): Orbits of bricks of finite dimensional algebras
Abstract: In 2014, Adachi-Iyama-Reiten introduced the tau-tilting theory of finite dimensional algebras to fix the deficiency of mutation of tilting modules. The subject soon received a lot of attention and developed in various directions. Around the same time, Chindris-Kinser-Weyman studied the moduli spaces of quiver representations, in particular the behaviour of Schur representations (bricks) of finite dimensional algebras. In this talk, we try to relate these two lines of research. More specifically, motivated by a conjectural geometric counterpart for the algebraic notion of tau-tilting finiteness, we treat module varieties of finite dimensional algebras, in particular the orbit of bricks. We show that, unlike arbitrary bricks, those used for the labelling of the lattice of functorially finite torsion classes always admit open orbits. From this, we obtain a conceptual proof of the counterpart of the first Brauer-Thrall conjecture for bricks. We push these results further in the treatment of another Brauer-Thrall type conjecture which is still open in full generality.
This is based on my Joint work with Charles Paquette (Royal Military College, Canada).
For information on earlier talks please check the complete seminar archive.
