Last
updated: May 13, 2013, 22:28 EST
Conference Titles, Abstracts, and Slides
Vladimir Bavula,
New criteria for a ring to have a
semisimple left quotient ring.
Abstract: Goldie's Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring (i.e. a semisimple left ring of fractions). The aim of my talk is to give four new criteria (using a completely different approach and new ideas). The first one is based on the recent fact that for an arbitrary ring R the set of maximal left denominator sets of R is a non-empty set. The Second Criterion is given via the minimal primes of R and goes further than the First one and Goldie's Theorem in the sense that it describes explicitly the maximal left denominator sets via the minimal primes of R. The Third Criterion is close to Goldie's Criterion but it is easier to check in applications (basically, it reduces Goldie's Theorem to the prime case). The Fourth Criterion is given via certain left denominator sets.
Charlie Beil, Noncommutative resolutions of non-Gorenstein singularities.
Abstract: I will describe how the dg algebra obtained from the derived endomorphism ring of a suitable perfect complex may be viewed as a noncommutative resolution of its center. Moreover, I will explain how this dg algebra reduces to the usual notion of a noncommutative crepant resolution when its center is a Gorenstein singularity.
Frauke Bleher, Closures of orbits of dimension 2.
Abstract: This talk is about joint work with Ted Chinburg and Birge Huisgen-Zimmermann. Let k be an algebraically closed field, let Lambda be a finite dimensional k-algebra, let P be a projective indecomposable Lambda-module, and let C be a submodule of rad(P). We concentrate on the case in which the orbit of Aut_Lambda(P) acting on C in the appropriate Grassmannian is an affine plane over k. Our goal is to bound the geometry of the orbit closure of such a C, using "good blow ups" of relatively minimal smooth projective surfaces, such that the bounds only depend on k and the k-dimension of C.
İlke Çanakçı, On surface cluster algebras: snake and band graph calculus.
Abstract: I will report on a joint work with Ralf Schiffler in which we introduce the notion of abstract snake graphs and develop a graphical calculus for surface cluster algebras. Moreover, I will talk about how to extend this results to abstract band graphs.
Andrew Carroll, Modules of constant Jordan type.
Abstract: Motivated by the work of Carlson, Friedlander and Pevtsova in the modular representation theory of elementary abelian p-groups, we introduce the class of modules of constant Jordan type over the path algebra of a quiver. As in the case of elementary abelian p-groups, we show that such modules give rise to vector bundles over various projective varieties (namely, the toric quiver varieties of Hille). We will discuss various examples of these modules, and under mild hypotheses construct large subclasses that have rich homological structure. This is joint work with Calin Chindris and Zongzhu Lin.
Giovanni Cerulli Irelli, Desingularization of quiver Grassmannians for Dynkin quivers.
Abstract: To every Dynkin quiver Q, we associate an algebra B(Q) which is derived equivalent to the Auslander algebra of kQ. We also construct a functor Lambda: mod kQ --> mod B(Q) with many remarkable properties, e.g. Lambda(M) is rigid and of injective and projective dimension less than 2, for every Q--representation M. We use these constructions to desingularize every quiver Grassmannian Gr_e(M) associated with M. Moreover the whole representation variety Rep(B(Q),dim(Lambda(M))) fibers over the orbit closure of M via a quotient map. We conjecture that Rep(B(Q),dim(Lambda(M))) is irreducible. This is joint work with Evgeny Feigin and Markus Reineke.
Calin Chindris, On the invariant theory for tame algebras. (Lecture cancelled)
Abstract: In this talk, I will describe an approach aimed at characterizing tame algebras via invariant theory. In particular, we will talk about fields of rational invariants and moduli spaces of modules over finite dimensional algebras along with some general reduction techniques for dealing with such objects. We will then apply these techniques to acyclic gentle algebras. This is based on joint work with Andy Carroll.
Jiarui Fei, Counting rational points of quiver representations of relations.
Abstract: In this talk, I will explain how to count the rational points of the GIT quotients of quiver representations with relations. I will focus on two types of algebras – one is extended from a quiver Q, and the other is Dynkin A_2 tensored with Q. For both, explicit (recursive) formulas will be given. For application in the quantum algebra, we study when they are polynomial-count. We follow the similar line as quiver without relations using the Hall algebra. However, algebraic manipulations in Hall algebra will be replaced by corresponding geometric constructions. You will see many examples.
Michael Gekhtman, Exotic cluster structure in GL(n). (Lecture cancelled)
Abstract: I will discuss a cluster structure in GL(n) compatible with a non-standard Poisson-Lie bracket. The differences between this structure and the standard one will be emphasized, in particular, the fact that the cluster algebra and the upper cluster algebra do not coincide in the non-standard case. This is joint work with M. Shapiro and A. Vainshtein.
Stephen Hermes, On the homology of the Ginzburg algebra.
Abstract: The Ginzburg algebra of a quiver with potential (Q,W) is a 3-Calabi-Yau differential graded algebra instrumental in the construction of cluster categories. If the quiver Q under consideration is an orientation of a Dynkin diagram, then the homology of the Ginzburg algebra is a "twisted" polynomial ring with coefficients in the corresponding preprojective algebra. We use this to compute the minimal A∞-structure of the Ginzburg algebra in Dynkin type by exploiting a connection with an augmentation of the derived category of kQ-modules.
Ivo Herzog, Diohantine supports of coherent functors.
Abstract: Let k be a field of characteristic 0 and L = sl(2,k) the Lie algebra of 2-by-2 traceless matrices over k. The Lie algebra L acts by derivations on the affine k-plane k[x,y], which decomposes as a direct sum of homogeneous components L(n), indexed by the natural numbers n. We will prove that if C is a coherent functor on the category U(L)-mod of finitely generated representations of the universal enveloping algebra, then the support of C in the natural numbers is diophantine. This is joint work with S. L'Innocente.
Ivan Horozov, Cohomology of GL(4,Z) with nontrivial coefficients. (Lecture cancelled)
Abstract: We compute the cohomology groups of GL(4,Z) with coefficients in symmetric powers of the standard representation twisted by the determinant. This problem arises in Goncharov's approach to the study of motivic multiple zeta values of depth 4. We use Borel-Serre compactification, a result of Harder on Eisenstein cohomology and a computationally effective version for the homological Euler characteristic of arithmetic groups.
Tom Howard, Constructing equivalences between singularity categories of finite dimensional algebras.
Abstract: The singularity category of a finite dimensional algebra, also known as the stable derived category, is the quotient of the bounded derived category by the thick subcategory of perfect complexes. The singularity category appears in the algebraic geometry of Orlov, in Rouquier's examples of arbitrary representation dimension, and in Keller's work on cluster categories. While it is easy to see that every derived equivalence induces an equivalence of singularity categories, I will discuss several methods for constructing singularity equivalences which are not induced by derived equivalences. As an application, I will show that every Nakayama algebra is singularity equivalent to a self-injective Nakayama algebra.
Miodrag Iovanov, An infinite version of the pure semisimplicity conjecture.
Abstract: The category of locally finite modules over an arbitrary algebra can be thought of as the infinite dimensional generalization of the category of modules over a finite dimensional one. We thus consider the following (infinite) version of the pure semisimplicity conjecture: if every locally finite left A-module ovrer an arbitrary algebra A decomposes as a direct sum of indecomposable modules, does the same follow for every right locally finite A-module? We study algebras with the property that finite dimensional representations are serial, and show that a theory that parallels in good part the theory of infinite primary abelian groups can be developed. Using this, we find counterexamples to the above infinite version of pure semisimplicity conjecture. One will be the path algebra of the half line infinite quiver, and also its complete path algebra; in particular, the tools used give insightes to nontrivial phenomena for representation theory of some quite simple infinite quivers or monomial algebras. We also find a classification of algebras (or linear categories) which are one-sided serial in the locally finite sense (i.e. locally finite injectives are serial).
David Jorgensen, On support varieties over complete intersections.
Abstract: The notion of support variety is a means of encoding homological behavior of a module over a local ring into an algebraic variety. There is well-developed theory of support varieties for modules over complete intersection local rings. We will discuss this theory, giving a brief survey of old results, and some new results.
Ellen Kirkman, The invariant theory of Artin-Schelter regular algebras: a survey.
Abstract: click here
Daniel Labardini Fragoso, Tagged triangulations, quivers with potentials, and Jacobian algebras.
Abstract: Every ideal triangulation of a surface with marked points gives rise to a quiver with potential in a natural way. It was proved over five years ago that the QPs associated to ideal triangulations related by a flip are related by Derksen-Weyman-Zelevinsky's QP-mutation. From the cluster algebra perspective, ideal triangulations constitute just a small stratum of the cluster complex, as Fomin-Shapiro-Thurston showed. To obtain the whole cluster complex, one needs the more general notion of tagged triangulation. It can be seen that tagged triangulations also have QPs associated in a natural way, but only recently has it been shown that tagged triangulations related by a flip have QPs related by QP-mutation. In this talk I will say a few words on this result (which obviously implies the non-degeneracy of the alluded QPs) and its proof. I will also present some results concerning uniqueness of non-degenerate potentials and the representation type of Jacobian algebras. The first part of the talk is based on work done by the speaker, the second part on joint work in progress with Christof Geiss and Jan Schröer.
Bernard Leclerc, Nakajima varieties and orbit closures of Dynkin quivers.
Abstract: The geometry of orbit closures of quivers of Dynkin type A,D,E is a very classical topic. For instance, it was shown by Lusztig that the local intersection cohomology of these varieties gives a canonical basis for the positive part of the quantum enveloping algebra of the same Dynkin type. In 2011, in a joint paper with D. Hernandez, we have shown that these orbit closures can be realized as some particular graded Nakajima varieties. This result has been extended to orbit closures of the repetitive algebras of Dynkin type in a joint recent paper with Plamondon, and further generalized by Keller and Scherotzke. It is also strongly related to recent work of Cerulli-Feigin-Reineke on the desingularization of quiver Grassmannians of type A,D,E. In my lecture, I will give an introduction to these ideas.
Helmut Lenzing, In memoriam of Michael Butler.
Liping Li, Representations of modular skew group algebras.
Abstract: In this paper we study representations of skew group algebras AG, where A is a connected, basic, finite-dimensional algebra (or a locally finite graded algebra) over an algebraically closed field k with characteristic p >=0, and G is an arbitrary finite group each element of which acts as an algebra automorphism on A. We characterize skew group algebras with finite global dimension or finite representation type, and classify the representation types of transporter categories for $p \neq 2,3$. When A is a locally finite graded algebra and the action of G on A preserves grading, we show that AG is a generalized Koszul algebra if and only if so is A.
Shiping Liu, Standard components of a Krull-Schmidt category.
Abstract: This is joint work with Charles Paquette. We first provide, for a general Krull-Schmidt category, some criteria for an Auslander-Reiten component with sections to be standard. Specializing to the category of finitely presented representations of a strongly locally finite quiver and its bounded derived category, we obtain many new types of standard Auslander-Reiten components. Applied to the module category of a finite-dimensional algebra, our criteria become particularly nice and yield some new results.
Frank Marko, Derived representation type of Schur superalgebras.
Abstract: click here
David Meyer, Do universal deformation rings recognize fusion?
Abstract: Let Gamma be a finite group, and let V be an absolutely irreducible module for Gamma over F_p. By B. Mazur, V has a universal deformation ring R(Gamma,V) whose structure is closely related to the cohomology groups H^i(Gamma,Hom_{F_p}(V,V)) for i=1,2. In this talk, we consider the case when Gamma is an extension of an abelian or dihedral group G whose order is relatively prime to p by an elementary abelian p-group N. We discuss to which extent R(Gamma,V) and H^i(Gamma,Hom_{F_p}(V,V)) for i=1,2 can see the fusion of N in Gamma.
Steffen Oppermann, d-Dimensional Geigle-Lenzing spaces and canonical algebras.
Abstract: This is a report on ongoing joint work with Martin Herschend, Osamu Iyama, and Hiroyuki Minamoto. Weighted projective lines were introduced by Geigle and Lenzing. One key property of these is that they give rise to hereditary categories with tilting objects, whose endomorphism rings are canonical algebras. Both weighted projective lines and canonical algebras have proven to be interesting objects in representation theory, and been studied intensively. In my talk I will introduce the notion of d-dimensional Geigle-Lenzing spaces, generalizing the concept of weighted projective lines. Also in this case we obtain a nice tilting bundle, whose endomorphism ring we call a d-canonical algebra. I will then focus on some properties of weighted projective lines which generalize nicely to the d-dimensional setup.
Daiva Pucinskaite, Bruhat order, Ext-quiver and Verma-multiplicities.
Abstract: An integral regular block of Bernstein-Gelfand-Gelfand category O(g) of a semisimple complex Lie algebra g is equivalent to the category of finite dimensional modules over a finite dimensional C-algebra A given by a quiver Q and relations. The structure of Q is related to the Bruhat order of the Weyl group W of g: The points in the quiver correspond to the Weyl group elements and all neighboring points (with respect to the Bruhat order) are connected in Q with two arrows going in opposite directions. The question when there is an arrow between two non-neighboring points is generally open. In this talk, we characterize those points of W that are only connected with their neighbors. In this case all non-zero Verma-multiplicities of the corresponding projective A-modules as well as projective R(A)-modules are one (here R(A) is the Ringel dual of A).
Claus Michael Ringel, The Auslander bijections.
Abstract: Let A be an artin algebra. The lecture will report on the work of M. Auslander in his seminal Philadelphia Notes (published already in 1978), where he exhibited his theory of morphisms determined by modules. This theory has to be considered as an exciting frame for describing the poset structure of the category of A-modules. The main idea is to identify parts of the category with the submodule lattice of a finite length B-module, where B is again an artin algebra. As a consequence, we can use the Jordan-Hoelder theorem for finite length modules in order to deal with factorizations of morphisms.
Dylan Rupel, Two perspectives on mutations.
Abstract: A cluster algebra is a subalgebra of a rational function field where generators and relations are obtained by a recursive combinatorial process known as "mutation". The combinatorial data needed to begin this process is a valued quiver. There are two perspectives one can take on the mutation operations: they can be viewed as "internal mutations" (a recursive process occurring within a fixed field of rational functions) or as "external mutations" (iterated application of isomorphisms between different fields). In this talk I will present a description of internal and external mutations of acyclic quantum cluster algebras using classical tools from the representation theory of the initial acyclic valued quiver.
Sarah Scherotzke, Graded quiver varieties and derived categories.
Abstract: Quiver varieties have been invented by Nakajima to give a geometric construction of representations of quantum groups. They have also been used recently to construct monodial categorifications of cluster algebras. Quiver varieties associated with a quiver Q are defined by geometric invariant theory. In my talk, I will explain how they can be described explicitly as modules of certain mesh categories. We show that there is a bijection between their strata and the isomorphism classes of objects in the derived category of the quiver Q. Furthermore, using the desingularisation map between graded quiver varieties, we show how to construct a desingularisation map for Grassmannians of representations of tilted algebras of Dynkin type. This is joint work with Bernhard Keller.
Ralf Schiffler, Positivity in cluster algebras.
Abstract: The positivity conjecture states that for every cluster variable the coefficients in the Laurent expansion with respect to any cluster are positive. In this talk I report on recent developments on this conjecture.
Jeanne Scott, Postnikov diagrams.
Abstract: In this talk I'll explain what Postnikov diagrams are and their relevance for the cluster algebra structure of the Grassmannian's homogeneous coordinate ring.
Ahmet Seven, Quasi-Cartan companions of cluster-tilted quivers.
Abstract: Quasi-Cartan companions are certain symmetrizable matrices introduced by Barot, Geiss and Zelevinksy to characterize finite type cluster algebras. In this talk, we will discuss basic properties of quasi-Cartan companions for infinite types. In particular, we will show that any skew-symmetric matrix associated to a cluster tilted quiver has a canonical type of a quasi-cartan companion and discuss applications.
Michael Shapiro, Cluster structure, compatible Poisson brackets, and integrability.
Abstract: We introduce a notion of compatible Poisson structure for cluster algebra, discuss cluster algebra structure in Postnikov's networks (planar oriented graphs with weighted edges). Then I will describe example of pentagram map when cluster transformation determines a completely integrable discrete system. The talk is based on joint papers with M.Gekhtman, A.Vainshtein, and S.Tabachnikov.
Evgeny Smirnov, Schubert calculus and Gelfand-Zetlin polytopes.
Abstract: We describe a new approach to the Schubert calculus on full flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope. We also prove a formula for all Demazure characters of a given representation of GL(n) via exponential sums over integral points in faces of the Gelfand-Zetlin polytope associated with the representation. Time permitting, I will also discuss a (mostly conjectural) generalization of our approach that would allow to describe the K-theory of a full flag variety. The talk is based on joint work with V. Kiritchenko and V. Timorin.
Salvatore Stella, Wonder of sine-Gordon Y-systems (Joint with T. Nakanishi).
Abstract: The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were introduced by Tateo in the 90’s in the study of the integrable deformation of conformal field theory by the thermodynamic Bethe ansatz method. The periodicity property and the dilogarithm identities concerning these Y-systems were conjectured by Tateo, and recently proved using cluster algebras. In this talk we explain how these Y-systems can be understood using triangulations of polygons and how this provides automatically a proof of both periodicity and dilogarithm identities in full generality.
Hugh Thomas, Reflection groups and representations of quivers.
Abstract: Let Q be a quiver without oriented cycles. It has been understood for a long time that the associated root system plays an important role in the representation theory of Q. Associated to the root system, there is a Weyl group W. It is already clear classically that W has some intrinsic connection to the representation theory of Q, because reflection functors act on dimension vectors of indecomposable representations (essentially) as simple reflections in W. However, the connection between W and the representation theory of Q goes deeper than this. I will discuss some other features of W (in particular weak order, and Reading's theory of c-sortable elements), leading up to the correspondence between torsion classes containing finitely many indecomposable representations and the c-sortable elements, shown by Ingalls-Thomas and Amiot-Iyama-Reiten-Todorov. Time permitting, I will include some speculation about how to extend this correspondence to torsion classes generated by a rigid representation.
Alexandra Zvonarëva, On the derived Picard groupoid of a Brauer tree algebra.
Abstract: The notion of silting mutation was introduced by Aihara and Iyama several years ago. We will discuss the connection between silting mutation and multiplication in the derived Picard groupoid of a symmetric algebra. Using this connection we will describe some braid-like relations in the derived Picard groupoid of a Brauer tree algebra.
Abstract: Goldie's Theorem (1960), which is one of the most important results in Ring Theory, is a criterion for a ring to have a semisimple left quotient ring (i.e. a semisimple left ring of fractions). The aim of my talk is to give four new criteria (using a completely different approach and new ideas). The first one is based on the recent fact that for an arbitrary ring R the set of maximal left denominator sets of R is a non-empty set. The Second Criterion is given via the minimal primes of R and goes further than the First one and Goldie's Theorem in the sense that it describes explicitly the maximal left denominator sets via the minimal primes of R. The Third Criterion is close to Goldie's Criterion but it is easier to check in applications (basically, it reduces Goldie's Theorem to the prime case). The Fourth Criterion is given via certain left denominator sets.
Charlie Beil, Noncommutative resolutions of non-Gorenstein singularities.
Abstract: I will describe how the dg algebra obtained from the derived endomorphism ring of a suitable perfect complex may be viewed as a noncommutative resolution of its center. Moreover, I will explain how this dg algebra reduces to the usual notion of a noncommutative crepant resolution when its center is a Gorenstein singularity.
Frauke Bleher, Closures of orbits of dimension 2.
Abstract: This talk is about joint work with Ted Chinburg and Birge Huisgen-Zimmermann. Let k be an algebraically closed field, let Lambda be a finite dimensional k-algebra, let P be a projective indecomposable Lambda-module, and let C be a submodule of rad(P). We concentrate on the case in which the orbit of Aut_Lambda(P) acting on C in the appropriate Grassmannian is an affine plane over k. Our goal is to bound the geometry of the orbit closure of such a C, using "good blow ups" of relatively minimal smooth projective surfaces, such that the bounds only depend on k and the k-dimension of C.
İlke Çanakçı, On surface cluster algebras: snake and band graph calculus.
Abstract: I will report on a joint work with Ralf Schiffler in which we introduce the notion of abstract snake graphs and develop a graphical calculus for surface cluster algebras. Moreover, I will talk about how to extend this results to abstract band graphs.
Andrew Carroll, Modules of constant Jordan type.
Abstract: Motivated by the work of Carlson, Friedlander and Pevtsova in the modular representation theory of elementary abelian p-groups, we introduce the class of modules of constant Jordan type over the path algebra of a quiver. As in the case of elementary abelian p-groups, we show that such modules give rise to vector bundles over various projective varieties (namely, the toric quiver varieties of Hille). We will discuss various examples of these modules, and under mild hypotheses construct large subclasses that have rich homological structure. This is joint work with Calin Chindris and Zongzhu Lin.
Giovanni Cerulli Irelli, Desingularization of quiver Grassmannians for Dynkin quivers.
Abstract: To every Dynkin quiver Q, we associate an algebra B(Q) which is derived equivalent to the Auslander algebra of kQ. We also construct a functor Lambda: mod kQ --> mod B(Q) with many remarkable properties, e.g. Lambda(M) is rigid and of injective and projective dimension less than 2, for every Q--representation M. We use these constructions to desingularize every quiver Grassmannian Gr_e(M) associated with M. Moreover the whole representation variety Rep(B(Q),dim(Lambda(M))) fibers over the orbit closure of M via a quotient map. We conjecture that Rep(B(Q),dim(Lambda(M))) is irreducible. This is joint work with Evgeny Feigin and Markus Reineke.
Calin Chindris, On the invariant theory for tame algebras. (Lecture cancelled)
Abstract: In this talk, I will describe an approach aimed at characterizing tame algebras via invariant theory. In particular, we will talk about fields of rational invariants and moduli spaces of modules over finite dimensional algebras along with some general reduction techniques for dealing with such objects. We will then apply these techniques to acyclic gentle algebras. This is based on joint work with Andy Carroll.
Jiarui Fei, Counting rational points of quiver representations of relations.
Abstract: In this talk, I will explain how to count the rational points of the GIT quotients of quiver representations with relations. I will focus on two types of algebras – one is extended from a quiver Q, and the other is Dynkin A_2 tensored with Q. For both, explicit (recursive) formulas will be given. For application in the quantum algebra, we study when they are polynomial-count. We follow the similar line as quiver without relations using the Hall algebra. However, algebraic manipulations in Hall algebra will be replaced by corresponding geometric constructions. You will see many examples.
Michael Gekhtman, Exotic cluster structure in GL(n). (Lecture cancelled)
Abstract: I will discuss a cluster structure in GL(n) compatible with a non-standard Poisson-Lie bracket. The differences between this structure and the standard one will be emphasized, in particular, the fact that the cluster algebra and the upper cluster algebra do not coincide in the non-standard case. This is joint work with M. Shapiro and A. Vainshtein.
Stephen Hermes, On the homology of the Ginzburg algebra.
Abstract: The Ginzburg algebra of a quiver with potential (Q,W) is a 3-Calabi-Yau differential graded algebra instrumental in the construction of cluster categories. If the quiver Q under consideration is an orientation of a Dynkin diagram, then the homology of the Ginzburg algebra is a "twisted" polynomial ring with coefficients in the corresponding preprojective algebra. We use this to compute the minimal A∞-structure of the Ginzburg algebra in Dynkin type by exploiting a connection with an augmentation of the derived category of kQ-modules.
Ivo Herzog, Diohantine supports of coherent functors.
Abstract: Let k be a field of characteristic 0 and L = sl(2,k) the Lie algebra of 2-by-2 traceless matrices over k. The Lie algebra L acts by derivations on the affine k-plane k[x,y], which decomposes as a direct sum of homogeneous components L(n), indexed by the natural numbers n. We will prove that if C is a coherent functor on the category U(L)-mod of finitely generated representations of the universal enveloping algebra, then the support of C in the natural numbers is diophantine. This is joint work with S. L'Innocente.
Ivan Horozov, Cohomology of GL(4,Z) with nontrivial coefficients. (Lecture cancelled)
Abstract: We compute the cohomology groups of GL(4,Z) with coefficients in symmetric powers of the standard representation twisted by the determinant. This problem arises in Goncharov's approach to the study of motivic multiple zeta values of depth 4. We use Borel-Serre compactification, a result of Harder on Eisenstein cohomology and a computationally effective version for the homological Euler characteristic of arithmetic groups.
Tom Howard, Constructing equivalences between singularity categories of finite dimensional algebras.
Abstract: The singularity category of a finite dimensional algebra, also known as the stable derived category, is the quotient of the bounded derived category by the thick subcategory of perfect complexes. The singularity category appears in the algebraic geometry of Orlov, in Rouquier's examples of arbitrary representation dimension, and in Keller's work on cluster categories. While it is easy to see that every derived equivalence induces an equivalence of singularity categories, I will discuss several methods for constructing singularity equivalences which are not induced by derived equivalences. As an application, I will show that every Nakayama algebra is singularity equivalent to a self-injective Nakayama algebra.
Miodrag Iovanov, An infinite version of the pure semisimplicity conjecture.
Abstract: The category of locally finite modules over an arbitrary algebra can be thought of as the infinite dimensional generalization of the category of modules over a finite dimensional one. We thus consider the following (infinite) version of the pure semisimplicity conjecture: if every locally finite left A-module ovrer an arbitrary algebra A decomposes as a direct sum of indecomposable modules, does the same follow for every right locally finite A-module? We study algebras with the property that finite dimensional representations are serial, and show that a theory that parallels in good part the theory of infinite primary abelian groups can be developed. Using this, we find counterexamples to the above infinite version of pure semisimplicity conjecture. One will be the path algebra of the half line infinite quiver, and also its complete path algebra; in particular, the tools used give insightes to nontrivial phenomena for representation theory of some quite simple infinite quivers or monomial algebras. We also find a classification of algebras (or linear categories) which are one-sided serial in the locally finite sense (i.e. locally finite injectives are serial).
David Jorgensen, On support varieties over complete intersections.
Abstract: The notion of support variety is a means of encoding homological behavior of a module over a local ring into an algebraic variety. There is well-developed theory of support varieties for modules over complete intersection local rings. We will discuss this theory, giving a brief survey of old results, and some new results.
Ellen Kirkman, The invariant theory of Artin-Schelter regular algebras: a survey.
Abstract: click here
Daniel Labardini Fragoso, Tagged triangulations, quivers with potentials, and Jacobian algebras.
Abstract: Every ideal triangulation of a surface with marked points gives rise to a quiver with potential in a natural way. It was proved over five years ago that the QPs associated to ideal triangulations related by a flip are related by Derksen-Weyman-Zelevinsky's QP-mutation. From the cluster algebra perspective, ideal triangulations constitute just a small stratum of the cluster complex, as Fomin-Shapiro-Thurston showed. To obtain the whole cluster complex, one needs the more general notion of tagged triangulation. It can be seen that tagged triangulations also have QPs associated in a natural way, but only recently has it been shown that tagged triangulations related by a flip have QPs related by QP-mutation. In this talk I will say a few words on this result (which obviously implies the non-degeneracy of the alluded QPs) and its proof. I will also present some results concerning uniqueness of non-degenerate potentials and the representation type of Jacobian algebras. The first part of the talk is based on work done by the speaker, the second part on joint work in progress with Christof Geiss and Jan Schröer.
Bernard Leclerc, Nakajima varieties and orbit closures of Dynkin quivers.
Abstract: The geometry of orbit closures of quivers of Dynkin type A,D,E is a very classical topic. For instance, it was shown by Lusztig that the local intersection cohomology of these varieties gives a canonical basis for the positive part of the quantum enveloping algebra of the same Dynkin type. In 2011, in a joint paper with D. Hernandez, we have shown that these orbit closures can be realized as some particular graded Nakajima varieties. This result has been extended to orbit closures of the repetitive algebras of Dynkin type in a joint recent paper with Plamondon, and further generalized by Keller and Scherotzke. It is also strongly related to recent work of Cerulli-Feigin-Reineke on the desingularization of quiver Grassmannians of type A,D,E. In my lecture, I will give an introduction to these ideas.
Helmut Lenzing, In memoriam of Michael Butler.
Liping Li, Representations of modular skew group algebras.
Abstract: In this paper we study representations of skew group algebras AG, where A is a connected, basic, finite-dimensional algebra (or a locally finite graded algebra) over an algebraically closed field k with characteristic p >=0, and G is an arbitrary finite group each element of which acts as an algebra automorphism on A. We characterize skew group algebras with finite global dimension or finite representation type, and classify the representation types of transporter categories for $p \neq 2,3$. When A is a locally finite graded algebra and the action of G on A preserves grading, we show that AG is a generalized Koszul algebra if and only if so is A.
Shiping Liu, Standard components of a Krull-Schmidt category.
Abstract: This is joint work with Charles Paquette. We first provide, for a general Krull-Schmidt category, some criteria for an Auslander-Reiten component with sections to be standard. Specializing to the category of finitely presented representations of a strongly locally finite quiver and its bounded derived category, we obtain many new types of standard Auslander-Reiten components. Applied to the module category of a finite-dimensional algebra, our criteria become particularly nice and yield some new results.
Frank Marko, Derived representation type of Schur superalgebras.
Abstract: click here
David Meyer, Do universal deformation rings recognize fusion?
Abstract: Let Gamma be a finite group, and let V be an absolutely irreducible module for Gamma over F_p. By B. Mazur, V has a universal deformation ring R(Gamma,V) whose structure is closely related to the cohomology groups H^i(Gamma,Hom_{F_p}(V,V)) for i=1,2. In this talk, we consider the case when Gamma is an extension of an abelian or dihedral group G whose order is relatively prime to p by an elementary abelian p-group N. We discuss to which extent R(Gamma,V) and H^i(Gamma,Hom_{F_p}(V,V)) for i=1,2 can see the fusion of N in Gamma.
Steffen Oppermann, d-Dimensional Geigle-Lenzing spaces and canonical algebras.
Abstract: This is a report on ongoing joint work with Martin Herschend, Osamu Iyama, and Hiroyuki Minamoto. Weighted projective lines were introduced by Geigle and Lenzing. One key property of these is that they give rise to hereditary categories with tilting objects, whose endomorphism rings are canonical algebras. Both weighted projective lines and canonical algebras have proven to be interesting objects in representation theory, and been studied intensively. In my talk I will introduce the notion of d-dimensional Geigle-Lenzing spaces, generalizing the concept of weighted projective lines. Also in this case we obtain a nice tilting bundle, whose endomorphism ring we call a d-canonical algebra. I will then focus on some properties of weighted projective lines which generalize nicely to the d-dimensional setup.
Daiva Pucinskaite, Bruhat order, Ext-quiver and Verma-multiplicities.
Abstract: An integral regular block of Bernstein-Gelfand-Gelfand category O(g) of a semisimple complex Lie algebra g is equivalent to the category of finite dimensional modules over a finite dimensional C-algebra A given by a quiver Q and relations. The structure of Q is related to the Bruhat order of the Weyl group W of g: The points in the quiver correspond to the Weyl group elements and all neighboring points (with respect to the Bruhat order) are connected in Q with two arrows going in opposite directions. The question when there is an arrow between two non-neighboring points is generally open. In this talk, we characterize those points of W that are only connected with their neighbors. In this case all non-zero Verma-multiplicities of the corresponding projective A-modules as well as projective R(A)-modules are one (here R(A) is the Ringel dual of A).
Claus Michael Ringel, The Auslander bijections.
Abstract: Let A be an artin algebra. The lecture will report on the work of M. Auslander in his seminal Philadelphia Notes (published already in 1978), where he exhibited his theory of morphisms determined by modules. This theory has to be considered as an exciting frame for describing the poset structure of the category of A-modules. The main idea is to identify parts of the category with the submodule lattice of a finite length B-module, where B is again an artin algebra. As a consequence, we can use the Jordan-Hoelder theorem for finite length modules in order to deal with factorizations of morphisms.
Dylan Rupel, Two perspectives on mutations.
Abstract: A cluster algebra is a subalgebra of a rational function field where generators and relations are obtained by a recursive combinatorial process known as "mutation". The combinatorial data needed to begin this process is a valued quiver. There are two perspectives one can take on the mutation operations: they can be viewed as "internal mutations" (a recursive process occurring within a fixed field of rational functions) or as "external mutations" (iterated application of isomorphisms between different fields). In this talk I will present a description of internal and external mutations of acyclic quantum cluster algebras using classical tools from the representation theory of the initial acyclic valued quiver.
Sarah Scherotzke, Graded quiver varieties and derived categories.
Abstract: Quiver varieties have been invented by Nakajima to give a geometric construction of representations of quantum groups. They have also been used recently to construct monodial categorifications of cluster algebras. Quiver varieties associated with a quiver Q are defined by geometric invariant theory. In my talk, I will explain how they can be described explicitly as modules of certain mesh categories. We show that there is a bijection between their strata and the isomorphism classes of objects in the derived category of the quiver Q. Furthermore, using the desingularisation map between graded quiver varieties, we show how to construct a desingularisation map for Grassmannians of representations of tilted algebras of Dynkin type. This is joint work with Bernhard Keller.
Ralf Schiffler, Positivity in cluster algebras.
Abstract: The positivity conjecture states that for every cluster variable the coefficients in the Laurent expansion with respect to any cluster are positive. In this talk I report on recent developments on this conjecture.
Jeanne Scott, Postnikov diagrams.
Abstract: In this talk I'll explain what Postnikov diagrams are and their relevance for the cluster algebra structure of the Grassmannian's homogeneous coordinate ring.
Ahmet Seven, Quasi-Cartan companions of cluster-tilted quivers.
Abstract: Quasi-Cartan companions are certain symmetrizable matrices introduced by Barot, Geiss and Zelevinksy to characterize finite type cluster algebras. In this talk, we will discuss basic properties of quasi-Cartan companions for infinite types. In particular, we will show that any skew-symmetric matrix associated to a cluster tilted quiver has a canonical type of a quasi-cartan companion and discuss applications.
Michael Shapiro, Cluster structure, compatible Poisson brackets, and integrability.
Abstract: We introduce a notion of compatible Poisson structure for cluster algebra, discuss cluster algebra structure in Postnikov's networks (planar oriented graphs with weighted edges). Then I will describe example of pentagram map when cluster transformation determines a completely integrable discrete system. The talk is based on joint papers with M.Gekhtman, A.Vainshtein, and S.Tabachnikov.
Evgeny Smirnov, Schubert calculus and Gelfand-Zetlin polytopes.
Abstract: We describe a new approach to the Schubert calculus on full flag varieties using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach allows us to compute the intersection products of Schubert cycles by intersecting faces of a polytope. We also prove a formula for all Demazure characters of a given representation of GL(n) via exponential sums over integral points in faces of the Gelfand-Zetlin polytope associated with the representation. Time permitting, I will also discuss a (mostly conjectural) generalization of our approach that would allow to describe the K-theory of a full flag variety. The talk is based on joint work with V. Kiritchenko and V. Timorin.
Salvatore Stella, Wonder of sine-Gordon Y-systems (Joint with T. Nakanishi).
Abstract: The sine-Gordon Y-systems and the reduced sine-Gordon Y-systems were introduced by Tateo in the 90’s in the study of the integrable deformation of conformal field theory by the thermodynamic Bethe ansatz method. The periodicity property and the dilogarithm identities concerning these Y-systems were conjectured by Tateo, and recently proved using cluster algebras. In this talk we explain how these Y-systems can be understood using triangulations of polygons and how this provides automatically a proof of both periodicity and dilogarithm identities in full generality.
Hugh Thomas, Reflection groups and representations of quivers.
Abstract: Let Q be a quiver without oriented cycles. It has been understood for a long time that the associated root system plays an important role in the representation theory of Q. Associated to the root system, there is a Weyl group W. It is already clear classically that W has some intrinsic connection to the representation theory of Q, because reflection functors act on dimension vectors of indecomposable representations (essentially) as simple reflections in W. However, the connection between W and the representation theory of Q goes deeper than this. I will discuss some other features of W (in particular weak order, and Reading's theory of c-sortable elements), leading up to the correspondence between torsion classes containing finitely many indecomposable representations and the c-sortable elements, shown by Ingalls-Thomas and Amiot-Iyama-Reiten-Todorov. Time permitting, I will include some speculation about how to extend this correspondence to torsion classes generated by a rigid representation.
Alexandra Zvonarëva, On the derived Picard groupoid of a Brauer tree algebra.
Abstract: The notion of silting mutation was introduced by Aihara and Iyama several years ago. We will discuss the connection between silting mutation and multiplication in the derived Picard groupoid of a symmetric algebra. Using this connection we will describe some braid-like relations in the derived Picard groupoid of a Brauer tree algebra.