Seminar
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.
For a regular email announcement please contact birep.
Future Talks
Friday, 10 January 2020
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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.
Friday, 17 January 2020
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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.
Monday, 27 January 2020
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16:15
Dirk Kussin (Berlin): tba
Friday, 31 January 2020
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Room V2-200
Martin Kalck (Freiburg): tba
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Room V2-200
Tashi Walde (Bonn): tba
Seminar Archive
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.
Thursday, 26 September 2019
Wednesday, 25 September 2019
Tuesday, 24 September 2019
Monday, 16 September 2019
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14:15, Room V5-227 (60 minutes)
Marc Stephan (Bonn/Augsburg): Interactions between elementary abelian p-group actions in topology and algebra
Abstract: I will provide a selective overview about rank conjectures for actions of elementary abelian p-groups. They estimate the size of the total dimension in homology over a field of characteristic p for free actions on finite CW complexes or finite chain complexes. Recently, Iyengar and Walker found algebraic examples with smaller homology than predicted, while joint work with Henrik Rüping shows that these counterexamples can not be realized topologically.
To establish bounds for the total dimension in homology, it is still interesting to consider the algebraic version and connect it to problems in commutative algebra. I will explain such a connection from joint work with Jeremiah Heller, and how it is related to constructions of vector bundles on projective space from modules of constant Jordan type due to Benson and Pevtsova.
Monday, 09 September 2019
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BiBo Seminar in Bielefeld
11:15, Room U2-200 (60 minutes)
Biao Ma (Bielefeld): Combinatorics of faithfully balanced modules
Abstract: In this talk, I will give a combinatorial characterization of faithfully balanced modules for the path algebra of the quiver An with linear orientation. By using this characterization one can deduce that the number of basic faithfully balanced modules is the nth 2-factorial number. Among them are n! modules with exactly n indecomposable summands which form a lattice (with respect to some appropriate partial order) – this extends the lattice of tilting modules. This is joint work with William Crawley-Boevey, Baptiste Rognerud and Julia Sauter.
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Bielefeld-Bochum-Seminar
13:30, Room U2-200 (60 minutes)
Alexander Pütz (Bochum): Degenerate Affine Flag Varieties and Quiver Grassmannians
Abstract: We study degenerate flag varieties where certain projections replace the identity maps in the inclusion relations for the chains of the spaces in the geometric interpretation of the flag variety. Quiver Grassmannians are projective varieties parametrising subrepresentations of a quiver representation.
We show that certain quiver Grassmannians for the equioriented cycle provide finite dimensional approximations of the degenerate affine flag variety of type GL_n. These quiver Grassmannians admit a finite cellular decomposition parametrised by affine Dellac configurations. Their irreducible components are normal, Cohen-Macaulay, have rational singularities and are parametrised by grand Motzkin paths. The Poincaré polynomials of the approximations admit a description based on the affine Dellac configurations. This research links the theory of quiver Grassmannians with the representation theory of affine Kac-Moody groups.
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Bielefeld-Bochum-Seminar
15:00, Room U2-200 (30 minutes)
Julia Sauter (Bielefeld): Relative geometry of representations - II
Abstract: We have a closer look at F-hereditary structures. Every algebra of representation dimension at most three admits an F-hereditary structure. All relative representation spaces are smooth if and only if the relative structure is F-hereditary. Furthermore, for relative quiver Grassmannians we can show that they are smooth if we have an F-hereditary structure and an F-rigid module.
Friday, 05 July 2019
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14:15, Room T2-208
Estanislao Herscovich (Grenoble): The cohomology of the Fomin-Kirillov algebra on 3 generators
Abstract: The aim of the talk is to present an elementary computation of the algebra structure of the Yoneda algebra of the Fomin-Kirillov algebra on 3 generators, based on a bootstrap technique built over the (non-acyclic) Koszul complex.
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15:30, Room T2-208
Grzegorz Bobiński (Torun): A characterization of representation infinite quiver settings
Abstract: We characterize pairs (Q,d) consisting of a quiver Q and a dimension vector d, such that over a given algebraically closed field k there are infinitely many representations of Q of dimension vector d. We also present an application of this result to the study of algebras with finitely many orbits with respect to the action of (the double product) of the group of units.
Friday, 07 June 2019
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14:15, Room T2-208
Philipp Lampe (Canterbury): The growth of real seeds and a determinant from group representation theory
Abstract: This talk provides a taster of an ingredient that we added to a proof in a joint work with Anna Felikson. In particular, we will look at a determinant assembled from the characters of a finite group. First, we give an overview of its long and colourful history going back to Catalan, Dedekind, Burnside and Frobenius. Second, we explain how the determinant helped to estimate growth rates of real seeds in cluster theory.
Friday, 24 May 2019
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14:15, Room T2-208
Janina Carmen Letz (Salt Lake City): Local to global principles for generation time over Noether algebras
Abstract: In the derived category of modules over a Noether algebra a complex G is said to generate a complex X if the latter can be obtained from the former by taking finitely many summands and cones. The number of cones needed in this process is the generation time of X. I will present some local to global type results for computing this invariant, and also discuss some applications.
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15:30, Room T2-208
Zhengfang Wang (Bonn): Tate-Hochschild cohomology and B-infinity algebra
Abstract: Tate-Hochschild cohomology was implicitly defined in Buchweitz’ unpublished manuscript in 1986, using his stable derived category. Analogous to Hochschild cohomology, it is interesting to ask whether there is a Gerstenhaber algebra structure on Tate-Hochschild cohomology.
In this talk, we will give an affirmative answer to the above question. For this, we first construct a natural complex to compute Tate-Hochschild cohomology. Then we show that there is a so-called B-infinity algebra structure on this complex by giving an explicit action of the little 2-discs operad on it. In particular, passing to cohomology, we get a Gerstenhaber algebra structure. If time permits, we will also talk about Keller’s very recent result and conjecture.
Friday, 17 May 2019
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Bielefeld-Münster Representation Theory Seminar
13:15, Room T2-208
Biao Ma (Bielefeld): Faithfully balanced modules and applications in relative homological algebra
Abstract: For a finite-dimensional algebra we revisit faithfully balanced modules and introduce the relative version of them. As applications, we establish the relative version of Brenner-Butler's tilting theorem and (higher) Auslander correspondence. Examples will be given to explain the main results. This is joint work with Julia Sauter.
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Bielefeld-Münster Representation Theory Seminar
14:30, Room T2-208
Haydee Lindo (Williamstown, MA): Endomorphism invariant modules and ring classifications
Abstract: I will speak on modules which are invariant under endomorphisms of their envelopes. This will include connections to the general theory of trace modules with some preliminary applications to ring classifications and conjectures involving modules with no self-extensions.
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Bielefeld-Münster Representation Theory Seminar
16:00, Room T2-208
Lutz Hille (Münster): Tilting Modules for the Auslander Algebra with a View to Derived Categories
Abstract: We consider the Auslander algebra of the truncated polynomial ring and classify exceptional modules and spherical modules. Using a recent result of Geuenich, we can describe all tilting modules as universal extensions of full exceptional sequences. Then we use spherical twists to construct also tilting complexes in the derived category, which have a very explicit description.
It is still open in general whether this is already the full classification, so we discuss the known results and the open problems.
This is joint work with David Ploog.
Saturday, 04 May 2019
Friday, 03 May 2019
Thursday, 02 May 2019
Friday, 12 April 2019
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13:30, Room T2-208
Markus Linckelmann (London): On Picard groups of finite group algebras
Abstract: The Picard group of self Morita equivalences of a finite-dimensional algebra over an algebraically closed field k is an algebraic group. By contrast, the Picard group of a finite group algebra over a p-adic ring with finite residue field is a finite group. The structure of the automorphism group of a finite group algebra over a p-local domain with an algebraically closed residue field is largely unknown; it seems to be unknown in general whether this group is finite. A recent result by F. Eisele shows that this group is also an algebraic group over the residue field. In joint work with R. Boltje and R. Kessar, we identify a `large' subgroup of the Picard group of a block algebra in terms of the fusion systems of the blocks and the Dade groups of its defect groups. This is partly motivated by the - to date open - question whether Morita equivalent block algebras have isomorphic defect groups and fusion systems. Another motivation comes from recent work of Eaton, Eisele, and Livesey, where the above results on Picard groups play a role in the proof of special cases of Donovan's finiteness conjecture.
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14:45, Room T2-208
Dave Benson (Aberdeen): Some exotic symmetric tensor categories in characteristic two
Abstract: This talk is about joint work with Pavel Etingof. A theorem of Deligne says that in characteristic zero, any symmetric tensor category "of moderate growth" admits a tensor functor to vector spaces or to super (i.e., Z/2-graded) vector spaces. In prime characteristic, this is not true, but one may ask whether there is a good list of "incompressible" symmetric tensor categories to which they they do all map. We construct an infinite ascending chain of finite symmetric tensor categories in characteristic two, all of which are incompressible. The constructions are based on the theory of tilting modules over the finite groups SL(2,2^n). Similar examples should exist in other prime characteristics, but the details have not yet been worked out.
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16:00, Room T2-208
Jon Carlson (Athens, Georgia): Lots of categories for the Green correspondence
Abstract: This is joint work with Lizhong Wang and Jiping Zhang.
The object is to establish a Green correspondence for categories of complexes of modules as well as their homotopy categories and derived categories. There is a categorical expression of the Green correspondence that is similar to a construction of Benson and Wheeler. At a key point in the constuction, we must assume that one of the categories has idempotent completion. This condition holds provided the category has countable direct summands. But under that assumption there are many categories that satisfy the hypothesis.
For information on earlier talks please check the complete seminar archive.
