Seminar
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Seminar Archive
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Abstract: Tate-Hochschild cohomology was implicitly defined in Buchweitz’ unpublished manuscript in 1986, using his stable derived category.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.
Wednesday, 06 February 2019
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14:15, Room U2-205
Teresa Conde (Stuttgart): Gabriel-Roiter measure and finiteness of the representation dimension
Abstract: The induction scheme used in Roiter's proof of the first Brauer-Thrall conjecture prompted Gabriel to introduce an invariant, known as the Gabriel–Roiter measure. The usefulness of the Gabriel-Roiter measure is not limited to the first Brauer-Thrall conjecture: Ringel has used it to give new proofs of results established by Auslander in the 70's. In this talk, we use the Gabriel-Roiter measure to provide a new proof of the finiteness of the representation dimension for Artin algebras, a result originally proved by Iyama in 2002.
Friday, 01 February 2019
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14:15, Room U2-205
Severin Barmeier (Bonn): Diagrams of algebras, categories of coherent sheaves and deformations
Abstract: Given a complex algebraic variety X, the restriction of its structure sheaf to a finite cover of affine open sets can be viewed as a diagram of (commutative) algebras. Deformations of a diagram obtained in this way correspond precisely to deformations of the category of
(quasi)coherent sheaves as an Abelian category (after W. Lowen and M. Van den Bergh).
We describe the higher deformation theory explicitly via L-infinity algebras for X covered by two affine opens and explain the connection to
"classical" deformations of the complex structure and deformation quantizations by means of examples. This is joint work with Y. Frégier.
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15:30, Room U2-205
Wassilij Gnedin (Bochum): A homological characterization of ribbon graph orders
Abstract: Recently, finite-dimensional algebras which can be related to surfaces have attracted a lot of interest.
My talk is concerned with certain 'non-commutative curve singularities' arising from ribbon graphs on closed surfaces.
These so-called ribbon graph orders can be viewed as infinite-dimensional versions of Brauer graph algebras as well as gentle algebras.
Although defined by particular combinatorial conditions, it turns out that ribbon graph orders possess a unique blend of purely homological features (such as a Calabi-Yau property of the perfect derived category and semisimplicity of the singularity category).
Using their homological characterization I will show that ribbon graph orders as well as certain finite-dimensional special biserial algebras are preserved under derived equivalences.
Friday, 25 January 2019
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14:15, Room U2-205
Sebastian Opper (Paderborn): On auto-equivalences and derived invariants of gentle algebras
Abstract: This talk will be about derived equivalences of gentle algebras and the group of auto-equivalences of their derived categories. In joint work with Plamondon and Schroll, we attached a surface to every gentle algebra and showed that its geometry encodes of the triangulated structure of the derived category. I will explain how a derived equivalence of gentle algebras gives rise to a homeomorphism between their associated surfaces and how this leads to a complete derived invariant of gentle algebras which generalizes the combinatorial invariant of Avella-Alaminos and Geiss. Finally, I will talk about applications to groups of auto-equivalences of gentle algebras and their connection to mapping class groups.
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15:30, Room U2-205
Frederik Marks (Stuttgart): Flat ring epimorphisms and localisations of commutative noetherian rings
Abstract: We study different types of localisations of a commutative noetherian ring. In particular, we are interested in the following questions: When is a flat ring epimorphism a universal localisation in the sense of Schofield? And when is such a universal localisation a classical ring of fractions? We approach these questions using the theory of support and local cohomology, and by analysing the specialisation closed subset of the spectrum associated with a flat ring epimorphism. As for the first question, we show that all flat ring epimorphisms are universal localisations when the underlying ring is either locally factorial or of Krull dimension one. If time permits, we will also comment on the situation for more general rings, which turns out to be significantly more complicated and diverse. Finally, we show that an answer to the question of when universal localisations are classical depends on the structure of the Picard group of the underlying ring. This talk is based on joint work with Lidia Angeleri Hügel, Jan Stovicek, Ryo Takahashi and Jorge Vitória.
Friday, 18 January 2019
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14:15, Room U2-205
Catharina Stroppel (Bonn): Semiinfinite highest weight categories
Abstract: We briefly recall the classical highest weight categories theory (following Cline-Parshall-Scott, Donkin and Ringel) for finite dimensional algebras in a language which allows generalizations to stratified algebras and infinite situations. In particular we formulate aspects of tilting theory and Ringel duality in a semi-infinite setting. If time allows we will mention some explicit examples for this construction related to diagram algebras and categorifications.
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BGTS Colloquium
16:15, Lecture Hall H5
Sabine Jansen (München): Condensation, big jump and heavy tails: from phase transitions to probability
Abstract: Ice melts, water evaporates - these are everyday experiences of phase transitions. The explanation of this macroscopic phenomenon from microscopic laws belongs to the realm of statistical physics, which treats matter as a composite system made up of many individual "agents" with random behavior. From a mathematician's point of view, a fully rigorous understanding still eludes us. The search for it leads to questions in probability that open up surprising connections: toy models for surface tension of liquid droplets build on heavy-tailed variables used in insurance mathematics; a big jump made by a random walker is a condensation phenomenon in disguise. The talk explains some of these connections and presents open problems and partial answers.
Friday, 11 January 2019
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14:15, Room U2-205
William Crawley-Boevey (Bielefeld): Decomposition of persistence modules
Abstract: I shall discuss the decomposition of pointwise finite-dimensional persistence modules. A persistence module indexed by the real plane is said to be middle exact if for each rectangle in the plane, the associated 3-term exact sequence of vector spaces is exact in the middle. I shall outline a new proof of a theorem of Cochoy and Oudot classifying such middle exact modules. They arise in the study of interlevel set persistence homology, answering a question of Botnan and Lesnick. This is joint work with Magnus Botnan.
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15:45, Room U2-205
Jörg Schürmann (Münster): Degenerate affine Hecke algebras and Chern classes of Schubert cells
Abstract: We explain in the context of complete flag varieties X=G/B the relation between Chern classes of Schubert cells and convolution actions of degenerate affine Hecke-algebras as in the work of Ginzburg. This is based on the Lagrangian approach via characteristic cycles. As an application we show that the two cohomological Weyl group actions constructed by Ginzburg and Aluffi-Mihalcea coincide. These Weyl group actions permute the (equivariant) Chern classes of the corresponding Schubert cells. This is joint work with P. Aluffi, L. Mihalcea and C. Su.
Friday, 23 November 2018
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BiBo Seminar in Bochum
14:00, Room IA 1-71 (60 minutes)
Arif Dönmez (Bochum): Moduli of representations of one-point extensions
Abstract: We study the moduli spaces of (semi-)stable representations of one-point extensions of quivers by rigid representations and derive results on their geometric properties with homological methods.
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BiBo Seminar in Bochum
15:30, Room IA 1-71 (30 minutes)
Julia Sauter (Bielefeld): An invitation to relative geometry of representations
Abstract: Following Auslander and Solberg, relative homological algebra replaces Ext^1 by a subfunctor Ext^1_F. In this set-up it is natural to replace the representation space of quiver representations by locally closed subsets where (certain) Hom-dimensions are fixed (their closures are usually referred to as rank varieties). I would like to use relative homological algebra to study these spaces and explain as a first step the relative Voigt's lemma. I would end with many conjectures and would hope to find some interested people who would like to work on this with me.
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BiBo Seminar in Bochum
16:45, Room IA 1-71 (60 minutes)
Christof Geiss (Mexico City): Real Schur roots and rigid representations
Abstract: This is a report on joint work with B. Leclerc and J. Schröer.
Let F be a field. In previous work we constructed, associated to a symmetrizable generalized Cartan matrix C with symmetrizer D and and orientation Ω, an 1-Iwanga-Gorenstein F-algebra H:=H(C,D,Ω), defined in terms of a quiver with relations.
We show that the rigid indecomposable locally free H-modules are parametrized, via their rank vector, by the real Schur roots associated to (C,Ω). Moreover, if M is such a module then it is free as if module over its endomorphism ring, and this ring is of the form F[x]/(x^n) for some n. This allows us to classify the left finite bricks of H in terms of the real Schur roots associated to (C^t,Ω). The main tool to prove our results is a F[[x]]-order, which permits us to relate locally free H-modules with modules over the canonical F((x))-species associated to the combinatorial data (C,D,Ω).
Friday, 16 November 2018
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13:15, Room U2-205
Olaf Schnürer (Paderborn): Smoothness of derived categories of algebras
Abstract: We report on joint work with Alexey Elagin and Valery Lunts where we prove smoothness in the dg sense of the bounded derived category of finitely generated modules over any finite-dimensional algebra over a perfect field. More generally, we prove this statement for any algebra over a perfect field that is finite over its center and whose center is finitely generated as an algebra. These results are deduced from a general sufficient criterion for smoothness.
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14:30, Room U2-205
Lidia Angeleri Hügel (Verona): Silting complexes over hereditary rings
Abstract: I will report on joint work with Michal Hrbek. Given a hereditary ring, we use the lattice of homological ring epimorphisms to construct compactly generated t-structures in its derived category. This leads to a classification of all (not necessarily compact) silting complexes over the Kronecker algebra.
Friday, 09 November 2018
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14:15, Room U2-205
Kevin Coulembier (Sydney): Tensor categories in positive characteristic
Abstract: Tensor categories are abelian k-linear monoidal categories satisfying some natural additional properties. Archetypical examples are the representation categories over affine (super)group schemes. P. Deligne gave very succinct intrinsic criteria for a tensor category to be equivalent to such a representation category, over fields k of characteristic zero. These descriptions are known to fail badly in prime characteristics. In this talk, I will present analogues in prime characteristic of these criteria, admittedly less succinct, but still intrinsic. Time permitting, I will comment on the link with a recent conjecture of V. Ostrik extending Deligne’s work in a different direction.
Friday, 19 October 2018
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14:15, Room U2-205
Sarah Scherotzke (Münster): The Chern character and categorification
Abstract: The Chern character is a central construction which appears in topology, representation theory and algebraic geometry. In algebraic topology it is for instance used to probe algebraic K-theory which is notoriously hard to compute, in representation theory it takes the form of classical character theory. Recently, Toen and Vezzosi suggested a construction, using derived algebraic geometry, which allows to unify the various Chern characters. We will categorify this Chern character. In the categorified picture algebraic K-theory is replaced by the category of non-commutative motives.
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15:30, Room U2-205
Jörg Feldvoss (Mobile, Alabama): Cohomological Vanishing Theorems for Leibniz Algebras
Abstract: Leibniz cohomology was introduced by Bloh and Loday as a non-commutative analogue of Chevalley-Eilenberg cohomology of Lie algebras. It turned out that Leibniz cohomology works more generally for Leibniz algebras which are a non-(anti)commutative version of Lie algebras. Many results for Lie algebras have been proven to hold in this more general context.
In the talk I will start from scratch and define Leibniz algebras, Leibniz (bi)modules, and their cohomology. Then I will explain the Leibniz analogues of vanishing theorems for the Chevalley-Eilenberg cohomology of semisimple and solvable Lie algebras due to Whitehead, Dixmier, and Barnes. In particular, we obtain the second Whitehead lemma for Leibniz algebras and the rigidity of semisimple Leibniz algebras in characteristic zero. The latter results were conjectured to hold for quite some time. Our main tools are the cohomological analogues of two spectral sequencesof Pirashvili for Leibniz homology and a spectral sequence due to Beaudouin.
All this is joint work with Friedrich Wagemann.
For information on earlier talks please check the complete seminar archive.
