| Northeastern University Mathematics Department | ||
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Maurice Auslander Distinguished Lectures and International Conference*
April 23 - 28, 2009 Woods Hole, Massachusetts, USA Speck Auditorium, Rowe Laboratory Building, (formerly Whitman), MBL | ||
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Poster |
| Edward L. Green: Resolutions, finite generation of Ext, and generalizations of Koszul Abstract: Let K be a field and let R be a not necessarily finite dimensional K-algebra. For most of the talk we will assume that R = R0 &oplus R1 &oplus R2 &oplus ... is positively Z-graded with R0 a semisimple K-algebra, R1 of finite length over R0, and R generated in degrees 0 and 1. The Ext-algebra of R is E(R) = ExtR(R0,R0). The goal of the talk is show some strong relationships between the structure of a minimal graded projective resolution of R0 as an R-module to the ring structure of E(R). For example, how does finite generation of E(R) relate to the structure of a minimal graded projective resolution of R0? Moreover, we show that in some special cases, the close connections between resolutions and Ext-algebras lead to results that relate certain R-modules to certain E(R)-modules. Koszul algebras and Koszul duality is possibly the most well known such relationship. I will very briefly review this and spend the remainder of the talk surveying some generalizations of this, including d-Koszul, 2-d-Koszul, almost Koszul, and T-Koszul. |
Dieter Happel: Piecewise hereditary algebras Abstract: A finite dimensional algebra A over an algebraically closed field k is said to be piecewise hereditary if it is derived equivalent to a hereditary, abelian category. The hereditary categories occuring in this situation are up to derived equivalence either module categories over finite dimensional hereditary algebras or coherent sheaves over weighted projective lines in the sense of Geigle and Lenzing. In this talk we will discuss homological properties of piecewise hereditary algebras. In particluar we will state a homological characterization in terms of the strong global dimension of A, which was obtained in joint work with Dan Zacharia. Also we will discuss special types of Nakayama algebras and give a list when they are piecewise hereditary. In this situation a major tool is played by the Coxeter polynomial associated with the algebra A. |
Claire Amiot: A generalization of cluster categories Abstract: In 2005 Buan, Marsh, Reineke, Reiten and Todorov have introduced the cluster category associated with an acyclic quiver. Their aim was to categorify acyclic cluster algebras. In this talk I will define these categories and show how it is possible to generalize this construction replacing an acyclic quiver by a finite dimensional algebra of global dimension 2, or by a quiver with potential. |
Lidia Angeleri Hugel: Homological Dimensions in Cotorsion Pairs Abstract Abstract: Two classes A and B of modules over a ring R are said to form a cotorsion pair (A,B) if A=Ker Ext1R( ,B) and B=Ker Ext1R(A, ). We will investigate relative homological dimensions in cotorsion pairs. This will be applied to study the big and the little finitistic dimension of R. The talk relies on joint work with Jan Trlifa j and Octavio Mendoza. |
Frauke Bleher: Finiteness theorems for deformations of complexes Abstract: This talk is about joint work with Ted Chinburg on a generalization to complexes of Mazur's deformation theory for modules for a profinite group. A new question arises when deforming complexes: Can the versal deformation be specified by a finite amount of linear algebra information with coefficients in the versal deformation ring? In this talk, I will describe some evidence that this is the case when the complex arises from arithmetic in a suitable sense. |
Thomas Brustle: From Christoffel words to Markoff numbers Abstract: For a pair (a,b) of relatively prime natural numbers, the Christoffel word C(a,b) is defined by the path with integral vertices which is closest to the line segment from (0,0) to (a,b). Viewing this line segment as an arc in the once-punctured torus, we define a J-module M(a,b) for each Christoffel word. Here J is the Jacobian algebra of the once-punctured torus. We show that one obtains the Markoff number associated with C(a,b) by counting submodules of M(a,b). |
Aslak Bakke Buan: Cluster structures from tubes Abstract: A tube is a uniserial abelian category which e.g. can be realized as the nilpotent representations over a quiver which is an oriented cycle. We study the cluster category of a tube. It does not have tilting objects. However, we show that the set of maximal rigid object has a nice combinatorial structure, namely that of a cluster algebra of type B. We point out that this is a special case of a more general construction of cluster structures from sets of maximal rigid objects in cluster categories. Based on joint work with Marsh and Vatne |
Giovanni Cerulli Irelli: Euler-Poincare' characteristic of quiver grassamannians associated with thinly graded modules Abstract: We discuss a class of quiver representations called thinly graded. For some thinly graded representaions we consider the associated quiver grassmannians and we produce a cellular decomposition of them. As a consequence of this, we give a technique to compute combinatorially their Euler--Poincare' characteristic. The generalization of this construction to every thinly graded module is a work in progress with Francesco Esposito. |
Calin Chindris: Cluster fans for quivers Abstract: The cluster fan of a quiver without oriented cycles is the (possibly infinite) fan on the set of almost positive real Schur roots whose cones are generated by the so-called compatible subsets. In this talk, I will present a description of the cluster fan of a quiver in terms of certain stability conditions of the quiver in question. I will also explain how our results can be used to derive the Igusa-Orr-Todorov-Weyman's description of the (N-1)-skeleton of the cluster fan of a Dynkin quiver with N vertices. | Gabriella D'Este: Partial tilting modules and complexes Abstract: We investigate some "combinatorial" (and sometimes "topological") properties of rather big proper partial tilting or cotilting modules with respect to - classes of modules with more or less or left good closure properties; - right bounded complexes (of projective modules) with morphisms up to homotopy; - possible embeddings into bimodules with prescribed left or right injective support. |
Ernst Dieterich: Real division algebras, restricted quiver representations, and Euclidean configurations Abstract Abstract: A real division algebra is a non-zero real vector space A, endowed with an R-bilinear multiplication A × A &rarr A, (x, y) &rarr xy such that for each a &isin A \ {0} both linear operators L: A &rarr A, x &rarr ax and R: A &rarr A, x &rarr xa are invertible. A famous theorem of Hopf (1940) and Bott, Milnor, Kervaire(1958) states that every finite dimensional real division algebra has dimension 1, 2, 4, or 8. The problem of classifying all finite dimensional real division algebras up to isomorphism is solved in the dimensions 1 and 2, but only partially solved in the dimensions 4 and 8. While these partial solutions historically emerged from diverse approaches and techniques, developed by numerous specialists during half a century, they are at present being understood to follow a common pattern that "locally" relates real division algebras in a first step to modules over an associative algebra, and in a second step to configurations in a Euclidean space, in terms of equivalences of categories. Thus unforeseen connections between non-associative algebras, modules over associative algebras, and Euclidean geometry emerge from the attempt to classify real division algebras. In return, these connections actually enable the classification of certain types of real division algebras. I will explain this common pattern and exemplify it by revisiting some of the known partial classifications of real division algebras under its unifying perspective. |
Audrey Doughty: The Auslander and Ringel-Tachikawa Theorem for submodules embeddings Abstract: Auslander and Ringel-Tachikawa have shown that for an artinian ring R of finite representation type, every R-module is the direct sum of finitely generated indecomposable R-modules. In this talk, we will adapt this result to finite representation type full subcategories of the module category of an artinian ring which are closed under subobjects and direct sums and contain all projective modules. In particular, the results in this paper hold for subspace representations of a poset, in case this category is of finite representation type. |
Christof Geiss: Tubular Cluster Algebras Abstract: The mutation finite cluster algebras associated to an elliptic root system of type D4(1,1), E6(1,1), E7(1,1) resp. E8(1,1) are categorified by the cluster category of coh X where X is a weighted projective line of weight type (2,2,2,2), (3,3,3), (4,4,2) resp. (6,3,2). These are precisely the (tame) tubular cases. We show that via a cluster character the cluster variables are in bijection with the corresponding Schur roots. In case D˜4(1,1) we present to this end an explicit bijection between Schur roots and tagged arcs on the 2-sphere with 4 punctures associated to this situation by Fomin-Shapiro-Thurston. (joint work with Michael Barot) |
Christof Geiss: Categorification of the Chamber Ansatz Abstract: for an adaptable element w of the Weyl group W the cluster algebra structure on the coordinate ring of the unipotent cell Uw is categorified by a subcategory Cw of the modules over the corresponding preprojctive algebra. Under the cluster character the initial seed consisting of cetain generalized minors corresponds to a canoncial cluster tilting object Tv in Cw. In order to solve for Cw Berenstein, Fomin and Zelevinsky introduced twisted minors. We show that these twisted minors correspond essentially to the inverse of Auslander-Reiten translate of the summunds of Tv. (joint work with B. Leclerc and Jan Schoeer) |
Ellen Kirkman: Invariant Subrings of Regular Algebras under Hopf Algebra Actions Abstract: The Shephard-Todd-Chevalley Theorem states that if a finite group G acts on a commutative polynomial ring A = k[V] as elements of GLn(V), then the ring of invariants AG is a polynomial ring if and only if G is generated by reflections. In the same context Watanabe’s Theorem states that if G acts on A as elements of SLn(V), then the ring of invariants AG is a Gorenstein ring. We consider general- izations of these theorems to the noncommutative setting where A is a noetherian Artin-Schelter regular algebra with a finite group G acting linearly on A. More gen- erally we consider actions on A by a finite dimensional semi-simple Hopf algebra H, where each homogeneous component Aj is an H-module and A is an H-module algebra. (with James Kuzmanovich and James Zhang) |
Mark Kleiner: Finite Coxeter groups and preprojective roots Abstract: The talk will present results and constructions in the theory of Coxeter groups that are inspired by the representation theory of quivers. For example, let c be a fixed Coxeter element of a Coxeter group W. A positive root of W is called c-preprojective if, for some positive integer m, applying the m-th power of c to the root gives a negative root. Then W is finite if and only if all positive roots of W are c-preprojective. | Helmut Lenzing: Stable categories of vector bundles on weighted projective lines Abstract: We extend the range of hereditary representation theory by investigating the stable category of vector bundles on a weighted projective line. This is joint work with Kussin, Meltzer and de la Pena. In more detail, the line bundles on a weighted projective line form the system of indecomposable projective-injective objects for an exact structure on its category of vector bundles, turning this category into a Frobenius category. By work of Happel the attached stable category is triangulated; in the most interesting cases it turns out to be Calabi-Yau of fractional CY-dimension, a dimension we then explicitly determine. For particular weight types we further discuss the existence of tilting objects and draw interesting conclusions on the shape of the derived categories of their endomorphism rings. |
Dag Oskar Madsen: T- Koszul algebras Abstract Abstract: Let &Lambda = &Lambda0 &oplus &Lambda1 &oplus . . . &oplus &Lambdas be a finite dimensional graded algebra over a field and suppose &Lambda has finite global dimension. We do not assume that &Lambda0 is semi-simple. Let T be a graded &Lambda -module concentrated in degree zero. In this talk I will propose the following new definition of T-Koszul algebras: &Lambda is a T-Koszul algebra if both (1) and (2) hold. (1) T is a tilting &Lambda0-module. (2) T is graded self-orthogonal as a &Lambda -module |
Gregg Musiker: Positivity results for cluster algebras from surfaces Abstract: We give combinatorial formulas for cluster algebras with principal coefficients coming from triangulated surfaces (with or without punctures), as well as some cluster algebras obtained by ``folding''. In particular, this proves the positivity conjecture of Fomin and Zelevinsky for such cluster algebras, including those of classical type. This is joint work with Ralf Schiffler and Lauren Williams. |
Ralf Schiffler: Clusters, exceptional sequences and reduced expressions. Abstract: To an indecomposable object in a cluster category of finite type one can associate a simple reflection in the corresponding Weyl group, and, if one fixes an order on the indecomposable objects of the cluster category, one can can associate to any object of the cluster category a sequence of simple reflections. This defines a map from cluster-tilting objects to reduced expressions of the longest element of the Weyl group. In this talk, we will describe this map, show how it behaves with respect to mutations and sketch what can be generalized to the non finite type case. This is joint work with Kiyoshi Igusa. |
Markus Schmidmeier: The Entries in the LR-Tableau Abstract: Let &Gamma be the Littlewood-Richardson tableau corresponding to an embedding M of a subgroup in a finite abelian p-group. Each individual entry in &Gamma yields information about the module structure of subquotients of M, and about the position of M within the category of embeddings. |
Jeanne Scott: Laurent expansions for twisted Plücker coordinates via perfect matchings Abstract: I will explain how to compute Laurent expansions of twisted Plücker coordinates with respect to a cluster of the homogeneous coordinate ring of the Grassmannian Grk,n associated to a Postnikov diagram. This expansion formula is described in terms of (weighted) perfect matchings in an appropriate bipartite graph dual to the Postnikov diagram. |
Hugh Thomas: Higher Auslander algebras, cyclic polytopes, and analogues of tropical cluster algebras Abstract: Consider two simple models for the An cluster complex: triangulations of an (n + 3)-gon, and tilting objects for the path algebra of a linearly-oriented An+1 quiver. We show that there are higher-dimensional analogues of both these sets of objects, and that they are naturally in bijection. These higher dimensional analogues are: triangulations of a cyclic polytope of dimension 2d with n+2d+1 vertices, and basic tilting objects over the (d-1)-fold higher Auslander algebra of the path algebra of the linearly-oriented An+1 quiver (satisfying an additional condition). The analogue of the cluster variables in the two models are the internal d-dimensional simplices of the polytope and the non-projective-injective summands of the tilting objects. While we do not have anything like a cluster algebra on this set of variables, we show the existence of an analogue of the tropical cluster algebra structure associated to a lamination. This is joint work with Steffen Oppermann. ** | Helene Tyler: Shortest Annihilating Sequences for Preprojective Quiver Representations Abstract: This talk is a survey of joint work with Mark Kleiner, which has focused on preprojective representations of a quiver without oriented cycles and extensions of the results to representations of valued quivers. For each such representation, there is a shortest sequence of reflection functors that annihilates it, which is unique up to a certain equivalence. This single invariant descriptor depends only on the geometry of the quiver and the set of equivalence classes of these sequences is a partially ordered set that contains a great deal of information about the preprojective component of the Auslander-Reiten quiver". |
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Sponsors: Bernice Auslander and Northeastern University
| Department of Mathematics | Office: | 531 Lake Hall | Messages: | (617) 373-2450 | GGT Home |
| Northeastern University | Phone: | (617) 373-2450 | Fax: | (617) 373-5658 | Started: March 4, 2009 |
| Boston, MA, 02115 | Email: | g.todorov@neu.edu | Directions | to Northeastern | Last modified: March 4, 2009 |