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updated: May 23, 2015, 15:21 CET
Conference Titles, Abstracts, and Slides
Murad Alim, Quivers and mirror symmetry.
Abstract: I will describe what a BPS quiver is and how it can be assigned to deformation families of supersymmetric physical theories. These theories can be obtained geometrically using two different string theories on two mirror families of Calabi-Yau threefolds, which are identified by mirror symmetry. The nodes of the quivers are assigned to generating sets of objects on both sides of mirror symmetry corresponding to stable coherent sheaves and special Lagrangian submanifolds. Quiver representation theory together with a physically motivated stability condition encodes the BPS spectra of these theories and their wall-crossing behavior in the moduli space.
Seidon Alsaody, Composition algebras, triality and algebraic groups.
Abstract: A composition algebra is a (not necessarily associative or unital) non-zero algebra endowed with a non-degenerate multiplicative quadratic form. Over any field, finite-dimensional composition algebras can only exist in dimensions 1, 2, 4 and 8. The classification problem for these algebras remains open and is especially difficult in dimension eight. Lately this problem has been approached using e.g. representation theory of Lie algebras.
In this talk I will give an overview of the classification problem, with emphasis on recent results establishing an equivalence between the category of eight-dimensional composition algebras, and a category where the objects are certain pairs of outer automorphisms of algebraic groups. The Principle of Triality, originally due to Elie Cartan, plays a central role in this correspondence.
Lidia Angeleri Hügel, Silting modules.
Abstract: Silting modules generalise tilting modules over an arbitrary ring, as well as the support τ-tilting modules over a finite dimensional algebra recently introduced by Adachi, Iyama and Reiten. They are the module-theoretic counterpart of (possibly non-compact) two-term silting complexes, and they are closely related to certain t-structures and co-t-structures in the derived category. Several well known results from tilting theory on existence of approximations and complements can be extended to silting modules. Moreover, it turns out that silting theory provides a suitable context for studying ring epimorphisms and localisations. We will focus on the case of a hereditary ring, where localisations turn out to be parametrised by silting modules. The talk is based on joint work with Frederik Marks and Jorge Vitória.
Chris Beasly, Semi-invariants of quivers and the quantum chiral ring.
Abstract: The notion of mutation for a quiver with potential can be understood as an instance of electric-magnetic duality a la Seiberg. In the physical setting, quiver invariants (mesons) vs semi-invariants (baryons) play distinct roles. I will discuss features of duality for quiver semi-invariants, along with some important open questions.
Frauke Bleher, On the module structure of relative differentials of curves.
Abstract: This talk is about joint work with Ted Chinburg and Aristides Kontogeorgis. Let k be an algebraically closed field of positive characteristic p, and let X be a non-singular projective curve over k. Suppose G is a finite group with non-trivial cyclic Sylow p-subgroups acting faithfully on X. Let M be the space of global sections of the sheaf of relative differentials of X over k. We show that the kG-module structure of M can be determined by so-called Boseck invariants. This generalizes previous results when G is cyclic or when the ramification is tame.
Thomas Brustle, On the non-leaving-face property for associahedra.
Abstract: D. Sleator, R. Tarjan and W. Thurston showed in 1988 that the associahedron satisfies the non-leaving-face property, that is, every geodesic connecting two vertices stays in the minimal face containing both. Recently, C. Ceballos and V. Pilaud established the non-leaving face property for generalized associahedra of types B,C,D, and some exceptional types including E6. The key ingredient in the proofs is a normalization, a sort of projection from the associahedron to a face. We use methods from cluster categories to define such a normalization, which allows us to establish the non-leaving face property at once for all finite cases that are modelled using cluster categories, namely the Dynkin diagrams. This talk reports on joint work with Jean-François Marceau.
Olgur Celikbas, Syzygies and tensor products of modules.
Abstract: click here
Calin Chindris, Locally semi-simple quiver representations.
Abstract: This talk is based on joint work with Dan Kline. When studying quiver representations within the general framework of invariant theory, one is naturally led to consider the class of locally semi-simple representations. Answering a question raised by Victor Kac, we show that a connected quiver is tame if and only if every representation with a semi-simple ring of endomorphisms is locally semi-simple.
Lars Christensen, Stable homology.
Abstract: Tate homology and cohomology originated in the realm of group algebras and evolved through a series of generalizations to the setting of Iwanaga-Gorenstein rings. The cohomological theory has a more far-reaching generalization to the setting of associative rings; it is now called stable cohomology, and it agrees with Tate homology over Iwanaga-Gorenstein rings. On the homological side, the picture has remained opaque. In the talk I will report on recent work---joint with Olgur Celikbas, Li Liang, and Grep Piepmeyer---that clears it up a bit.
Michele Del Zotto, The representation theory of BPS quivers.
Abstract: BPS quivers are a very special (yet humongous) class of quivers with potentials. The study of the representation theory of these algebras leads to very pleasant surprises. In particular, we will discuss a very natural generalization of the concept of Dlab-Ringel defects. These generalized defects are classified by simple Lie groups. The A1 case is the usual Dlab-Ringel defect, whose controlled subcategories correspond to homogenous and non-homogenous tubes in the representation theory of BPS quivers giving rise to tame algebras. All other cases occur for BPS quivers whose representation categories are wild, giving an interesting novel angle on these structures.
Ana Garcia, Syzygies over 2-Calabi-Yau tilted algebras.
Abstract: In this talk we consider the category of Cohen Macaulay modules over 2-Calabi-Yau tilted algebras, CM(B). In this context the objects in the category are the syzygies over the algebra. We study homological properties of this category showing the relation between the Auslander-Reiten translation and the syzygy functor. We explore the connections with the representation dimension of algebras, the Igusa-Todorov functions, and periodic resolutions. For the particular case of Jacobian algebras arising from surfaces without punctures and from the punctured disk (Dynkin type D), we give a geometric description of CM(B) modules and describe the projective resolutions over these algebras.
Matt Garcia, Classifying triangulations of the continuous cluster category using quivers with multiplicity.
Abstract: Continuous cluster categories are triangulated topological categories introduced by Igusa and Todorov to generalize cluster categories of type A. One such continuous cluster category is Cπ, which is associated to the disk model of the hyperbolic plane. Since the hyperbolic plane is a covering space of many surfaces we expect to find that Cπ has the corresponding universal properties for cluster categories of surface type. In this talk we describe the possible triangulated structures on Cπ. To do so we must examine the covering categories of Cπ and their triangulations. Igusa and Todorov have shown that we must insist on at least 2 objects in each isomorphism class of the continuous Frobenius category F that when stabilized gives Cπ. Translated in the language of covering categories, this means we may only consider covers with an even number of sheets. By studying quivers with multiplicity (resulting in non-basic path algebras) and some associated combinatorial data we can classify all the possible triangulations of a continuous cluster category Cπ. In particular, for the case of exactly two objects in each isomorphism class of F we recover the triangulation described by Igusa and Todorov, the triangulation described by Orlov and a third possible triangulation. This is joint work with Kiyoshi Igusa.
Al Garver, Combinatorics of Exceptional Sequences in Type A.
Abstract: Exceptional sequences are certain ordered sequences of quiver representations with applications to noncrossing partitions, factorizations of Coxeter elements, and cluster algebras. We introduce a class of combinatorial objects called strand diagrams that we use to classify exceptional sequences of representations of type A Dynkin quivers. We also use variations of the model to classify c-matrices of type A Dynkin quivers, to interpret exceptional sequences as linear extensions of certain posets, and to give an elementary bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. This is joint work with Kiyoshi Igusa, Jacob Matherne, and Jonah Ostroff.
Sira Gratz, Cluster algebras of infinite rank as colimits.
Abstract: Assem, Dupont and Schiffler introduced the category of rooted cluster algebras, which has as objects pairs (A,S), where A is a cluster algebra (of possibly infinite rank) and S a distinguished initial seed of A. After explaining these notions we will show that, although the category of rooted cluster algebras does not in general admit colimits, every rooted cluster algebra can be written as a directed colimit of rooted cluster algebras of finite rank. Directed colimits of rooted cluster algebras of Dynkin type A have a geometric interpretation as triangulations of the closed disc with possibly infinitely many marked points on the boundary. They are related to infinite discrete cluster categories of type A, respectively the continuous cluster category of type A as introduced by Igusa and Todorov.
Edward L. Green, Brauer configuration algebras.
Abstract: This is joint work with Sibylle Schroll. I introduce Brauer configurations which are generalizations of Brauer graphs which are generalizations of Brauer trees. Associated to a Brauer configuration is a Brauer configuration algebra. I will explain how to construct a Brauer configuration algebra from its configuration. Brauer configuration algebras are a class of finite dimensional symmetric algebas having special properties. In Sibylle Schroll's talk, a number of the properties will be discussed.
Stephen Hermes, Semi-invariant pictures and maximal Green sequences.
Abstract: Maximal green sequences were introduced by B. Keller to give a combinatorial construction of Donaldson-Thomas invariants. T. Brüstle, G. Dupont and M. Pérotin proved that acyclic tame quivers admit only finitely many maximal green sequences. Building off of recent work by K. Igusa and G. Todorov interpreting green mutation in terms of semi-invariant pictures, we prove two conjectures about maximal green sequences. First, we extend the results of Brüstle-Dupont-Pérotin to quivers mutation equivalent to tame ones. Second, we prove a conjecture of D. Xie that for acyclic quivers, maximal green sequences mutate at the target of any infinite type arrow before the source.
Gustavo Jasso, The d-abelian property of d-cluster-tilting subcategories.
Abstract: d-cluster-tilting subcategories of module categories were introduced by Iyama as the natural playground for higher Auslander--Reiten theory. In this talk, we will review some properties of these subcategories in order to motivate the introduction of d-abelian categories. I will explain how to recover basic results in higher Auslander--Reiten theory from this perspective. The talk contains joint work with Sondre Kvamme and Theo Raedschelders.
Peter Jørgensen, Co-t-structures: the first decade.
Abstract: Co-t-structures were introduced independently by Bondarko and Pauksztello in 2007 as a mirror image of t-structures. The two types of structures are related by a "looking glass principle", in the sense that many of their properties are analogous, but translation from one side to the other is rarely mechanical. We will recall the canonical t-structure in the derived category, show an analogous co-t-structure in the homotopy category, and show an example of a category which is skewed in the sense that it has many t-structures and few co-t-structures, or vice versa. We will also mention the close relation between t-structures, co-t-structures, simple minded collections, and silting objects.
Ryan Kinser, Hopf actions on path algebras.
Abstract: There is a notion of a Hopf algebra acting on another algebra which generalizes the classical notion of a group acting on an algebra by automorphisms. We classify (by explicit formulas) the Hopf actions of Taft algebras on path algebras of loopless, Schurian quivers. Then, we apply this to give some actions of the Frobenius-Lusztig kernel u_q(sl_2) and Drinfeld doubles of Taft algebras on path algebras. This talk is a summary of joint work with Chelsea Walton.
Helmut Lenzing, Preprojective algebras, revisited.
Abstract: In my talk I will first review the various (equivalent) definitions of the preprojective algebra associated to the path algebra of a finite (connected) quiver Q without oriented cycles. The preprojective algebra will be viewed as a positively graded algebra, correspondingly we will investigate its category of graded modules and derive from this its ring-theoretical and homological properties. A key aspect will be to treat graded modules over the preprojective algebra as functors on the mesh category of the quiver Q. The focus will be on the tame hereditary case where the features of the preprojective algebra are especially nice because of its link to the theory of Kleinian (or simple) singularities.
Charles Paquette, Accumulation points of real Schur roots.
Abstract: Let Q be a finite and acyclic quiver and let rep(Q) denote the category of finite dimensional representations of Q over some algebraically closed field. In this talk, we consider the set S_Q of real Schur roots of Q, that is, the set of dimension vectors of the so-called exceptional representations. If we identify dimension vectors up to scaling, we see that when Q is of infinite representation type, the elements of S_Q will give rise to accumulation points (more formally, of accumulation rays). In this talk, we will study these accumulation points. This is closely related to the notion of canonical decomposition of a dimension vector.
Chrysostomos Psaroudakis, Silting objects, t-structures and derived equivalences.
Abstract: Silting objects were introduced by Keller-Vossieck in their study of t-structures for the bounded derived category of representations over Dynkin quivers. They generalize tilting objects and they can be considered more generally in abstract triangulated categories. It is known that compact tilting objects give rise to derived equivalences between rings. Passing to the context of derived equivalences between abelian categories, it is natural to study the associated heart of a tilting object. In this talk, we consider (not necessarily compact) silting objects in the derived category of a Grothendieck category A. We show that the realization functor associated to a silting object is a triangle equivalence between the bounded derived category of the heart and the bounded derived category of A if and only if the object is tilting. Furthermore we discuss the dual notions of cosilting and cotilting. This is joint work with Jorge Vitoria.
Dylan Rupel, Companion cluster algebras to a generalized cluster algebra.
Abstract: Cluster algebras are a class of recursively defined algebras that have found connections to many areas of mathematics. The construction of a cluster algebra proceeds from the initial data of a skew-symmetrizable matrix called the exchange matrix, a collection of elements from a semifield called the coefficients, and a collection of algebraically independent rational functions called the cluster variables. At each step of the recursion each piece of data mutates according to concrete prescribed rules, in particular the cluster variables mutate according to binomial exchange relations whereby a product of a known cluster variable and an unknown cluster variable is given by a sum of two monomials in the remaining known cluster variables. This mutation process can be reduced to a pair of recursions involving integer vectors called c-vectors and g-vectors together with a recursion on collections of polynomials called F-polynomials.
In this talk I will describe a joint work with Tomoki Nakanishi in which we extend these constructions to the case of generalized cluster algebras where the binomial exchange relations above are replaced by polynomial exchange relations with coefficients satisfying a normalization condition and a power condition. Our main results exhibit a relationship between the c-vectors, g-vectors, and F-polynomials of these generalized cluster algebras and their left- and right-companion cluster algebras.
Ralf Schiffler, Cluster-tilted algebras.
Abstract: Let A be a hereditary algebra, T a tilting module and C the corresponding tilted algebra. Thus C is the endomorphism algebra of T. Then the trivial extension B of C by the bimodule Ext^2(DC,C) is called a cluster-tilted algebra. In this expository talk, we shall review the relation between the algebras C and B as well as their module categories.
Marcus Schmidmeier, Crossing and noncrossing partitions of the disk.
Abstract: Partitions of the disk can be categorified as short exact sequences for nilpotent linear operators. Symmetry operations on the disk give insight about Littlewood-Richardson tableaux and the geometry of representation spaces for short exact sequences. This talk is about a joint project with Justyna Kosakowska.
Sibylle Schroll, Multiserial algebras.
Abstract: This talk is based on joint work with Ed Green. We start by defining multiserial and special multiserial algebras and we show their relationship with the Brauer configuration algebras defined in Ed Green's talk. We give examples of (special) multiserial algebras in the literature, and, in particular, we relate to them the symmetric algebras with Jacobson radical cubed zero.
Chelsea Walton, On actions of Hopf algebras.
Abstract: In this talk, I will provide motivation and an overview of recent work on Hopf algebra actions on several classes of algebras. Many examples and several open questions will also be discussed.
He Wang, Resonance varieties, Hilbert series and Chen ranks.
Abstract: Associated to finitely generated group G there are two types of cohomology jump loci: the characteristic varieties, which sit inside the representation variety Hom(G,C*), and the resonance varieties, which sit inside H^1(G,C). The Hilbert series of the Alexander invariants of G and the Chen ranks of G are closely related to these varieties. In this talk, I will introduce these concepts, explain various relations among them, and show two applications. The first application is to certain basis-conjugating subgroups of the automorphism group of a free group, called the McCool groups. The second application is to the picture groups of type An, introduced by Igusa, Orr, Todorov and Weyman. This talk is about joint work with Alex Suciu.
Harold Williams, Cluster algebras as Legendrian knot invariants.
Abstract: We explain a new connection between the theories of cluster algebras and Legendrian knots. The starting point is the recent construction by Shende-Treumann-Zaslow of a new Legendrian knot invariant. This invariant is a moduli space of objects with various interpretations, for example as sheaves, quiver representations, or objects in a Fukaya category. For specific choices of Legendrian knot, it turns out that these invariants recover essentially all examples of cluster varieties related to the group SLn. Cluster varieties are spaces that have been realized to play a wide range of roles in geometry and representation theory in the past decade, and include character varieties of punctured surfaces and strata of algebraic groups and homogeneous spaces. Our main result is that on such a variety the cluster structure, which can be regarded as an atlas of open toric subsets with special properties and transition functions, is in a sense determined by the structure of the set of exact Lagrangian fillings of the associated Legendrian knot. This is joint work in progress with Vivek Shende, David Treumann, and Eric Zaslow.
Jinde Xu, Maximal rigid objects and cluster-tilting objects in 2-CY categories.
Abstract: In this talk, we want to give a solution to the Conjecture II.1.9 of [A. B. Buan, O. Iyama, I. Reiten and J. Scott, Cluster structures for 2-Calabi–Yau categories and unipotent groups, Compos. Math. 145(4) (2009) 1035–1079], which said that any maximal rigid object without loops or 2-cycles in its quiver is a cluster-tilting object in a connected Hom-finite triangulated 2-CY category C.
Yichao Yang, A relation between tilting graphs and cluster-tilting graphs of hereditary algebras.
Abstract: We give the condition of isomorphisms between tilting graphs and cluster-tilting graphs of hereditary algebras. As a conclusion, it is proved that a graph is a skeleton graph of Stasheff polytope if and only if it is both the tilting graph of a hereditary algebra and also the cluster-tilting graph of another hereditary algebra. At last, when comparing such uniformity, the geometric realizations of simplicial complexes associated with tilting modules and cluster-tilting objects are discussed respectively. This is joint work with Professor Fang Li.
Alexandra Zvonareva, Derived Picard groups of selfinjective Nakayama algebras.
Abstract: This talk is based on joint work with Yury Volkov. In this talk I will describe the derived Picard group of a selfinjective Nakayama algebra. I will try to explain how silting mutations, orbit categories and spherical objects are used to obtain this description.
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Abstract: I will describe what a BPS quiver is and how it can be assigned to deformation families of supersymmetric physical theories. These theories can be obtained geometrically using two different string theories on two mirror families of Calabi-Yau threefolds, which are identified by mirror symmetry. The nodes of the quivers are assigned to generating sets of objects on both sides of mirror symmetry corresponding to stable coherent sheaves and special Lagrangian submanifolds. Quiver representation theory together with a physically motivated stability condition encodes the BPS spectra of these theories and their wall-crossing behavior in the moduli space.
Seidon Alsaody, Composition algebras, triality and algebraic groups.
Abstract: A composition algebra is a (not necessarily associative or unital) non-zero algebra endowed with a non-degenerate multiplicative quadratic form. Over any field, finite-dimensional composition algebras can only exist in dimensions 1, 2, 4 and 8. The classification problem for these algebras remains open and is especially difficult in dimension eight. Lately this problem has been approached using e.g. representation theory of Lie algebras.
In this talk I will give an overview of the classification problem, with emphasis on recent results establishing an equivalence between the category of eight-dimensional composition algebras, and a category where the objects are certain pairs of outer automorphisms of algebraic groups. The Principle of Triality, originally due to Elie Cartan, plays a central role in this correspondence.
Lidia Angeleri Hügel, Silting modules.
Abstract: Silting modules generalise tilting modules over an arbitrary ring, as well as the support τ-tilting modules over a finite dimensional algebra recently introduced by Adachi, Iyama and Reiten. They are the module-theoretic counterpart of (possibly non-compact) two-term silting complexes, and they are closely related to certain t-structures and co-t-structures in the derived category. Several well known results from tilting theory on existence of approximations and complements can be extended to silting modules. Moreover, it turns out that silting theory provides a suitable context for studying ring epimorphisms and localisations. We will focus on the case of a hereditary ring, where localisations turn out to be parametrised by silting modules. The talk is based on joint work with Frederik Marks and Jorge Vitória.
Chris Beasly, Semi-invariants of quivers and the quantum chiral ring.
Abstract: The notion of mutation for a quiver with potential can be understood as an instance of electric-magnetic duality a la Seiberg. In the physical setting, quiver invariants (mesons) vs semi-invariants (baryons) play distinct roles. I will discuss features of duality for quiver semi-invariants, along with some important open questions.
Frauke Bleher, On the module structure of relative differentials of curves.
Abstract: This talk is about joint work with Ted Chinburg and Aristides Kontogeorgis. Let k be an algebraically closed field of positive characteristic p, and let X be a non-singular projective curve over k. Suppose G is a finite group with non-trivial cyclic Sylow p-subgroups acting faithfully on X. Let M be the space of global sections of the sheaf of relative differentials of X over k. We show that the kG-module structure of M can be determined by so-called Boseck invariants. This generalizes previous results when G is cyclic or when the ramification is tame.
Thomas Brustle, On the non-leaving-face property for associahedra.
Abstract: D. Sleator, R. Tarjan and W. Thurston showed in 1988 that the associahedron satisfies the non-leaving-face property, that is, every geodesic connecting two vertices stays in the minimal face containing both. Recently, C. Ceballos and V. Pilaud established the non-leaving face property for generalized associahedra of types B,C,D, and some exceptional types including E6. The key ingredient in the proofs is a normalization, a sort of projection from the associahedron to a face. We use methods from cluster categories to define such a normalization, which allows us to establish the non-leaving face property at once for all finite cases that are modelled using cluster categories, namely the Dynkin diagrams. This talk reports on joint work with Jean-François Marceau.
Olgur Celikbas, Syzygies and tensor products of modules.
Abstract: click here
Calin Chindris, Locally semi-simple quiver representations.
Abstract: This talk is based on joint work with Dan Kline. When studying quiver representations within the general framework of invariant theory, one is naturally led to consider the class of locally semi-simple representations. Answering a question raised by Victor Kac, we show that a connected quiver is tame if and only if every representation with a semi-simple ring of endomorphisms is locally semi-simple.
Lars Christensen, Stable homology.
Abstract: Tate homology and cohomology originated in the realm of group algebras and evolved through a series of generalizations to the setting of Iwanaga-Gorenstein rings. The cohomological theory has a more far-reaching generalization to the setting of associative rings; it is now called stable cohomology, and it agrees with Tate homology over Iwanaga-Gorenstein rings. On the homological side, the picture has remained opaque. In the talk I will report on recent work---joint with Olgur Celikbas, Li Liang, and Grep Piepmeyer---that clears it up a bit.
Michele Del Zotto, The representation theory of BPS quivers.
Abstract: BPS quivers are a very special (yet humongous) class of quivers with potentials. The study of the representation theory of these algebras leads to very pleasant surprises. In particular, we will discuss a very natural generalization of the concept of Dlab-Ringel defects. These generalized defects are classified by simple Lie groups. The A1 case is the usual Dlab-Ringel defect, whose controlled subcategories correspond to homogenous and non-homogenous tubes in the representation theory of BPS quivers giving rise to tame algebras. All other cases occur for BPS quivers whose representation categories are wild, giving an interesting novel angle on these structures.
Ana Garcia, Syzygies over 2-Calabi-Yau tilted algebras.
Abstract: In this talk we consider the category of Cohen Macaulay modules over 2-Calabi-Yau tilted algebras, CM(B). In this context the objects in the category are the syzygies over the algebra. We study homological properties of this category showing the relation between the Auslander-Reiten translation and the syzygy functor. We explore the connections with the representation dimension of algebras, the Igusa-Todorov functions, and periodic resolutions. For the particular case of Jacobian algebras arising from surfaces without punctures and from the punctured disk (Dynkin type D), we give a geometric description of CM(B) modules and describe the projective resolutions over these algebras.
Matt Garcia, Classifying triangulations of the continuous cluster category using quivers with multiplicity.
Abstract: Continuous cluster categories are triangulated topological categories introduced by Igusa and Todorov to generalize cluster categories of type A. One such continuous cluster category is Cπ, which is associated to the disk model of the hyperbolic plane. Since the hyperbolic plane is a covering space of many surfaces we expect to find that Cπ has the corresponding universal properties for cluster categories of surface type. In this talk we describe the possible triangulated structures on Cπ. To do so we must examine the covering categories of Cπ and their triangulations. Igusa and Todorov have shown that we must insist on at least 2 objects in each isomorphism class of the continuous Frobenius category F that when stabilized gives Cπ. Translated in the language of covering categories, this means we may only consider covers with an even number of sheets. By studying quivers with multiplicity (resulting in non-basic path algebras) and some associated combinatorial data we can classify all the possible triangulations of a continuous cluster category Cπ. In particular, for the case of exactly two objects in each isomorphism class of F we recover the triangulation described by Igusa and Todorov, the triangulation described by Orlov and a third possible triangulation. This is joint work with Kiyoshi Igusa.
Al Garver, Combinatorics of Exceptional Sequences in Type A.
Abstract: Exceptional sequences are certain ordered sequences of quiver representations with applications to noncrossing partitions, factorizations of Coxeter elements, and cluster algebras. We introduce a class of combinatorial objects called strand diagrams that we use to classify exceptional sequences of representations of type A Dynkin quivers. We also use variations of the model to classify c-matrices of type A Dynkin quivers, to interpret exceptional sequences as linear extensions of certain posets, and to give an elementary bijection between exceptional sequences and certain chains in the lattice of noncrossing partitions. This is joint work with Kiyoshi Igusa, Jacob Matherne, and Jonah Ostroff.
Sira Gratz, Cluster algebras of infinite rank as colimits.
Abstract: Assem, Dupont and Schiffler introduced the category of rooted cluster algebras, which has as objects pairs (A,S), where A is a cluster algebra (of possibly infinite rank) and S a distinguished initial seed of A. After explaining these notions we will show that, although the category of rooted cluster algebras does not in general admit colimits, every rooted cluster algebra can be written as a directed colimit of rooted cluster algebras of finite rank. Directed colimits of rooted cluster algebras of Dynkin type A have a geometric interpretation as triangulations of the closed disc with possibly infinitely many marked points on the boundary. They are related to infinite discrete cluster categories of type A, respectively the continuous cluster category of type A as introduced by Igusa and Todorov.
Edward L. Green, Brauer configuration algebras.
Abstract: This is joint work with Sibylle Schroll. I introduce Brauer configurations which are generalizations of Brauer graphs which are generalizations of Brauer trees. Associated to a Brauer configuration is a Brauer configuration algebra. I will explain how to construct a Brauer configuration algebra from its configuration. Brauer configuration algebras are a class of finite dimensional symmetric algebas having special properties. In Sibylle Schroll's talk, a number of the properties will be discussed.
Stephen Hermes, Semi-invariant pictures and maximal Green sequences.
Abstract: Maximal green sequences were introduced by B. Keller to give a combinatorial construction of Donaldson-Thomas invariants. T. Brüstle, G. Dupont and M. Pérotin proved that acyclic tame quivers admit only finitely many maximal green sequences. Building off of recent work by K. Igusa and G. Todorov interpreting green mutation in terms of semi-invariant pictures, we prove two conjectures about maximal green sequences. First, we extend the results of Brüstle-Dupont-Pérotin to quivers mutation equivalent to tame ones. Second, we prove a conjecture of D. Xie that for acyclic quivers, maximal green sequences mutate at the target of any infinite type arrow before the source.
Gustavo Jasso, The d-abelian property of d-cluster-tilting subcategories.
Abstract: d-cluster-tilting subcategories of module categories were introduced by Iyama as the natural playground for higher Auslander--Reiten theory. In this talk, we will review some properties of these subcategories in order to motivate the introduction of d-abelian categories. I will explain how to recover basic results in higher Auslander--Reiten theory from this perspective. The talk contains joint work with Sondre Kvamme and Theo Raedschelders.
Peter Jørgensen, Co-t-structures: the first decade.
Abstract: Co-t-structures were introduced independently by Bondarko and Pauksztello in 2007 as a mirror image of t-structures. The two types of structures are related by a "looking glass principle", in the sense that many of their properties are analogous, but translation from one side to the other is rarely mechanical. We will recall the canonical t-structure in the derived category, show an analogous co-t-structure in the homotopy category, and show an example of a category which is skewed in the sense that it has many t-structures and few co-t-structures, or vice versa. We will also mention the close relation between t-structures, co-t-structures, simple minded collections, and silting objects.
Ryan Kinser, Hopf actions on path algebras.
Abstract: There is a notion of a Hopf algebra acting on another algebra which generalizes the classical notion of a group acting on an algebra by automorphisms. We classify (by explicit formulas) the Hopf actions of Taft algebras on path algebras of loopless, Schurian quivers. Then, we apply this to give some actions of the Frobenius-Lusztig kernel u_q(sl_2) and Drinfeld doubles of Taft algebras on path algebras. This talk is a summary of joint work with Chelsea Walton.
Helmut Lenzing, Preprojective algebras, revisited.
Abstract: In my talk I will first review the various (equivalent) definitions of the preprojective algebra associated to the path algebra of a finite (connected) quiver Q without oriented cycles. The preprojective algebra will be viewed as a positively graded algebra, correspondingly we will investigate its category of graded modules and derive from this its ring-theoretical and homological properties. A key aspect will be to treat graded modules over the preprojective algebra as functors on the mesh category of the quiver Q. The focus will be on the tame hereditary case where the features of the preprojective algebra are especially nice because of its link to the theory of Kleinian (or simple) singularities.
Charles Paquette, Accumulation points of real Schur roots.
Abstract: Let Q be a finite and acyclic quiver and let rep(Q) denote the category of finite dimensional representations of Q over some algebraically closed field. In this talk, we consider the set S_Q of real Schur roots of Q, that is, the set of dimension vectors of the so-called exceptional representations. If we identify dimension vectors up to scaling, we see that when Q is of infinite representation type, the elements of S_Q will give rise to accumulation points (more formally, of accumulation rays). In this talk, we will study these accumulation points. This is closely related to the notion of canonical decomposition of a dimension vector.
Chrysostomos Psaroudakis, Silting objects, t-structures and derived equivalences.
Abstract: Silting objects were introduced by Keller-Vossieck in their study of t-structures for the bounded derived category of representations over Dynkin quivers. They generalize tilting objects and they can be considered more generally in abstract triangulated categories. It is known that compact tilting objects give rise to derived equivalences between rings. Passing to the context of derived equivalences between abelian categories, it is natural to study the associated heart of a tilting object. In this talk, we consider (not necessarily compact) silting objects in the derived category of a Grothendieck category A. We show that the realization functor associated to a silting object is a triangle equivalence between the bounded derived category of the heart and the bounded derived category of A if and only if the object is tilting. Furthermore we discuss the dual notions of cosilting and cotilting. This is joint work with Jorge Vitoria.
Dylan Rupel, Companion cluster algebras to a generalized cluster algebra.
Abstract: Cluster algebras are a class of recursively defined algebras that have found connections to many areas of mathematics. The construction of a cluster algebra proceeds from the initial data of a skew-symmetrizable matrix called the exchange matrix, a collection of elements from a semifield called the coefficients, and a collection of algebraically independent rational functions called the cluster variables. At each step of the recursion each piece of data mutates according to concrete prescribed rules, in particular the cluster variables mutate according to binomial exchange relations whereby a product of a known cluster variable and an unknown cluster variable is given by a sum of two monomials in the remaining known cluster variables. This mutation process can be reduced to a pair of recursions involving integer vectors called c-vectors and g-vectors together with a recursion on collections of polynomials called F-polynomials.
In this talk I will describe a joint work with Tomoki Nakanishi in which we extend these constructions to the case of generalized cluster algebras where the binomial exchange relations above are replaced by polynomial exchange relations with coefficients satisfying a normalization condition and a power condition. Our main results exhibit a relationship between the c-vectors, g-vectors, and F-polynomials of these generalized cluster algebras and their left- and right-companion cluster algebras.
Ralf Schiffler, Cluster-tilted algebras.
Abstract: Let A be a hereditary algebra, T a tilting module and C the corresponding tilted algebra. Thus C is the endomorphism algebra of T. Then the trivial extension B of C by the bimodule Ext^2(DC,C) is called a cluster-tilted algebra. In this expository talk, we shall review the relation between the algebras C and B as well as their module categories.
Marcus Schmidmeier, Crossing and noncrossing partitions of the disk.
Abstract: Partitions of the disk can be categorified as short exact sequences for nilpotent linear operators. Symmetry operations on the disk give insight about Littlewood-Richardson tableaux and the geometry of representation spaces for short exact sequences. This talk is about a joint project with Justyna Kosakowska.
Sibylle Schroll, Multiserial algebras.
Abstract: This talk is based on joint work with Ed Green. We start by defining multiserial and special multiserial algebras and we show their relationship with the Brauer configuration algebras defined in Ed Green's talk. We give examples of (special) multiserial algebras in the literature, and, in particular, we relate to them the symmetric algebras with Jacobson radical cubed zero.
Chelsea Walton, On actions of Hopf algebras.
Abstract: In this talk, I will provide motivation and an overview of recent work on Hopf algebra actions on several classes of algebras. Many examples and several open questions will also be discussed.
He Wang, Resonance varieties, Hilbert series and Chen ranks.
Abstract: Associated to finitely generated group G there are two types of cohomology jump loci: the characteristic varieties, which sit inside the representation variety Hom(G,C*), and the resonance varieties, which sit inside H^1(G,C). The Hilbert series of the Alexander invariants of G and the Chen ranks of G are closely related to these varieties. In this talk, I will introduce these concepts, explain various relations among them, and show two applications. The first application is to certain basis-conjugating subgroups of the automorphism group of a free group, called the McCool groups. The second application is to the picture groups of type An, introduced by Igusa, Orr, Todorov and Weyman. This talk is about joint work with Alex Suciu.
Harold Williams, Cluster algebras as Legendrian knot invariants.
Abstract: We explain a new connection between the theories of cluster algebras and Legendrian knots. The starting point is the recent construction by Shende-Treumann-Zaslow of a new Legendrian knot invariant. This invariant is a moduli space of objects with various interpretations, for example as sheaves, quiver representations, or objects in a Fukaya category. For specific choices of Legendrian knot, it turns out that these invariants recover essentially all examples of cluster varieties related to the group SLn. Cluster varieties are spaces that have been realized to play a wide range of roles in geometry and representation theory in the past decade, and include character varieties of punctured surfaces and strata of algebraic groups and homogeneous spaces. Our main result is that on such a variety the cluster structure, which can be regarded as an atlas of open toric subsets with special properties and transition functions, is in a sense determined by the structure of the set of exact Lagrangian fillings of the associated Legendrian knot. This is joint work in progress with Vivek Shende, David Treumann, and Eric Zaslow.
Jinde Xu, Maximal rigid objects and cluster-tilting objects in 2-CY categories.
Abstract: In this talk, we want to give a solution to the Conjecture II.1.9 of [A. B. Buan, O. Iyama, I. Reiten and J. Scott, Cluster structures for 2-Calabi–Yau categories and unipotent groups, Compos. Math. 145(4) (2009) 1035–1079], which said that any maximal rigid object without loops or 2-cycles in its quiver is a cluster-tilting object in a connected Hom-finite triangulated 2-CY category C.
Yichao Yang, A relation between tilting graphs and cluster-tilting graphs of hereditary algebras.
Abstract: We give the condition of isomorphisms between tilting graphs and cluster-tilting graphs of hereditary algebras. As a conclusion, it is proved that a graph is a skeleton graph of Stasheff polytope if and only if it is both the tilting graph of a hereditary algebra and also the cluster-tilting graph of another hereditary algebra. At last, when comparing such uniformity, the geometric realizations of simplicial complexes associated with tilting modules and cluster-tilting objects are discussed respectively. This is joint work with Professor Fang Li.
Alexandra Zvonareva, Derived Picard groups of selfinjective Nakayama algebras.
Abstract: This talk is based on joint work with Yury Volkov. In this talk I will describe the derived Picard group of a selfinjective Nakayama algebra. I will try to explain how silting mutations, orbit categories and spherical objects are used to obtain this description.
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