Schedule-at-a-Glance

46 PRIMA 2013 AbstractsOppermann. The first half will be given by Steffen Oppermann.Weighted projective lines were introduced by Geigle andLenzing. One key property of these is that they give riseto hereditary categories with tilting objects, whose endomorphismrings are canonical algebras. Both weightedprojective lines and canonical algebras have proven tobe interesting objects in representation theory, and beenstudied intensively. In our talks we will introduce the notionof d-dimensional Geigle-Lenzing spaces, generalizingthe concept of weighted projective lines. Also in this casewe obtain a nice tilting bundle, whose endomorphism ringwe call a d-canonical algebra. We will then focus on someproperties of weighted projective lines which generalizenicely to the d-dimensional setup.Cohomological length functionsHenning KrauseUniversität Bielefeld, Germanyhkrause@math.uni-bielefeld.deA cohomological functor from a triangulated categoryinto the category of finite length modules over some ringgives rise to an integer valued function on the set of all objects.One might ask: What are the characteristic propertiesof such a function, and can one recover the functorfrom the corresponding function? Somewhat surprisingly,we can offer fairly complete answers to both questions.Examples from representation theory will illustrate theseanswers, and we discuss some cases when all cohomologicallength functions can be classified.Invariant flags for nilpotent operators, andweighted projective linesHelmut LenzingPaderborn University, Germanyhelmut@math.uni-paderborn.deThis is joint work with Dirk Kussin and Hagen Meltzerextending a previous analysis of the invariant subspaceproblem for nilpotent operators (of finite dimensional vectorspaces over an algebraically closed field k). The problemitself goes back to G. Birkhoff (1934) and was treatedin detail by C.M. Ringel and M. Schmidmeier (2006-2008); it is further related to research by D. Simson (2007)and Pu Zhang (2011). In previous work with Kussin andMeltzer we did establish a relationship of the invariantsubspace problem to singularity theory, more specificallywe showed a link to the theory of weighted projectivelines X of special weight type (2, 3, c). In this talk, weshow that for arbitrary triple weight type (a, b, c), a suitablefactor category S of the category of vector bundleson X, is equivalent to the category of graded representationsof nilpotency degree bounded by c, equipped withtwo invariant flags of graded sub-representations , wherethe flag lengths are determined by the integers a and b.The category S carries a natural exact structure which isalmost-Frobenius, but in general not Frobenius. It’s associatedstable category S is triangulated and shown tobe equivalent to the singularity category of the (suitablygraded) triangle singularity x a + y b + z c .Cluster categories and independence resultsfor exchange graphsPierre-Guy PlamondonUniversity of Paris-Sud, Francepierre-guy.plamondon@math.u-psud.frCluster categories are triangulated categories that havebeen used to study S.Fomin and A.Zelevinsky’s cluster algebras.They come equipped with special objects, calledcluster-tilting objects, that are obtained from one anotherby a process called mutation; this gives rise to an exchangegraph. In this talk, we will study properties ofthis exchange graph, and see how it is related to the exchangegraph of a cluster algebra. We will then applythese results to a conjecture of Fomin-Zelevinsky statingthat the exchange graph of a cluster algebra does not dependon the choice of the ring of coefficients over whichthe algebra is defined. (This is joint work with G.CerulliIrelli, B.Keller and D.Labardini-Fragoso.)Triangulated categories and tau-tilting theoryIdun ReitenNorwegian University of Science and Technology, Norway)idun.reiten@math.ntnu.noThis talk is based upon work with Adachi and Iyama.We explain how work on triangulated categories (clustercategories and more generally, 2-Calabi-Yau categories)inspired an extension of some aspects of classical tiltingtheory, which we have called tau-tilting theory.From submodule categories to preprojectivealgebrasClaus Michael RingelBielefeld University, Germany and Shanghai Jiao TongUniversity, Chinaringel@math.uni-bielefeld.deLet S(n) be the category of invariant subspaces of nilpotentoperators with nilpotency index at most n. Suchsubmodule categories have been studied already in 1936by Birkhoff, they have attracted a lot of attention in recentyears, for example in connection with some weightedprojective lines (Kussin, Lenzing, Meltzer). On the otherhand, we consider the preprojective algebra of type A n;the preprojective algebras were introduced by Gelfandand Ponomarev, they are now of great interest, for examplethey form an important tool to study quantum groups(Lusztig) or cluster algebras (Geiss, Leclerc, Schroeer).Direct connections between the submodule category S(n)and the module category of the preprojective algebra oftype A n−1 have been established quite a long time ago byAuslander and Reiten, and recently also by Li and Zhang,but apparently this remained unnoticed. The lecture isbased on joint investigations with Zhang Pu and will providedetails on this relationship. As a byproduct we seethat here we deal with ideals I in triangulated categoriesT such that I is generated by an idempotent and T/I isabelian.Exact categories over Cohen–Macaulay ringsRyo TakahashiNagoya University, Japantakahashi@math.nagoya-u.ac.jpIt is known that the following full subcategories of finitelygenerated modules are Frobenius categories:• the category of maximal Cohen–Macaulay modulesover a Gorenstein ring,• the category of totally reflexive ( = finitely generatedGorenstein projective) modules over a noetherianring,• the category of special Cohen–Macaulay modulesover a rational surface singularity.In this talk, we give a systematic generalization of thisfact over a Cohen–Macaulay ring. More precisely, westudy certain full subcategories of finitely generated modules,and investigate when they are exact categories withenough projectives and enough injectives. This talk isbased on joint work with Osamu Iyama.Toward repetitive equivalences