Talks in 2008

  • 18.12.2008, 17.15, Stefan Wolf (Paderborn)
    Geometry of Quiver Flag Varieties
    Abstract: Fix a finite dimensional K-representation of a quiver Q, or, more generally, a finite dimensional module of a K-algebra A, for K some field. Then, one has a natural generalisation of the flag variety by additionally demanding that each subspace is also a submodule of the given module.
      In general these schemes are neither reduced nor irreducible. I will explain how to calculate the tangent space to this variety and then show, that, under some additional conditions, it is smooth and irreducible. Then I will interpret this result as some coefficient of a Hall polynomial.
  • 04.12.2008, 15.45, Daniel Murfet (Canberra)
    Auslander-Reiten translation for singular projective varieties
    Abstract: If X is a smooth projective variety over a field, then it is well-known that the bounded derived category of coherent sheaves on X admits a Serre functor. When X is singular this is no longer the case, and in the talk I will explain a replacement for the Serre functor, provided by Auslander-Reiten translation in the homotopy category of injective quasi-coherent sheaves on X. The formalism will then be used to prove that the stable derived category of X is Hom-finite if and only if X is Gorenstein.
  • 04.12.2008,14.15, Helmut Lenzing (Paderborn)
    Phenomenology of singularities
    The talk will be of an introductory nature and focus on an analysis of surface singularities.  The classes of simple, parabolic and unimodular singularities will be discussed in some detail, in particular their relationship to symmetry groups related to tilings of the spherical, Euclidean or hyperbolic plane. The talk will advocate the aspect that the singularities themselves are best understood through the properties of the resulting quotients of the actions of such symmetry groups. The arising quotients form compact 2-orbifolds, their Euler characteristic being a measure of the complexity for the singularity in question. We will show that taking the orbifolds as a starting point, will provide naturally the links between singularities, group actions, group representations, Cohen-Macaulay modules over the singularity, vector bundles on the orbifold, etc. This will, In particular, be illustrated for the case of simple singularities,  again  ‘explaining’ McKay correspondence. Throughout, emphasis will be given to explain and illustrate the links between the various concepts. There will be only few proofs.
  • 27.11.2008, 15.45,  Xiao-Wu Chen (Hefei/Paderborn)
    Rational singularities and almost split sequences
    Abstract: The paper "Rational singularities and almost split sequences" of Auslander relates almost split sequences to singularity theory by showing that the McKay quiver built from the finite-dimensional representations of a finite subgroup G of GL(2,C), where C is the complex numbers, is isomorphic to the AR quiver of the reflexive modules of the quotient singularity associated with G.
  • 27.11.2008, 14.15, Dirk Kussin (Paderborn)
    Introduction to Cohen-Macaulay modules
    Abstract: This talk provides an elementary introduction to the theory of Cohen-Macaulay modules. Basic reference is the book of Yuji Yoshino with title "Cohen-Macaulay modules over Cohen-Macaulay rings".
  • 20.11.2008, 15.45, Markus Schmidmeier (Florida Atlantic University)
    Nilpotent Linear Operators
    Abstract: Linear operators and their invariant subspaces are ubiquitous in modern  mathematics.  The Invariant Subspace Problem, for example, is considered one  of the most prominent open problems in Functional Analysis.  And the  corresponding problem for Abelian groups has attracted attention since Garrett Birkhoff in 1934 challenged us to classify the isomorphism types of  pairs consisting of a finite abelian group and a subgroup.
     In my talk I will speak about joint work with Claus Michael Ringel on linear  operators which act nilpotently on finite dimensional vector spaces, and on
    their invariant subspaces.   It turns out that for those linear operators  for which the nilpotency index is at most six, a complete classification can  be given.
    I will introduce the slope as an isomorphism invariant of an invariant  subspace of a linear operator; often this invariant can be computed by  taking just the barycenter. The slope positions the invariant space within  the category, in the sense that homomorphisms in general ``flow'' in the  direction of the slope.
    It turns out that the invariant subspaces of operators acting with  nilpotency index at most six form a tubular category; such categories carry  another coordinate system, the tubular index.  This index and the slope  coincide, and hence the indecomposable objects in the tubular category carry  their coordinate system into any  subspace category.  Besides the slope, also the Auslander-Reiten translation defines a direction  for the ``flow'' of homomorphisms.  Through slope and translation, the  concept of curvature is defined for subspace categories.  In return,  curvature distinguishes the tubular category of operators of nilpotency  index at most six as the only subspace category of parabolic type.
  • 20.11.2008, 14.15, Karl-Heinz Kiyek (Paderborn)
    Weakly cotorsion modules
    Abstract: In the first part, we introduce the notion of a weakly cotorsion module (over a commutative ring) and prove some properties of such modules. In the second part, we apply these results to the theory of one-dimensional local CM-rings. In the third part, we recall the notion of cotorsion modules, and show that a cortosion module is a weakly cotorsion module. Is the converse true?
  • 06.11.2008, 14.15,  Stefan Wolf (Paderborn)
    Reflections, Hall Polynomials at q=0 and the Composition Monoid
    Abstract:Let Q be a finite quiver without oriented cycles and w a word in the simples of this quiver. A quiver flag of type w of a Q-representation M is a composition series having the simples in the prescribed order of w. For Dynkin and extended Dynkin quivers the number of flags of type w of a fixed module M over a finite field F_q is given by a polynomial in q.
    We show how one can calculate this polynomial at q=0 using BGP reflection functors and use this to give a morphism from the generic composition algebra of Q to the composition monoid of Q for Q a Dynkin quiver and say what can be said about the general extended Dynkin case.
  • 23.10.2008, 15.15, Jue Le (Hefei/Paderborn)
    Auslander-Reiten Theory for the Homotopy Category of Projective
    Abstract: In the talk I will give an Auslander-Reiten formula for the homotopy category of complexes of projective modules. This formula guarantees the explicit description of Auslander-Reiten triangles in the homotopy category. Furthermore, almost split sequences in a module category can be deduced from these Auslander-Reiten triangles.
  • 15.10.2008, 16.45, Hugh Thomas (New Brunswick)
    m-cluster tilting objects and m-noncrossing partitions
    Abstract: m-noncrossing partitions (m a positive integer) are objects which come to us from Coxeter theory, but they admit a representation-theoretic reformulation.  I will explain this reformulation, and then go on to explain the resulting simple and conceptual bijection between the m-noncrossing partitions and the m-cluster tilting objects associated to a quiver Q without oriented cycles.  (Such bijections have previously been known only in the m=1 case.)  Time permitting, I will also touch on the negative cluster categories.
    This is joint work with Aslak Bakke Buan and Idun Reiten.
  • 15.10.2008, 15.15, Xiao-Wu Chen (Hefei/Paderborn)
    On the extensions of covariantly finite subcategories and applications
    Abstract:  The talk will consist of four parts: (1). I will give the proof to a result by Gentle and Todorov, which says that in an abelian category with enough projective objects, the extension subcategory of two covariantly finite subcategories is still covariantly finite; (2). I will give an example to show that Gentle-Todorov's theorem may fail in an arbitrary abelian category; (3). I will prove  that a triangulated version of Gentle-Todorov's theorem holds; (4). I will make applications of Gentle-Todorov's theorem to obtain short proofs to a classical result by Ringel and a recent result by Krause and Solberg.

  • 09.10.2008, 16.15, George Ciprian Modoi (Cluj-Napoca)
    A reformulation of Brown representability theorem and some consequences
    Abstract: It is well-known that if a triangulated category T satisfies Brown representability theorem (shortly BRT) then every triangulated coproduct preserving functor with the domain T has a right adjoint. But, what about the converse? We shall provide a reformulation of BRT having some similarities with that converse, more precisely: T satisfies BRT if and only if every exact coproduct preserving functor starting from the abelianization of T into an abelian AB4 category with enough injectives has a right adjoint. We also discuss some consequences of this reformulation, for example we give another proof of a Freyd-style representability theorem due to Neeman.
  • 23.09.2008, 16.00, Raum D1.312, Henning Krause (Paderborn)
    Ext-orthogonal pairs for hereditary rings
    Abstract: For the module category of a hereditary ring, the Ext-orthogonal pairs of subcategories are studied. For each Ext-orthogonal pair that is generated by a single module, a 5-term exact sequence is constructed. The pairs of finite type are characterized and two consequences for the class of hereditary rings are established: homological epimorphisms and universal localizations coincide, and the telescope conjecture for the derived category holds true.

  • 23.09.2008, 14.15, Raum D1.312, Nikolay Dichev und Yu Ye (Paderborn)
    Thick subcategories of representations of a quiver
    Abstract: Let k be an algebraically closed field and Q a Dynkin or an Euclidean quiver. In this talk, we will prove that any thick subcategory C of mod(kQ) is closed under taking arbitrary kernels, or equivalently, C is an abelian category and the embedding functor is exact. Recall that by a thick subcategory we mean a full additive subcategory which is closed under taking direct summands, kernels of epis, cokernels of monos and extensions.

  • 18.09.2008, 14.15, Raum D1.320, Lesya Bodnarchuk (Paderborn)
    One application of the theory of boxes
    Abstract: A famous result in the representation theory of algebras and boxes is Drozd's dichotomy Theorem saying that any representation infinte Roiter box is either tame or wild. There is a conjecture (due to Ringel) that an analogous statement should hold for bricks (schurian objects). This is still  not proven in general; however our study of vector bundles on degenerations of an ellipic curve via matrix problems leads to a wide class of boxes which are representation wild but brick-tame.

  • 07.08.2008, 11.00, Yu Ye (Paderborn)
    Thick subcategories of a tame hereditary algebra
    Abstract:  Let A be a finite dimensional tame hereditary algebra over an algebraically closed field and C a full additive subcategory of mod(A). In this talk, we will show that if C is closed under taking direct summands, kernels of epimorphisms, cokernels of monomorphisms and arbitrary extensions, then C is closed under taking arbitrary kernels and cokernels. The motivation for this work is to find connections between the subcategories of mod(A) and thick subcategories of the (bounded) derived category of mod(A). This is a joint work with Nikolay Dichev.

  • 17.07.2008, 15.30, Tim Römer (Osnabrück)
    Boij-Söderberg Conjectures
    Abstract: Motivated by the Multiplicity Conjectures of Herzog, Huneke and Srinivasan, Boij and Söderberg presented conjectures which describe (up to scalar multiple) which tables of natural numbers can be Betti tables of Cohen-Macaulay graded modules over standard graded polynomial rings. Recently all these conjectures have been proved by  Boij, Eisenbud, Floeystad, Schreyer, Söderberg, Weyman in a series of articles. In this talk I present an overview of the conjectures, their proofs and some related questions.

  • 17.07.2008, 14.00, Hagen Meltzer (Stettin)
    Stable vector bundle categories
    Abstract: This is a report on joint work with Dirk Kussin and Helmut Lenzing. We study stable categories of vector bundles (mainly over weighted projective lines) with respect to suitable systems of line bundles. This category is in general triangulated. In the domestic case we give an explicit construction of a tilting object and conclude some consequences of this.

  • 10.07.2008, 16.15, Hans-Christian Herbig (Greifswald)
    On the deformation quantization of singular symplectic quotient spaces
    Abstract: We explain how the Batalin-Fradkin-Vilkovisky construction can be used to find continuous star products on certain singular symplectic quotients.

  • 26.06.2008, 16.15, Kay Jin Lim (Aberdeen)
    The variety for the Specht Module of the p by p partition
    Abstract: Carlson introduced the cohomological and rank variety for a module over a finite group algebra. Let p be an odd prime. We give a general form for the largest component of the variety for the Specht module of the p by p partition restricted to a maximal elementary abelian p-subgroup of rank p. Also, we give an upper bound for the degree of the projectivized variety.

  • 11.06.2008, 14.15, Raum C4.204, Lesya Bodnarchuk (Paderborn)
    On the representation theory of boxes (bocses)
    Abstract: The technique of boxes was worked out in 70s by the Kiev representation school to formalize and generalize the matrix reduction algorithm, which is the main tool known for dealing with matrix problems. The proof of Drozd's Tame and Wild dichotomy theorem is based on the technique of boxes.This theorem asserts that any representation infinite  triangular box is either tame of wild. For a long time there were no approach to study wild boxes. However, there is an observation that some wild boxes behave tamely with respect to bricks. We emphasize a class of such boxes arising naturally in geometry.

  • 06.06.2008, 16.45, Raum A3.301, Dirk Kussin (Paderborn)
    The tilting graph of the cluster category of a tubular algebra
    Abstract: We show that the tilting graph of the cluster category of a tubular
    algebra is connected. (Joint work with M. Barot and H. Lenzing.)

  • 06.06.2008, 15.30, Raum A3.301, Igor Burban (Bonn)
    Cluster-tilting on hypersurface curve singularities
    Abstract: My talk is based on a joint work with O.Iyama, B.Keller and I.Reiten, see Adv. Math.,  vol. 217, no 6, 2443-2484 (2008) and arxiv:0704.1249. Let (R,m) be an isolated Gorenstein singularity, then the stable category of maximal Cohen-Macaulay modules over R is a Hom-finite triangulated category over k = R/m. Moreover, it is  2-Calabi-Yau  if dim(R) = 3. Using ideas coming from the theory of cluster algebras, Keller and Reiten have defined the  notion of a cluster-tilting object  in a 2-CY category. In the geometric situation of maximal Cohen-Macaulay modules  it turns out  to be  closely related with non-commutative crepant resolutions of van den Bergh. In my talk I am going to describe the cluster-tilting objects and the corresponding cluster-tilted algebras in the stable category of maximal Cohen-Macaulay modules over simple and minimally elliptic curve singularities.  In the last case we obtain an interesting class of representation-tame symmetric algebras studied earlier by Erdmann, Holm, Bialkowski and Skowronski.

  • 29.05.2008, 16.45, Sefi Ladkani (MPI Bonn)
    Derived equivalences of posets, with applications to cluster tilting objects
    Abstract: Diagrams over a finite partially ordered set (poset) X can be identified with sheaves over a suitable topology on X and sometimes also with modules over the incidence algebra of X. Two posets are said to be derived equivalent if their bounded derived categories of diagrams are equivalent as triangulated categories. This leads to an equivalence relation between posets, which is strictly coarser than isomorphism, but still fine enough to be interesting. However, there is no known algorithm that determines for two posets whether they are derived equivalent or not. In the first part of the talk I will explain the above notions and present several new constructions producing derived equivalences between posets, and more generally, between triangular matrix rings. In the second part I will outline applications to the posets of tilting modules and cluster tilting objects of path algebras of quivers without oriented cycles.

  • 29.05.2008, 15.15, Grzegorz Bobiński (Torun)
    Directing modules are regular in codimension one
    Abstract: Given a finite dimensional algebra and a dimension vector, one defines the variety of modules of this dimension vector. Geometric properties of points of module varieties are closely related to their homological properties, when viewed as modules. In my talk I will try to illustrate this phenomena by considering a distinguished component of a module variety for the dimension vector of a (not necessarily indecomposable) directing module.

  • 14.05.2008, 14.15, Raum C4.204, Yu Ye (Paderborn)
    An introduction to Hochschild cohomology of categories
    Abstract:  In this talk we will give the definition of Hochschild cohomology of abelian categories and differential graded categories, and try to explain some facts and applications. This is a report on the work of Keller and  Lowen-Van den Bergh.

  • 08.05.2008, 16.45, Manuel Saorin (Murcia)
    Recollements of right bounded derived categories: an unbounded approach
    Abstract: We study the problem of lifting and restricting TTF triples (equivalently, recollement data) between a triangulated category with coproducts and certain full subcategories. We show how to use that to generalize and revisit some well-known results on recollements of right bounded derived categories of algebras. In particular, we shall give necessary and sufficient criteria for the right bounded derived category of a differential graded (dg) category to be a recollement of two right bounded derived categories of dg categories. The result extends and corrects thecorresponding earlier result for ordinary algebras.

  • 08.05.2008, 15.15, Jan Stovicek (Trondheim)
    On well-generatedness of homotopy categories of complexes
    Abstract: Well generated triangulated categories, as defined by Neeman, provide us with a useful generalization of compactly generated triangulated categories. One may ask when some naturally arising categories such as K(Mod-R), K(Flat-R) or K(Inj-R) are well-generated. For this purpose, I will define the concept of a locally well-generated triangulated category, and I will characterize when each of the categories above is well-generated using well-known conditions on the ring R.

  • 24.04.2008, 16.45, Apostolos Beligiannis (Ioannina)
    Tilting theory and quasi-abelian categories.
    Abstract: Quasi-abelian categories were introduced by Yoneda in the late fifties. They form an important class of exact categories in the sense of Quillen and appear to be  the proper framework for the study of various non-abelian categories which appear in nature, e.g. categories of interest in topological algebra and functional analysis. Recently have been studied extensively  by Schneiders, Rump, Bondal-Van den Bergh and others. In the talk I will present some basic constructions and examples of quasi-abelian categories, and then I will concentrate on the connections between quasi-abelian categories and one-dimensional tilting theory in an arbitrary abelian category.

  • 24.04.2008, 15.15, Dirk Kussin (Paderborn)
    Representation dimension III: Exterior algebras (after Rouquier)

  • 17.04.2008, 16.45, Marcel Wiedemann (Leeds)
    Constructions with real root representations of quivers
    Abstract: Let Q be a quiver. V. Kac showed that the dimension vectors of indecomposable representations correspond to the positive roots of the associated Kac-Moody algebra. Moreover, there exists a unique indecomposable representation corresponding to each positive real root, called a real root representation. In this talk we shall discuss the following question: "How can one 'construct' real root representations and what are their 'properties'? We introduce the maximal rank type property for representations of quivers and use this property of real root representations to construct all real root representations of a certain class of quivers using Universal Extension Functors, introduced by C. Ringel. Moreover, we discuss examples of representations which cannot be constructed using Universal Extension Functors.

  • 17.04.2008, 15.15, Karsten Dietrich (Paderborn)
    Representation dimension II: Finitistic dimension (after Igusa/Todorov)

  • 10.04.2008, 16.45, Lesya Bodnarchuk (Kaiserslautern)
    Simple coherent sheaves on plane degenerations of an elliptic curve.
    Abstract: Indecomposable vector bundles on smooth elliptic curves were classified in 1957 (50 years ago) by Atiyah. In my talk I present a generalization of Atiyah's classification for simple coherent sheaves, for any plane degeneration of an elliptic curve. An approach for studying coherent sheaves on singular projective curves was suggested in works of Burban, Drozd and Greuel. In particular, it was shown that the category of coherent sheaves on a nodal cubic curve and on cycles of projective lines is tame, and all other degenerations of elliptic curves are wild. However, it turns out that all plane cubic curves are brick-tame, and there we can give an explicit description of simple coherent sheaves. The main technical tool we use to prove this result is the representation theory of boxes. Moreover, our approach leads to an interesting class of wild matrix problems which are wild but brick-tame.

  • 10.04.2008, 15.15, Stefan Wolf (Paderborn)
    Representation dimension I: Finiteness (after Iyama)

  • 06.03.2008, 16.15, Karsten Dietrich (Paderborn)
    Funktorielle Operationen auf präprojektiven Komponenten von zahmen Bimoduln
    Abstrakt: Wir betrachten eine Klasse von speziellen (4,1)-Bimoduln. Aus der Kategorie der Darstellungen eines solchen Bimoduls erhalten wir auf leichte Weise eine abelsche, erbliche Kategorie, auf der die inverse Auslander-Reiten Translation eine Autoäquivalenz induziert. Diese stimmt auf Objekten mit gewissen tubularen Shifts  überein. Wir untersuchen inwiefern diese Übereinstimmung auch als Funktoren gilt.

  • 28.02.2008, 15.45, Srikanth Iyengar (Lincoln, Nebraska)
    Lower bounds for dimensions of triangulated categories via Koszul objects
    Abstract: This talk concerns the dimension of a triangulated category, in the sense of Bondal and Van den Bergh. Rouquier has demonstrated the utility of this notion to research in representation theory, by his work on the representation dimension of exterior algebras. In my talk, I will describe an on-going  project with Henning Krause, where we obtain lower bounds on dimensions for fairly general triangulated categories. It builds on recent work of Bergh and Oppermann, and some elementary constructions from commutative algebra.

  • 28.02.2008, 14.15, Stefan Wolf (Paderborn)
    On Quiver Grassmannians
    Abstract: Quiver Grassmannians are varieties given by submodules of a fixed dimension vector for a representation of a quiver. Their Euler characteristic plays an important role in the theory of cluster algebras, especially with respect to the positivity conjecture. I will discuss this and other geometric properties like irreducibility and smoothness and their application to representation theory.

  • 14.02.2008, 16.45, Giordano Favi (Basel)
    Triangular geometry, Rickard idempotents and the telescope conjecture
    Abstract: Triangular geometry is the study of an essentially small tensor triangulated category K via its spectrum Spc(K) as introduced by Balmer. We will explain some features of this topological space and describe a canonical presheaf of triangulated categories on it. We will then move on and assume that K is the category of compact objects of some compactly generated category T, itself equipped with a tensor structure. Standard Bousfield techniques applied to T allow us to define idempotent objects e(V) in T attached to any subset V of Spc(K), generalizing a construction of Rickard. Finally we use this construction and the above presheaf to explain the local behaviour of the so-called telescope conjecture.
  • 14.02.2008, 15.15, Claudia Köhler (Paderborn)
    Gabriel-Roiter Maß und Überlagerungstheorie
    Abstrakt: Ein Verfahren zur Berechnung des Gabriel-Roiter Maß für Fadenmoduln wird diskutiert. Dabei erweist sich die Überlagerungstheorie als nützlich.

  • 07.02.2008, 16.15, Dirk Kussin (Paderborn)
    On the classification of function fields of genus zero
    Abstract: We classify those tame bimodules for which the generic module of the corresponding tame hereditary algebra has a commutative endomorphism ring. This extends former results to arbitrary characteristic; in characteristic two inseparable cases occur. Our treatment also shades new light on the classical classification of (commutative) function fields of genus zero.

  • 17.01.2008, 14.45, Michael Barot (UNAM)
    Cluster-tilting objects - a case study
    Abstract: This is a report on a joint work with Christof Geiss. We investigate the cluster category of a canonical algebra of weight type (2,2,2,2) in detail. In particular we describe its cluster tilting objects and derive the result that the exchange graph is connected.

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