Last
updated: May 17, 2016, 11:18 EDT
Conference Titles, Abstracts, and Slides
Emily Barnard, The Canonical Join Complex. 
Abstract: The canonical join representation of an element of a finite lattice is a unique, "smallest" join-representation in terms of join-irreducible elements. In upcoming work of Iyama, Reading, Reiten, and Thomas on the modules of the preprojective algebra corresponding to a finite Weyl group, the authors make many interesting connections to the lattice theory of the weak order on W. Specifically, they characterize the torsion-free class corresponding to an element of W by its canonical joinands in the weak order. In this talk, we study the combinatorics of canonical join representations in certain finite lattices.
Charlie Beil, Nonnoetherian Dimer Algebras and Noncommutative Crepant Resolutions.
Abstract: It is known that every cancellative dimer algebra is a noncommutative crepant resolution (NCCR), and every 3-dimensional affine toric Gorenstein singularity admits an NCCR given by a cancellative dimer algebra. However, dimer algebras which are cancellative are quite rare, and we consider the question: how close are nonnoetherian (homotopy) dimer algebras to being NCCRs? To address this question, I will propose a generalization of NCCRs to nonnoetherian tiled matrix rings. I will then describe a class of dimer algebras, as well as a class of noncommutative blowups, which are nonnoetherian NCCRs.
Andrew Carroll, Minimal Inclusions of Torsion Classes.
Abstract: In an effort to better understand the poset structure of torsion classes of modules over a finite-dimensional associative algebra (ordered by set inclusion), we investigate the following question: given a fixed torsion class T, how can we parameterize the torsion classes that are covers of T? We show that any cover T’ of T is encoded by a unique indecomposable M with the property that every module in T’ has a filtration for which each sub factor either lies in T or is isomorphic to M. Given this result, starting from the torsion class T, we develop a trio of algebraic conditions that identify when an indecomposable module will be responsible for a cover T’ of T in the above sense. In this way, we can realize much of the study of the aforementioned poset structure in terms of combinatorics of sequences of these indecomposable modules. This is joint work with Gordana Todorov and Shijie Zhu.
Sam Dean, Purity in the Model Theory of Sheaves.
Abstract: In general it is not obvious how to approach the model theory of sheaves, since the axioms for sheaves cannot all be written down in a first order way. I will explain how to define the notion of a pp pair for modules over some sheaf of rings. These pp pairs give rise to functors which in turn give rise to a notion of purity for such modules. I will show that this notion of purity coincides with the one already studied by Estrada, Enochs, and Odabasi.
Gabriella D'Este, Auslander–Reiten Sequences and Intuition.
Abstract: At the beginning of Gabriel's paper "Auslander–Reiten sequences and representation-finite algebras" (Lecture Notes in Math. 831, Springer-Verlag, 1–71) he gives an abstract definition of an Auslander–Reiten sequence, ending at an indecomposable non-projective module. However, in contrast to this abstraction, he remarks at the end of his paper that "various specialists like Baustista, Brenner, Butler, Riedtmann ... have hoarded a few hundred examples in their dossiers, thus getting an intuition which no theoretical argument can replace". In the spirit of this remark, I will present some examples to illustrate the role of intuition when investigating Auslander–Reiten sequences.
Ivon Dorado, Representations of Algebraically Equipped Posets.
Abstract: (Joint work with Raymundo Bautista) Algebraically equipped posets are partially ordered sets with an additional structure given by a collection of vector subspaces of an algebra. Some particular cases of these, are generalized equipped posets and p-equipped posets, for a prime number p. We study their categories of representations and establish equivalences with some module categories, from which we obtain some properties, including the existence of almost split sequences.
Eleonore Faber, A McKay Correspondence for Reflection Groups.
Abstract: The classical McKay correspondence relates the geometry of so-called Kleinian surface singularities with the representation theory of finite subgroups of SL(2,C). M. Auslander observed an algebraic version of this correspondence: let G be a finite subgroup of SL(2,k) for a field k whose characteristic does not divide the order of G. The group acts linearly on the polynomial ring S=k[x,y] and then the so-called skew group algebra A=G*S can be seen as an incarnation of the correspondence. In particular, A is isomorphic to the endomorphism ring of S over the corresponding Kleinian surface singularity.
We want to establish an analogous result when G in GL(n,K) is a finite group generated by reflections, assuming that the characteristic of k does not divide the order of the group. Therefore we consider again the skew group algebra A=G*S, where S is the polynomial ring in n variables, and its quotient A/AeA, where e is the idempotent in A corresponding to the trivial representation. With D the coordinate ring of the discriminant of the group action on S, we show that the ring A/AeA is the endomorphism ring of the direct image of the coordinate ring of the associated hyperplane arrangement. In this way one obtains a noncommutative resolution of singularities of that discriminant, a hypersurface that is singular in codimension one. This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls.
Sira Gratz, Thick Subcategories and Non-Crossing Partitions.
Abstract: In joint work with Greg Stevenson we describe the lattice of thick subcategories in the bounded derived category of graded modules over the dual numbers. Despite the representation theory of the dual numbers being rather simple, classifying thick subcategories in this category proves to be combinatorially very interesting. In fact, there are relations to non-crossing partitions of an infinity-gon, thus providing an example of an infinite version of the classification of thick subcategories in bounded derived categories of Dynkin quivers via non-crossing partitions by Ingalls and Thomas.
Edward Green, Convex Algebras.
Abstract: Click here
Stephen Hermes, The No Gap Conjecture: A Proof by Pictures.
Abstract: Maximal green sequences are particular sequences of quiver mutations which have been gaining interest in recent years, due in no small part to their applicability to algebraic combinatorics, representation theory and theoretical physics. The "No Gap Conjecture", formulated by Brüstle-Dupont-Pérotin, states that the set of lengths of maximal green sequences for an acyclic quiver forms an interval of integers. We give a proof of the No Gap Conjecture in tame type using the semi-invariant pictures of Igusa-Orr-Todorov-Weyman. This talk is based off of joint work with K. Igusa, and joint work with Th. Brüstle, K. I. and G. Todorov.
Lutz Hille, Tilting Modules, Maximal Rigid Objects and Symmetric Groups.
Abstract: There are several algebras with only a finite number of tilting modules. For some algebras the classification leads to a bijection with elements in the symmetric group. We consider two such algebras, the first one Bn is a representation finite quotient of the preprojective algebra of type A. Here the classification is elementary. The second algebra is again a quotient of the quiver of the preprojective algebra, it is the Auslander algebra of the truncated polynomial ring. We compare this classification with a rather different category. The objects are triples (M,N,f) consisting of a homomorphism f: M ⟶ N of k[T]/Tn-modules. Vossieck has shown that for fixed modules M and N there is always a generic f, it corresponds to an open orbit in the space of all homomorphisms Hom(M,N).
In this talk I construct the generic homomorphism f explicitly using the classification of tilting modules for the algebras A and B above. Moreover, using a modified additive structure, we can show f is generic precisely when Ext1(f, f) = 0 in this category of triples. Eventually, this allows to classify all rigid objects in the category, where the maximal rigid objects are again in bijection to the elements of the symmetric group.
Michal Hrbek, Silting Modules over Commutative Rings.
Abstract: Silting modules, recently introduced by Angeleri-Marks-Vitória, provide a common generalization of (large) tilting modules over a general ring, and support τ-tilting modules over a finite dimensional algebra. In this joint work with Lidia Angeleri Hügel, we describe all equivalence classes of silting modules over an arbitrary commutative ring R, showing in particular that they correspond bijectively to Thomason subsets of the spectrum (=open sets in the Hochster dual topology on Spec(R)).
Miodrag Iovanov, Uniserial Representations of Finite Dimensional Algebras and Open Questions of Auslander, Reiten and Smalo.
Abstract: We introduce a new combinatorial framework for the study of finite dimensional uniserial representations of algebras and quivers. This is done by interpreting uniserial modules of a quiver as certain rational modules and finding combinatorial bases for them. We find complete invariants for uniserial modules, and give complete answers to several open questions posed by Auslander, Reiten and Smalo in their landmark '95 monograph [ARS], and others raised by the K. Bongartz and B. Huisgen-Zimmerman [BZ] in a series of five initiating papers on this topic. Among other, we answer the isomorphism problem for uniserial modules [ARS, Open Problem 1], completely describe the set of "masts" of a uniserial representation [BZ], characterize algebras of finite uniserial type completely in terms of generators and relations (finite uniserial type problem; [ARS, Open Problem 2]), completely describe and classify algebras that show up as endomorphisms of uniserial modules (endomorphism problem, [BZ]), and give several representation theoretic characterizations of monomial algebras as a consequence [ARS, Open Problem 5]. Time permitting, we show how this combinatorial setup can be extended to other types of finite dimensional representations, and can be applied to other open problems of [ARS].
Karin Marie Jacobsen, Abelian Quotients of Triangulated Categories.
Abstract: We study abelian quotient categories A=T/J, where T is a triangulated category and J is an ideal of T. We give technical criteria for when a representable functor is a quotient functor, and a criterion for when J gives rise to a cluster-tilting subcategory of T. As an application, we show that if T is a finite 2-Calabi-Yau category, then with very few exceptions, J is a cluster-tilting subcategory of T. In particular, this means that if T is a cluster category and X is an object in T, the functor from T to mod End(X) is only full and dense if X is a cluster-tilting object. This is joint work with Benedikte Grimeland.
Frank Marko, Linkage Principle for General Linear and Orthosymplectic Supergroups.
Abstract: We investigate the linkage principle for supergroups in positive characteristic p ≠ 2 using a modification of the approach of Doty. First we report on the linkage principle for the general linear supergroups. Then we discuss the linkage for orthosymplectic supergroups. In the case when the characteristic is zero, the linkage is determined by odd isotropic roots only. However, in the case of positive characteristic, non-isotropic roots play a role, too. We demonstarte this on the supergroup G=SpO(2|1). If char K=0, then the category of G-supermodules is semi-simple (because the root system of SpO(2|1) has no (odd) isotropic root). If char K = p >2, then this category is no longer semi-simple. This is a joint work in progress with Alexandr N. Zubkov.
Charles Paquette, Isotropic Schur Roots.
Abstract: Let Q be an acyclic quiver and let k be an algebraically closed field. A representation of Q is Schur if it has a trivial endomorphism algebra and a Schur root is the dimension vector of such a representation. A Schur root is isotropic if it is a zero of the Tits form of Q. In this talk, we will give a description of the perpendicular category of any isotropic Schur root and describe its (infinitely many) non-isomorphic simple objects. As a consequence, we will get a structural result for the ring of semi-invariants of any isotropic Schur root. Time permitting, we will also see how to construct all isotropic Schur roots of Q, using the action of some braid group. This is joint work with Jerzy Weyman.
Sasha Patotski, Representation Homology and Derived Character Maps.
Abstract: For an associative algebra A, its n-dimensional representation homology H•(A, n) is a natural homological invariant obtained by deriving a (non-abelian) moduli functor Rep_n(A) of n-dimensional representations of A. The homology groups H•(A, n) are hard to compute in general. However, representation homology can be related to more computable invariants such as the cyclic homology HC•(A) of A. There are natural maps
HC•(A) ⟶ H•(A, n) called the derived character maps, extending the usual characters of n-dimensional representations to higher cyclic homology. In this talk, I will explain how to compute derived character maps in a few interesting examples, the main being the symmetric algebra A=Sym(V) of a finite-dimensional vector space V. This is part of joint work with Y.Berest, G.Felder, A.Ramadoss and Th.Willwacher.
David Pauksztello, Silting pairs and stability conditions.
Abstract: This will be a report on joint work with Nathan Broomhead and David Ploog. The notion of a silting object is a generalisation of tilting object, which turns up in the context of derived equivalences. Silting objects come equipped with a rich combinatorial structure, which is related to mutation in cluster theory. In this talk, we shall discuss a CW complex arising from silting objects and their connection to the space of Bridgeland stability conditions for certain algebraic examples.
Mike Prest, Interpretations Between Definable Categories and Exact Functors Between Small Abelian Categories.
Abstract: There is a natural anti-equivalence between, on the one hand, the category of small abelian categories with exact functors and, on the other, the category of definable additive categories with interpretation functors. I will describe this, some of its significance, a variety of examples and some recent results. For background and some references, see: M. Prest, Abelian categories and definable additive categories, arXiv:1202.0426.
Fan Qin, Triangular Bases of Quantum Cluster Algebras and Monoidal Categorification.
Abstract: In this talk, I will give an introduction of the cluster algebras arising from the representation theory of quantum affine algebras. I will construct a common triangular basis of such a cluster algebra, which is parametrized by the tropical points. The construction implies the monoidal categorification conjecture of Hernandez-Leclerc and, in this situation, the Fock-Goncharov conjecture.
Nathan Reading, Lattice Congruences of the Weak Order on a Finite Weyl Group.
Abstract: Finite Weyl groups provide the combinatorial underpinning of several different algebraic and geometric structures. As the name "Weyl group" suggests, typically the relevant combinatorics is the combinatorics of the group structure on the Weyl group. However, a Weyl group is also a lattice, when ordered by the weak order. The combinatorics of this alternate algebraic structure turns out to be just as interesting as the group structure, both intrinsically and in its applications to other algebraic structures. Specifically, the combinatorics of congruences on the weak order (and the corresponding quotients of the weak order) seems to "know" a lot about topics as varied as cluster algebras, combinatorial Hopf algebras, generalized associahedra, and preprojective and hereditary algebras.
This talk will mostly be an exposition of the combinatorics and geometry of lattice congruences of the weak order, but by way of motivation, I will mention some results of joint work with Iyama, Reiten, and Thomas relating to preprojective and hereditary algebras of Dynkin type.
Jeremy Russell, An Adjoint to the Auslander-Gruson-Jensen Functor.
Abstract: The Auslander-Gruson-Jensen functor is an exact functor from the category of finitely presented functors fp(Mod-R,Ab) into the category of functors (R-mod,Ab), mapping representable functors to tensor functors. In this talk, the Auslander-Gruson-Jensen is shown to admit an adjoint. This adjoint functor is shown to be fully faithful and hence embeds the functor category (R-mod,Ab) into the category fp(Mod-R,Ab).
Ralf Schiffler, Snake Graphs and Continued Fractions.
Abstract: Snake graphs arise naturally in the theory of cluster algebras (of surface type). Each generator of the cluster algebra is a Laurent polynomial that can be computed by a formula whose terms are parametrized by the perfect matchings of a snake graph. Forgetting the cluster algebras, one can also study snake graphs as abstract combinatorial objects. It turns out that the set of all snake graphs is in bijection with the set of all positive continued fractions, in such a way that the numerator of the continued fraction is equal to the number of perfect matchings of the snake graph. This gives a combinatorial interpretation of continued fractions and a new perspective on cluster algebras. This is joint work with Ilke Canakci.
Khrystyna Serhiyenko, Reflections of Local Slices in Cluster-Tilted Algebras.
Abstract: In the work of Assem-Brüstle-Schiffler, it was shown that cluster-tilted algebras are relation extensions of tilted algebras. Moreover, there is a close connection between these algebras given in terms of slices and local slices. For a cluster-tilted algebra B of tree type, the authors introduced an operation called reflection on the set of tilted algebras whose relation extension is B. Using local slices, we generalize this notion of reflections to arbitrary cluster-tilted algebras, and show that any two tilted algebras are connected by a sequence of reflections and coreflections. We also classify all transjective modules over a cluster-tilted algebra that do not lie on local slices. As a result, we obtain that the number of such modules is always finite. This is a joint work with Assem and Schiffler.
Alexander Sistko, On the Toeplitz-Jacobson Algebra and Direct Finiteness.
Abstract: We discuss the representation theory of the Jacobson algebra, a Leavitt path algebra with natural relevance to Kaplansky's Direct Finiteness Conjecture. In particular, we obtain some new structural and homological results, completely determine one-sided ideals, and outline a potential module-theoretic approach to the Direct Finiteness Conjecture. We reobtain results from the existing literature, from the specific work of V. Bavula's 2010 paper, A. Alahmedi et. al.'s 2013 paper, and the theory of Leavitt path algebras developed by, for instance, G. Abrams and P. Ara. Joint work with Miodrag Iovanov.
Alexander Slávik, Very Flat, Locally Very Flat, and Contraadjusted Modules.
Abstract: Very flat and contraadjusted modules arise in the context of contraherent cosheaves, i.e. certain objects dual to quasicoherent sheaves introduced by Positselski [arXiv:1209.2995]. We study the approximation properties of these classes in the setting of Noetherian rings, showing that if the class of all very flat modules is covering or the class of all contraadjusted modules enveloping, then the spectrum of the ring is finite. Locally very flat modules, an analogue to flat Mittag-Leffler modules, are introduced and shown not to form a precovering class unless the spectrum is finite. Joint work with Jan Trlifaj.
Sverre Smalø, My Cooperation with Maurice: Some Old Results.
Abstract:
Hugh Thomas, Stability Conditions for Preprojective Algebras and Reading's Shards.
Abstract: I will recall the notion of stability conditions in the sense of geometric invariant theory (as carried into the theory of finite-dimensional algebras by King). In order to explain the structure of the semi-stable subcategories of finite type preprojective algebras, I will introduce Reading's combinatorics of shards, defined based on the reflection arrangement of the corresponding type. This project is joint work with David Speyer, and also draws on a previous joint project with Nathan Reading, Idun Reiten, and Osamu Iyama.
Hipolito Treffinger, τ-Tilting Theory and τ-Slices.
Abstract: One of the main results in tilting theory is the Tilting Theorem, proved by Brenner and Butler in the early eighties. In this talk, we state a generalization of this theorem in the context of τ-tilting theory. Our theorem says that, for an arbitrary support τ-tilting A-module M, the classical functors given in the Brenner-Butler theorem induce equivalences of subcategories if and only if the Auslander-Reiten translation of M in two different module categories are isomorphic. Afterwards we introduce the notion of τ-slices. We show that τ-slices are examples of support τ-tilting modules that respect the condition given before. Also we state some results connecting τ-slices with the tilted algebras defined by Happel and Ringel.
Jan Trlifaj, Some Representation Theory Arising from Set-Theoretic Homological Algebra.
Abstract: Click here
Jose Velez-Marulanda, String and Band Complexes over Certain Algebra of Dihedral Type.
Abstract: We give a combinatorial description of a family of indecomposable objects in the triangulated category of perfect complexes $\mathcal{K}^b(\textup{proj-}\A)$, where $\A$ is a certain algebra of dihedral type (as introduced by K. Erdmann) with three simple modules. We then discuss the shape of the corresponding components of the Auslander-Reiten quiver of $\mathcal{K}^b(\textup{proj-}\A)$ containing these objects. This is a joint-work with Hern\'an Giraldo which was published online in Algebr. Represent. Theor. during November 2015.
Milen Yakimov, Noncommutative Factorial Algebras.
Abstract: This talk with be an exposition of the theory of noncommutative noetherian Unique Factorization Domains and their applications, initiated by Chatters more than 30 years ago. Over these years, large classes of such algebras were found in many settings: universal enveloping algebras of solvable Lie algebras in the 80s, subalgebras of quantum groups in the 90's, and quantum cluster algebras in the last 10 years. Nowadays, the majority of quantum cluster algebras that play a role in Lie theory fall in this class of algebras.
Dan Zacharia, A Representation Theoretic Approach to Exceptional Sheaves on the Projective n-Space.
Abstract: Click here
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Abstract: The canonical join representation of an element of a finite lattice is a unique, "smallest" join-representation in terms of join-irreducible elements. In upcoming work of Iyama, Reading, Reiten, and Thomas on the modules of the preprojective algebra corresponding to a finite Weyl group, the authors make many interesting connections to the lattice theory of the weak order on W. Specifically, they characterize the torsion-free class corresponding to an element of W by its canonical joinands in the weak order. In this talk, we study the combinatorics of canonical join representations in certain finite lattices.
Charlie Beil, Nonnoetherian Dimer Algebras and Noncommutative Crepant Resolutions.
Abstract: It is known that every cancellative dimer algebra is a noncommutative crepant resolution (NCCR), and every 3-dimensional affine toric Gorenstein singularity admits an NCCR given by a cancellative dimer algebra. However, dimer algebras which are cancellative are quite rare, and we consider the question: how close are nonnoetherian (homotopy) dimer algebras to being NCCRs? To address this question, I will propose a generalization of NCCRs to nonnoetherian tiled matrix rings. I will then describe a class of dimer algebras, as well as a class of noncommutative blowups, which are nonnoetherian NCCRs.
Andrew Carroll, Minimal Inclusions of Torsion Classes.
Abstract: In an effort to better understand the poset structure of torsion classes of modules over a finite-dimensional associative algebra (ordered by set inclusion), we investigate the following question: given a fixed torsion class T, how can we parameterize the torsion classes that are covers of T? We show that any cover T’ of T is encoded by a unique indecomposable M with the property that every module in T’ has a filtration for which each sub factor either lies in T or is isomorphic to M. Given this result, starting from the torsion class T, we develop a trio of algebraic conditions that identify when an indecomposable module will be responsible for a cover T’ of T in the above sense. In this way, we can realize much of the study of the aforementioned poset structure in terms of combinatorics of sequences of these indecomposable modules. This is joint work with Gordana Todorov and Shijie Zhu.
Sam Dean, Purity in the Model Theory of Sheaves.
Abstract: In general it is not obvious how to approach the model theory of sheaves, since the axioms for sheaves cannot all be written down in a first order way. I will explain how to define the notion of a pp pair for modules over some sheaf of rings. These pp pairs give rise to functors which in turn give rise to a notion of purity for such modules. I will show that this notion of purity coincides with the one already studied by Estrada, Enochs, and Odabasi.
Gabriella D'Este, Auslander–Reiten Sequences and Intuition.
Abstract: At the beginning of Gabriel's paper "Auslander–Reiten sequences and representation-finite algebras" (Lecture Notes in Math. 831, Springer-Verlag, 1–71) he gives an abstract definition of an Auslander–Reiten sequence, ending at an indecomposable non-projective module. However, in contrast to this abstraction, he remarks at the end of his paper that "various specialists like Baustista, Brenner, Butler, Riedtmann ... have hoarded a few hundred examples in their dossiers, thus getting an intuition which no theoretical argument can replace". In the spirit of this remark, I will present some examples to illustrate the role of intuition when investigating Auslander–Reiten sequences.
Ivon Dorado, Representations of Algebraically Equipped Posets.
Abstract: (Joint work with Raymundo Bautista) Algebraically equipped posets are partially ordered sets with an additional structure given by a collection of vector subspaces of an algebra. Some particular cases of these, are generalized equipped posets and p-equipped posets, for a prime number p. We study their categories of representations and establish equivalences with some module categories, from which we obtain some properties, including the existence of almost split sequences.
Eleonore Faber, A McKay Correspondence for Reflection Groups.
Abstract: The classical McKay correspondence relates the geometry of so-called Kleinian surface singularities with the representation theory of finite subgroups of SL(2,C). M. Auslander observed an algebraic version of this correspondence: let G be a finite subgroup of SL(2,k) for a field k whose characteristic does not divide the order of G. The group acts linearly on the polynomial ring S=k[x,y] and then the so-called skew group algebra A=G*S can be seen as an incarnation of the correspondence. In particular, A is isomorphic to the endomorphism ring of S over the corresponding Kleinian surface singularity.
We want to establish an analogous result when G in GL(n,K) is a finite group generated by reflections, assuming that the characteristic of k does not divide the order of the group. Therefore we consider again the skew group algebra A=G*S, where S is the polynomial ring in n variables, and its quotient A/AeA, where e is the idempotent in A corresponding to the trivial representation. With D the coordinate ring of the discriminant of the group action on S, we show that the ring A/AeA is the endomorphism ring of the direct image of the coordinate ring of the associated hyperplane arrangement. In this way one obtains a noncommutative resolution of singularities of that discriminant, a hypersurface that is singular in codimension one. This is joint work with Ragnar-Olaf Buchweitz and Colin Ingalls.
Sira Gratz, Thick Subcategories and Non-Crossing Partitions.
Abstract: In joint work with Greg Stevenson we describe the lattice of thick subcategories in the bounded derived category of graded modules over the dual numbers. Despite the representation theory of the dual numbers being rather simple, classifying thick subcategories in this category proves to be combinatorially very interesting. In fact, there are relations to non-crossing partitions of an infinity-gon, thus providing an example of an infinite version of the classification of thick subcategories in bounded derived categories of Dynkin quivers via non-crossing partitions by Ingalls and Thomas.
Edward Green, Convex Algebras.
Abstract: Click here
Stephen Hermes, The No Gap Conjecture: A Proof by Pictures.
Abstract: Maximal green sequences are particular sequences of quiver mutations which have been gaining interest in recent years, due in no small part to their applicability to algebraic combinatorics, representation theory and theoretical physics. The "No Gap Conjecture", formulated by Brüstle-Dupont-Pérotin, states that the set of lengths of maximal green sequences for an acyclic quiver forms an interval of integers. We give a proof of the No Gap Conjecture in tame type using the semi-invariant pictures of Igusa-Orr-Todorov-Weyman. This talk is based off of joint work with K. Igusa, and joint work with Th. Brüstle, K. I. and G. Todorov.
Lutz Hille, Tilting Modules, Maximal Rigid Objects and Symmetric Groups.
Abstract: There are several algebras with only a finite number of tilting modules. For some algebras the classification leads to a bijection with elements in the symmetric group. We consider two such algebras, the first one Bn is a representation finite quotient of the preprojective algebra of type A. Here the classification is elementary. The second algebra is again a quotient of the quiver of the preprojective algebra, it is the Auslander algebra of the truncated polynomial ring. We compare this classification with a rather different category. The objects are triples (M,N,f) consisting of a homomorphism f: M ⟶ N of k[T]/Tn-modules. Vossieck has shown that for fixed modules M and N there is always a generic f, it corresponds to an open orbit in the space of all homomorphisms Hom(M,N).
In this talk I construct the generic homomorphism f explicitly using the classification of tilting modules for the algebras A and B above. Moreover, using a modified additive structure, we can show f is generic precisely when Ext1(f, f) = 0 in this category of triples. Eventually, this allows to classify all rigid objects in the category, where the maximal rigid objects are again in bijection to the elements of the symmetric group.
Michal Hrbek, Silting Modules over Commutative Rings.
Abstract: Silting modules, recently introduced by Angeleri-Marks-Vitória, provide a common generalization of (large) tilting modules over a general ring, and support τ-tilting modules over a finite dimensional algebra. In this joint work with Lidia Angeleri Hügel, we describe all equivalence classes of silting modules over an arbitrary commutative ring R, showing in particular that they correspond bijectively to Thomason subsets of the spectrum (=open sets in the Hochster dual topology on Spec(R)).
Miodrag Iovanov, Uniserial Representations of Finite Dimensional Algebras and Open Questions of Auslander, Reiten and Smalo.
Abstract: We introduce a new combinatorial framework for the study of finite dimensional uniserial representations of algebras and quivers. This is done by interpreting uniserial modules of a quiver as certain rational modules and finding combinatorial bases for them. We find complete invariants for uniserial modules, and give complete answers to several open questions posed by Auslander, Reiten and Smalo in their landmark '95 monograph [ARS], and others raised by the K. Bongartz and B. Huisgen-Zimmerman [BZ] in a series of five initiating papers on this topic. Among other, we answer the isomorphism problem for uniserial modules [ARS, Open Problem 1], completely describe the set of "masts" of a uniserial representation [BZ], characterize algebras of finite uniserial type completely in terms of generators and relations (finite uniserial type problem; [ARS, Open Problem 2]), completely describe and classify algebras that show up as endomorphisms of uniserial modules (endomorphism problem, [BZ]), and give several representation theoretic characterizations of monomial algebras as a consequence [ARS, Open Problem 5]. Time permitting, we show how this combinatorial setup can be extended to other types of finite dimensional representations, and can be applied to other open problems of [ARS].
Karin Marie Jacobsen, Abelian Quotients of Triangulated Categories.
Abstract: We study abelian quotient categories A=T/J, where T is a triangulated category and J is an ideal of T. We give technical criteria for when a representable functor is a quotient functor, and a criterion for when J gives rise to a cluster-tilting subcategory of T. As an application, we show that if T is a finite 2-Calabi-Yau category, then with very few exceptions, J is a cluster-tilting subcategory of T. In particular, this means that if T is a cluster category and X is an object in T, the functor from T to mod End(X) is only full and dense if X is a cluster-tilting object. This is joint work with Benedikte Grimeland.
Frank Marko, Linkage Principle for General Linear and Orthosymplectic Supergroups.
Abstract: We investigate the linkage principle for supergroups in positive characteristic p ≠ 2 using a modification of the approach of Doty. First we report on the linkage principle for the general linear supergroups. Then we discuss the linkage for orthosymplectic supergroups. In the case when the characteristic is zero, the linkage is determined by odd isotropic roots only. However, in the case of positive characteristic, non-isotropic roots play a role, too. We demonstarte this on the supergroup G=SpO(2|1). If char K=0, then the category of G-supermodules is semi-simple (because the root system of SpO(2|1) has no (odd) isotropic root). If char K = p >2, then this category is no longer semi-simple. This is a joint work in progress with Alexandr N. Zubkov.
Charles Paquette, Isotropic Schur Roots.
Abstract: Let Q be an acyclic quiver and let k be an algebraically closed field. A representation of Q is Schur if it has a trivial endomorphism algebra and a Schur root is the dimension vector of such a representation. A Schur root is isotropic if it is a zero of the Tits form of Q. In this talk, we will give a description of the perpendicular category of any isotropic Schur root and describe its (infinitely many) non-isomorphic simple objects. As a consequence, we will get a structural result for the ring of semi-invariants of any isotropic Schur root. Time permitting, we will also see how to construct all isotropic Schur roots of Q, using the action of some braid group. This is joint work with Jerzy Weyman.
Sasha Patotski, Representation Homology and Derived Character Maps.
Abstract: For an associative algebra A, its n-dimensional representation homology H•(A, n) is a natural homological invariant obtained by deriving a (non-abelian) moduli functor Rep_n(A) of n-dimensional representations of A. The homology groups H•(A, n) are hard to compute in general. However, representation homology can be related to more computable invariants such as the cyclic homology HC•(A) of A. There are natural maps
HC•(A) ⟶ H•(A, n) called the derived character maps, extending the usual characters of n-dimensional representations to higher cyclic homology. In this talk, I will explain how to compute derived character maps in a few interesting examples, the main being the symmetric algebra A=Sym(V) of a finite-dimensional vector space V. This is part of joint work with Y.Berest, G.Felder, A.Ramadoss and Th.Willwacher.
David Pauksztello, Silting pairs and stability conditions.
Abstract: This will be a report on joint work with Nathan Broomhead and David Ploog. The notion of a silting object is a generalisation of tilting object, which turns up in the context of derived equivalences. Silting objects come equipped with a rich combinatorial structure, which is related to mutation in cluster theory. In this talk, we shall discuss a CW complex arising from silting objects and their connection to the space of Bridgeland stability conditions for certain algebraic examples.
Mike Prest, Interpretations Between Definable Categories and Exact Functors Between Small Abelian Categories.
Abstract: There is a natural anti-equivalence between, on the one hand, the category of small abelian categories with exact functors and, on the other, the category of definable additive categories with interpretation functors. I will describe this, some of its significance, a variety of examples and some recent results. For background and some references, see: M. Prest, Abelian categories and definable additive categories, arXiv:1202.0426.
Fan Qin, Triangular Bases of Quantum Cluster Algebras and Monoidal Categorification.
Abstract: In this talk, I will give an introduction of the cluster algebras arising from the representation theory of quantum affine algebras. I will construct a common triangular basis of such a cluster algebra, which is parametrized by the tropical points. The construction implies the monoidal categorification conjecture of Hernandez-Leclerc and, in this situation, the Fock-Goncharov conjecture.
Nathan Reading, Lattice Congruences of the Weak Order on a Finite Weyl Group.
Abstract: Finite Weyl groups provide the combinatorial underpinning of several different algebraic and geometric structures. As the name "Weyl group" suggests, typically the relevant combinatorics is the combinatorics of the group structure on the Weyl group. However, a Weyl group is also a lattice, when ordered by the weak order. The combinatorics of this alternate algebraic structure turns out to be just as interesting as the group structure, both intrinsically and in its applications to other algebraic structures. Specifically, the combinatorics of congruences on the weak order (and the corresponding quotients of the weak order) seems to "know" a lot about topics as varied as cluster algebras, combinatorial Hopf algebras, generalized associahedra, and preprojective and hereditary algebras.
This talk will mostly be an exposition of the combinatorics and geometry of lattice congruences of the weak order, but by way of motivation, I will mention some results of joint work with Iyama, Reiten, and Thomas relating to preprojective and hereditary algebras of Dynkin type.
Jeremy Russell, An Adjoint to the Auslander-Gruson-Jensen Functor.
Abstract: The Auslander-Gruson-Jensen functor is an exact functor from the category of finitely presented functors fp(Mod-R,Ab) into the category of functors (R-mod,Ab), mapping representable functors to tensor functors. In this talk, the Auslander-Gruson-Jensen is shown to admit an adjoint. This adjoint functor is shown to be fully faithful and hence embeds the functor category (R-mod,Ab) into the category fp(Mod-R,Ab).
Ralf Schiffler, Snake Graphs and Continued Fractions.
Abstract: Snake graphs arise naturally in the theory of cluster algebras (of surface type). Each generator of the cluster algebra is a Laurent polynomial that can be computed by a formula whose terms are parametrized by the perfect matchings of a snake graph. Forgetting the cluster algebras, one can also study snake graphs as abstract combinatorial objects. It turns out that the set of all snake graphs is in bijection with the set of all positive continued fractions, in such a way that the numerator of the continued fraction is equal to the number of perfect matchings of the snake graph. This gives a combinatorial interpretation of continued fractions and a new perspective on cluster algebras. This is joint work with Ilke Canakci.
Khrystyna Serhiyenko, Reflections of Local Slices in Cluster-Tilted Algebras.
Abstract: In the work of Assem-Brüstle-Schiffler, it was shown that cluster-tilted algebras are relation extensions of tilted algebras. Moreover, there is a close connection between these algebras given in terms of slices and local slices. For a cluster-tilted algebra B of tree type, the authors introduced an operation called reflection on the set of tilted algebras whose relation extension is B. Using local slices, we generalize this notion of reflections to arbitrary cluster-tilted algebras, and show that any two tilted algebras are connected by a sequence of reflections and coreflections. We also classify all transjective modules over a cluster-tilted algebra that do not lie on local slices. As a result, we obtain that the number of such modules is always finite. This is a joint work with Assem and Schiffler.
Alexander Sistko, On the Toeplitz-Jacobson Algebra and Direct Finiteness.
Abstract: We discuss the representation theory of the Jacobson algebra, a Leavitt path algebra with natural relevance to Kaplansky's Direct Finiteness Conjecture. In particular, we obtain some new structural and homological results, completely determine one-sided ideals, and outline a potential module-theoretic approach to the Direct Finiteness Conjecture. We reobtain results from the existing literature, from the specific work of V. Bavula's 2010 paper, A. Alahmedi et. al.'s 2013 paper, and the theory of Leavitt path algebras developed by, for instance, G. Abrams and P. Ara. Joint work with Miodrag Iovanov.
Alexander Slávik, Very Flat, Locally Very Flat, and Contraadjusted Modules.
Abstract: Very flat and contraadjusted modules arise in the context of contraherent cosheaves, i.e. certain objects dual to quasicoherent sheaves introduced by Positselski [arXiv:1209.2995]. We study the approximation properties of these classes in the setting of Noetherian rings, showing that if the class of all very flat modules is covering or the class of all contraadjusted modules enveloping, then the spectrum of the ring is finite. Locally very flat modules, an analogue to flat Mittag-Leffler modules, are introduced and shown not to form a precovering class unless the spectrum is finite. Joint work with Jan Trlifaj.
Sverre Smalø, My Cooperation with Maurice: Some Old Results.
Abstract:
Hugh Thomas, Stability Conditions for Preprojective Algebras and Reading's Shards.
Abstract: I will recall the notion of stability conditions in the sense of geometric invariant theory (as carried into the theory of finite-dimensional algebras by King). In order to explain the structure of the semi-stable subcategories of finite type preprojective algebras, I will introduce Reading's combinatorics of shards, defined based on the reflection arrangement of the corresponding type. This project is joint work with David Speyer, and also draws on a previous joint project with Nathan Reading, Idun Reiten, and Osamu Iyama.
Hipolito Treffinger, τ-Tilting Theory and τ-Slices.
Abstract: One of the main results in tilting theory is the Tilting Theorem, proved by Brenner and Butler in the early eighties. In this talk, we state a generalization of this theorem in the context of τ-tilting theory. Our theorem says that, for an arbitrary support τ-tilting A-module M, the classical functors given in the Brenner-Butler theorem induce equivalences of subcategories if and only if the Auslander-Reiten translation of M in two different module categories are isomorphic. Afterwards we introduce the notion of τ-slices. We show that τ-slices are examples of support τ-tilting modules that respect the condition given before. Also we state some results connecting τ-slices with the tilted algebras defined by Happel and Ringel.
Jan Trlifaj, Some Representation Theory Arising from Set-Theoretic Homological Algebra.
Abstract: Click here
Jose Velez-Marulanda, String and Band Complexes over Certain Algebra of Dihedral Type.
Abstract: We give a combinatorial description of a family of indecomposable objects in the triangulated category of perfect complexes $\mathcal{K}^b(\textup{proj-}\A)$, where $\A$ is a certain algebra of dihedral type (as introduced by K. Erdmann) with three simple modules. We then discuss the shape of the corresponding components of the Auslander-Reiten quiver of $\mathcal{K}^b(\textup{proj-}\A)$ containing these objects. This is a joint-work with Hern\'an Giraldo which was published online in Algebr. Represent. Theor. during November 2015.
Milen Yakimov, Noncommutative Factorial Algebras.
Abstract: This talk with be an exposition of the theory of noncommutative noetherian Unique Factorization Domains and their applications, initiated by Chatters more than 30 years ago. Over these years, large classes of such algebras were found in many settings: universal enveloping algebras of solvable Lie algebras in the 80s, subalgebras of quantum groups in the 90's, and quantum cluster algebras in the last 10 years. Nowadays, the majority of quantum cluster algebras that play a role in Lie theory fall in this class of algebras.
Dan Zacharia, A Representation Theoretic Approach to Exceptional Sheaves on the Projective n-Space.
Abstract: Click here
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