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.


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 Bruestle, 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 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, Euler-Poincaré 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-Poincaré 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.


Ernst Dieterich, Real division algebras, restricted quiver representations,and Euclidean configurations.

Abstract: A real division algebra is a non-zero real vector space A, endowed with an R-bilinear multiplication
A × A → A such that the left and right multiplications by nonzero elements 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, 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 U_w is categorified by a subcategory C_w of the modules over the corresponding preprojective algebra. Under the cluster character the initial seed consisting of certain generalized minors corresponds to a canonical cluster tilting object T^v in C_w. In order to solve for C_w Berenstein, Fomin and Zelevinsky introduced twisted minors. We show that these twisted minors correspond essentially to the inverse of Auslander-Reiten translate of the summands of T^v. (joint work with B. Leclerc and Jan Schroeer)


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 GL_n(V ), then the ring of invariants A^G 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 SL_n(V), then the ring of invariants A^G is a Gorenstein ring. We consider generalizations 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
generally, we consider actions on A by a finite dimensional semi-simple Hopf algebra H, where each homogeneous component A_j is an H-module and A is an H-module algebra. (with James Kuzmanovich and James Zhang)


Helmut Lenzing, Stable categories of vector bundles on weighted projective lines.

Abstract: TBA


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.


Markus Schmidmeier, The entries in the LR-Tableau.

Abstract: Let Γ be the Littlewood-Richardson tableau corresponding to an embedding M of a subgroup in a finite abelian p-group.  Each individual entry in Γ 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 Pluecker coordinates via perfect matchings.

Abstract: I will explain how to compute Laurent expansions of twisted Pluecker coordinates with respect to a cluster of the homogeneous coordinate ring of the Grassmannian Gr_{k,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 A_n cluster complex: triangulations of an (n + 3)-gon, and tilting objects for the path algebra of a linearly-oriented A_n+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 A_n+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.

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