Seminar
Friday, 17 July 2020
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14:15, Zoom (Please contact birep for login information.)
Mads Hustad Sandøy (Trondheim): Higher Koszul duality and connections with n-hereditary algebras
Abstract: Introduced by Iyama and others, n-hereditary algebras are an attempt to generalize the good properties of hereditary algebras to algebras of higher global dimension, and they come in two flavours: n-representation finite and n-representation infinite. For an n-representation infinite algebra satisfying some assumptions, there exists an equivalence between the stable graded category of its trivial extension and the bounded derived category of a category associated to its (n+1)-th preprojective algebra. This equivalence is reminiscent of the BGG-correspondence, which is itself known to descend from Koszul duality.
In this talk, based on joint work with Johanne Haugland, we describe how a higher version of the generalized Koszul duality introduced by Madsen and Green, Reiten and Solberg can be used to explain the existence of the aforementioned equivalence. Moreover, we describe results showing more general connections between n-hereditary algebras and certain kinds of Frobenius algebras satisfying higher Koszul duality.
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Future Talks
Friday, 30 October 2020
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Ulrich Thiel (Kaiserslautern): tba
Seminar Archive
Friday, 19 June 2020
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14:15, Zoom
Srikanth Iyengar (Salt Lake City): The Nakayama functor and its completion for Gorenstein algebras
Abstract: Buchweitz proved a duality theorem for the singularity category of certain noetherian algebras that are Iwanaga-Gorenstein and have isolated singularities. This result, which appears in his unpublished manuscript on maximal Cohen-Macaulay modules, unifies and extends two duality theorems of Auslander, one for artin algebras and one for local commutative Gorenstein rings. The starting point of this project, which is in collaboration with Henning Krause, was to explore what impact the Gorenstein property has on the singularity category of a ring that does not necessarily have isolated singularities. Grothendieck’s duality theorem for commutative algebra points to a path for such an exploration. This path has lead us to a notion of Gorenstein algebras, broader than the one considered by Buchweitz, and to the Nakayama functor and its lift to homotopy categories associated to the algebra. The plan for my talk is to describe some of what we found.
Friday, 12 June 2020
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14:15, Zoom
Zhengfang Wang (Stuttgart): On Keller’s conjecture for singular Hochschild cohomology
Abstract: Very recently, Keller shows that for a noetherian algebra A whose bounded dg derived category is smooth, the singular Hochschild cohomology (= Tate-Hochschild cohomology) is isomorphic, as a graded algebra, to the Hochschild cohomology of the dg singularity category of A. He conjectures that the above isomorphism lifts to an isomorphism in the homotopy category of B-infinity algebras at the complex level. In this talk, we will show a weakened version of this conjecture for algebras with radical square zero via Krause’s stable derived category. We will also show that Keller’s conjecture is invariant under one-point (co)extensions and certain singular equivalences. This is an ongoing joint work with Xiaowu Chen and Huanhuan Li.
Friday, 05 June 2020
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14:15, Zoom
Nicolas Berkouk (Paris): Derived methods towards stable invariants for persistence
Abstract: Multi-parameter persistence modules can be thought of as graded-modules over a certain polynomial ring. Nothing new under the sun one can be tempted to say. However, topological data analysis has shed a new light on these objects by introducing a notion of distance between them (the interleaving distance), which is crucial for applications. One important direction of research in this field is to seek for computable invariants of multi-parameter persistence modules which are stable with respect to this metric. Unfortunately, classical invariants from combinatorial algebraic geometry do not satisfy any form of stability in the general case. In this talk, we will illustrate how to equip the homotopy category of persistence modules with an interleaving distance, which allows to perform homological operations (such as taking minimal free resolutions) in a « stable » way. If times allow, I will also explain how to relate persistence modules and the interleaving distance with sheaves on a real vector space and the convolution distance (which was introduced by Kashiwara-Schapira). This last part is joint with François Petit.
Friday, 22 May 2020
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14:15, Zoom
Vincent Gelinas (Dublin): The depth, the delooping level and the finitistic dimension
Abstract: Our knowledge of the finitistic dimension varies drastically between the settings of Artin algebras and of commutative local Noetherian rings. In the commutative case, the Auslander-Buchsbaum formula shows it equals the depth of the ring, while no similar such result holds in the noncommutative case. In this talk, we will introduce a new invariant called the “delooping level” for Noetherian semiperfect rings, which is implicit in some proofs of the Auslander-Buchsbaum formula. We show that it recovers the depth in the commutative local case, and that it provides an upper bound for the finitistic dimension of Artin algebras, with agreement under certain cohomological conditions. This is based on the preprint arxiv:2004.04828.
Friday, 15 May 2020
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14:15, Zoom
Sira Gratz (Glasgow): SL(k)-friezes
Abstract: Classical frieze patterns are combinatorial structures which relate back to Gauss' Pentagramma Mirificum, and have been extensively studied by Conway and Coxeter in the 1970's.
A classical frieze pattern is an array of numbers satisfying a local (2 x 2)-determinant rule. Conway and Coxeter gave a beautiful and natural classification of SL(2)-friezes via triangulations of polygons. This same combinatorics occurs in the study of cluster algebras, and has revived interest in the subject. From this point of view, a natural way to generalise the notion of a classical frieze pattern is to ask of such an array to satisfy a (k x k)-determinant rule instead, for k bigger than 2, leading to the notion of higher SL(k)-friezes. While the task of classifying classical friezes yields a very satisfying answer, higher SL(k)-friezes are not that well understood to date.
In this talk, we'll discuss how one might start to fathom higher SL(k)-frieze patterns. The links between SL(2)-friezes and triangulations of polygons suggests a link to Grassmannian varieties under the Plücker embedding. We find a way to exploit this relation for higher SL(k)-friezes, and provide an easy way to generate a number of SL(k)-friezes via Grassmannian combinatorics, and suggest some ideas towards a complete classification using the theory of cluster algebras.
This talk is based on joint work with Baur, Faber, Serhiyenko and Todorov.
Friday, 08 May 2020
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14:15, Zoom
Manuel Flores Galicia (Bielefeld): Quasi-hereditary structures on path algebras of types A, D and E
Abstract: A quasi-hereditary algebra is an Artin algebra A together with a partial order on its set of isoclasses of simple modules satisfying certain conditions. A quasi-hereditary structure is an equivalence class of partial orders for an appropriate equivalence relation, all together giving rise to what we call the poset of quasi-hereditary structures on A. In this talk we will provide a full description of the poset of quasi-hereditary structures on a path algebra of Dynkin type A. For types D and E, we give a counting method for the number of quasi-hereditary structures. This talk is based on a joint paper with Yuta Kimura and Baptiste Rognerud (arXiv:2004.04726v2).
Friday, 31 January 2020
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13:15, Room V2-200
Sophiane Yahiatene (Bielefeld): Hurwitz action in extended Weyl groups with application to hereditary abelian categories
Abstract: The main object of this talk is the family of extended Weyl groups. These are reflection groups attached to categories of coherent sheaves over a weighted projective line. After stating important properties of these groups we focus on the set of minimal reflection factorizations of a class of distinguished elements, called Coxeter transformations. We prove that in almost all cases the so-called Hurwitz action is transitive on it. Afterwards, if time permitting we point out the connection to thick subcategories generated by exceptional sequences.
(Joint work with B. Baumeister and P. Wegener)
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14:30, Room V2-200
Martin Kalck (Freiburg): Relative singularity categories: dg-models and applications
Abstract: We'll try to explain how relative singularity categories can be used to gain insight into the structure of singularity categories. A key technique is a construction describing the relative singularity category as the perfect derived category of an "explicit" dg algebra.
We will explain this technique and show how it can be applied to unify and extend some results on singularity categories.
This talk is based on joint work with Dong Yang.
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16:00, Room V2-200
Tashi Walde (Bonn): Homotopy coherent theorems of Dold—Kan type
Abstract: The classical Dold—Kan correspondence is an equivalence of categories between connective chain complexes and simplicial objects in any abelian category. It is often implicit in key homological constructions such as the bar resolution and in fact forms the fundamental link between homotopical and homological algebra. In the last decades many variants of the Dold—Kan correspondence have been established and various axiomatic frameworks have been proposed to tie these equivalences together. Both in algebra and topology, situations arise where it is useful to work homotopy coherently (aka “derived"), i.e. study diagrams which don't just commute on the nose, but only up to specified (possibly higher) homotopies; the main examples being derived categories, stable module categories or stable homotopy theory.
In this talk we explain the key feature of the aforementioned "theorems of Dold—Kan type” and how they can be generalized to the homotopy coherent context by employing the theory of infinity-categories.
Monday, 27 January 2020
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16:15, Room T2-227
Dirk Kussin (Berlin): Cotilting sheaves and indecomposable pure-injective sheaves over noncommutative curves
Abstract: A quasicoherent sheaf is - by definition - of slope infinity, if it does not map to vector bundles. We classify all cotilting sheaves of slope infinity and all indecomposable pure-injective sheaves of slope infinity over any weighted noncommutative regular projective curve. (This includes all smooth projective curves, all weighted projective curves, etc.) This leads in the special cases where the curve is domestic, tubular or elliptic to a complete description of all non-coherent cotilting sheaves, and to a classification of the indecomposable pure-injective sheaves. In the tubular/elliptic cases we also address some still open problems for the irrational slopes. This is joint work with Rosanna Laking.
Friday, 17 January 2020
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14:15, Room V2-200
Daniel Labardini-Fragoso (Mexico City): Algebraic and combinatorial decompositions of Fuchsian groups
Abstract: The discrete subgroups of the real projective special linear group of degree two are often called 'Fuchsian groups'. For a Fuchsian group G whose action on the hyperbolic plane H is free, the orbit space H/G has a canonical structure of Riemann surface with a hyperbolic metric, whereas if the action of G is not free, then H/G has a structure of 'orbifold'. In the former case, there is a direct and very clear relation between G and the fundamental group of H/G: a theorem of the theory of covering spaces states that they are isomorphic. When the action of G is not free, the relation between G and the fundamental group of H/G is more subtle. A 1968 theorem of Armstrong states that the fundamental goup is the quotient of G modulo the subgroup E generated by elliptic elements. For G finitely generated, non-elementary and with at least one parabolic element, I will present full algebraic and combinatorial decompositions of G in terms of the fundamental group of H/G and a specific finitely generated subgroup of E, thus improving Armstrong's theorem.
This talk is based on an ongoing joint project with Sibylle Schroll and Yadira Valdivieso-Díaz that aims at describing the bounded derived categories of skew-gentle algebras in terms of curves on surfaces with orbifold points of order 2.
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15:30, Room V2-200
Igor Burban (Paderborn): Homological mirror symmetry for compact surfaces with boundary, tame non-commutative nodal curves and spherical objects on cycles of projective lines
Abstract: In my talk, I am going to explain, how tame non-commutative nodal curves appear as "holomorphic mirrors" of certain graded compact oriented surfaces with marked boundary. This part of the talk is based on a work of Yanki Lekili and Alexander Polishchuk, as well as on my joint works with Yuriy Drozd.
As a nice application of the homological mirror symmetry, I am going to explain the classification of spherical objects on a cycle of projective lines. This part of my talk is based on a part of the PhD thesis of Sebastian Opper.
Friday, 10 January 2020
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13:15, Room V2-200
Xiao-Wu Chen (Hefei): Leavitt path algebras, B-infinity algebras and Keller's conjecture
Abstract: Recently, Keller proves that the Tate-Hochschild cohomology algebra is isomorphic to the Hochschild cohomology algebra of the dg singularity category. He conjectures that the isomorphism lifts a B-infty-isomorphism on the cochain level. We verify his conjecture for an algebra with radical square zero, using the corresponding Leavitt path algebra. One ingredient of the proof is to enhance Krause's description of the singularity category to the dg level. This is joint with Zhengfang Wang and Huanhuan Li.
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Bielefeld-Paderborn-Seminar
14:30, Room V2-200
Henning Krause (Bielefeld): Local versus global for representations of algebras
Abstract: We consider some classes of finite dimensional algebras and discuss the classification of thick subcategories for their module categories. Typical examples are path algebras of quivers or group algebras of finite groups. This leads naturally to the study of derived categories. When a cohomology ring is acting, we may pass from global to local and obtain in some interesting cases a stratification of the module category.
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Bielefeld-Paderborn-Seminar
16:30, Room V2-200
Kai-Uwe Schmidt (Paderborn): Highly nonlinear functions
Abstract: The nonlinearity of a Boolean function in n variables is its Hamming distance to the set of all affine Boolean functions in n variables. Boolean functions with large nonlinearity are difficult to approximate by affine Boolean functions, which is of significant interest in cryptography. The largest possible nonlinearity of a Boolean function in n variables also equals the covering radius of the [2^n,n+1] Reed-Muller code, whose determination is subject to a famous conjecture from the 1980s.
In this talk, I will survey the history of this conjecture and then explain how the conjecture can be proved using a mixture of number-theoretic and probabilistic arguments. I will also discuss generalisations of this conjecture.
Friday, 13 December 2019
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13:15, Room V2-200
Maitreyee Kulkarni (Bonn): Infinite friezes and triangulations of an annulus
Abstract: In this talk I will introduce a combinatorial object called a frieze and describe its relations to triangulations and to representations of certain quivers. In particular, we will see that each periodic infinite frieze determines a triangulation of an annulus in a unique way. We will also study associated module categories and determine an invariant of friezes in terms of modules. This is joint work with Karin Baur, Ilke Canakci, Karin Jacobsen, and Gordana Todorov.
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14:30, Room V2-200
Baptiste Rognerud (Paris): Equivalences between blocks of cohomological Mackey algebras
Abstract: Mackey functors were introduced as a convenient tool for handling the induction theory of several objects having a similar behavior (group representations; representation rings, group cohomology, etc;). Later, it was proved by Thévenaz and Webb that the category of Mackey functors is equivalent to the category of modules over a finite dimensional algebra called the Mackey algebra. The proof is far from being difficult, but this result is of crucial importance : one can study Mackey functors using the ring and module theory. It turns out that the Mackey algebra is, in many aspects, similar to the group algebra.
In this talk, I will explain how the problems of constructing (derived) equivalences between categories representations of finite groups and between the corresponding categories of cohomological Mackey functors are related. We will see that the easy situation of Morita equivalences between blocks of group algebras may be much more interesting in the world of Mackey functors. This is part of a joint work with Markus Linckelmann.
Friday, 29 November 2019
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14:15, Room V2-200
Pedro Fernández (Bogotá): Matrix problems associated to some Brauer configuration algebras
Abstract: Bijections between solutions of the Kronecker problem and the four subspace problem with indecomposable projective modules over some Brauer configuration algebras are obtained by interpreting elements of some integer sequences as polygons of suitable Brauer configurations. This kind of configurations are also used to categorify (in the sense of Ringel and Fahr) some integer sequences.
This is joint work with Agustín Moreno Cañadas.
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15:30, Room V2-200
Helmut Lenzing (Paderborn): Algebraic theory of fuchsian singularities
Abstract: Fuchsian singularities are graded isolated singularities of Krull dimension two. Classically, they arise as rings of G-invariant differential forms (automorphic forms) with respect to a fuchsian group, a discrete cocompact subgroup G of the automophism group of the hyperbolic plane of complex numbers with positive imaginary part.
My talk has the following aims: (1) Generalize the concept of fuchsian singularities to algebraically closed fields of arbitrary characteristic. (2) Relate them to mathematical objects of a different nature. (3) Provide a purely ring theoretic characterization of fuchsian singularities.
We further determine their singularity categories together with relevant Grothendieck group related data.
Friday, 22 November 2019
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13:15, Room V2-200
Jenny August (Bonn): The Stability Manifold of a Contraction Algebra
Abstract: For a finite dimensional algebra, Bridgeland stability conditions can be viewed as a continuous generalisation of tilting theory, providing a geometric way to study the derived category. Describing this stability manifold is often very challenging but in this talk, I’ll look at a special class of symmetric algebras whose tilting theory is determined by a related hyperplane arrangement. This simple picture will then allow us to describe the stability manifold of such an algebra.
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14:30, Room V2-200
Magnus Hellstrøm-Finnsen (Trondheim): The spectrum for an additive and an exact monoidal category
Abstract: We will define and investigate some basic properties of the spectrum for an additive and a Quillen exact monoidal category. Further we will define a notion of support data on these categories and classify radical ideals with supported by primes.
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16:00, Room V2-200
Gustavo Jasso (Bonn): The symplectic geometry of higher Auslander algebras
Abstract: It follows from work of Bocklandt, Haiden-Katzarkov-Kontsevich and Lekili-Polishchuk that (appropriate versions of) the Fukaya categories associated to marked Riemann surfaces are equivalent to (appropriate versions of) the derived categories of graded locally gentle algebras.
In this talk I will explain the first steps in a higher-dimensional generalisation of the above. Natural higher-dimensional symplectic manifolds associated to Riemann surfaces are their symmetric products. In the simplest case of a marked disk, I will detail a description of the (partially wrapped) Fukaya categories of its symmetric products in terms of the derived categories of Iyama's higher-dimensional Auslander algebras of type A. Intrincate combinatorics (observed first by Auroux and Lipshitz-Ozsváth-Thurston) related to the Bruhat order of the symmetric group arise already in these simplest higher-dimensional examples.
This is a report on joint work with Tobias Dyckerhoff and Yanki Lekili.
Friday, 15 November 2019
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Bielefeld-Paderborn Seminar
14:15, Room V2-200
William Crawley-Boevey (Bielefeld): Clannish algebras revisited
Abstract: We are concerned with classifying the finitely generated indecomposable modules for a finite-dimensional associative algebra, or more generally a ring, or some related situation, such as objects in a derived category. There are a number of situations where classifications have been obtained in terms of so-called strings and bands. This includes string algebras, clannish algebras (introduced by the speaker in 1989), Dedekind-like rings and nodal algebras. I shall review some of this work, with examples from geometry, topology and arithmetic. In addition, I aim to describe some improvements to my earlier work on clannish algebras.
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Bielefeld-Paderborn Seminar
15:30, Room V2-200
Fabian Januszewski (Paderborn): A cohomological approach to characters of Lie groups
Friday, 25 October 2019
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14:15, Room V2-200
Fei Xie (Bielefeld): The derived category of a singular quintic del Pezzo surface
Abstract: I will give a semiorthogonal decomposition for the bounded derived category of coherent sheaves on a quintic del Pezzo surface with mild singularity (rational Gorenstein) over algebraically closed fields. The decomposition has three components. Two components are equivalent to derived categories of the base field. The remaining component is equivalent to the derived category of products of truncated polynomials with total length 5. The decomposition is obtained by studying the semiorthogonal decomposition of the minimal resolution of the surface. I will also briefly mention how to obtain a similar decomposition using Homological Projective Duality and how to obtain a decomposition over non-algebraically closed fields.
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15:30, Room V2-200
Paul Wedrich (Bonn): Quivers for SL(2) tilting modules
Abstract: I will explain how diagrammatic algebra can be used to give an explicit generators-and-relations presentation of all morphisms between indecomposable tilting modules for SL(2) over an algebraically closed field of positive characteristic. The result takes the form of a path algebra of an infinite, fractal-like quiver with relations, which can be considered as the (semi-infinite) Ringel dual of SL(2). Joint work with Daniel Tubbenhauer.
Friday, 18 October 2019
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14:15, Room V2-200
Alexander Slávik (Prague): On flat generators and Matlis duality for quasicoherent sheaves
Abstract: We show that for a quasicompact quasiseparated scheme X, the following assertions are equivalent: (1) the category QCoh(X) of all quasicoherent sheaves on X has a flat generator; (2) for every injective object E of QCoh(X), the internal hom functor into E is exact; (3) the scheme X is semiseparated. Joint work with Jan Šťovíček.
For information on earlier talks please check the complete seminar archive.
