Last updated: April 28, 2014, 21:30 EDT

Karin Baur, Asymptotic triangulations and their exchange graphs.

Abstract: We discuss asymptotic triangulations of the annulus, resp., of the infinity-gon. In the case of annuli, we show that they can be mutated as usual triangulation and describe the corresponding exchange graph. Asymptotic triangulations arise limits of triangulations under the action of the mapping class group. The strictly asymptotic arcs correspond to Pruefer and adic objects in the module category of \tilde{A}. In the case of the infinity-gon, we introduce cell-mutations as a new tool. With this, we obtain an (infinite) oriented exchange graph whose vertices are triangulations up to infinite sequence of mutations and whose oriented edges arise from irreversible cell mutation. Arcs in the infinity-gon can be viewed as representations for the linearly oriented quiver on \mathbb{Z}.


Vladimir Bavula, Left localizations of left Artinian rings.

Abstract: My talk will be about a description and structure of all the left denominator sets in an arbitrary left Artinian ring, about classification of all the maximal left denominator sets and left localizations.


Petter Andreas Bergh, The Grothendieck group of an n-angulated category.

Abstract: In a paper last year, Geiss, Keller and Oppermann defined higher analogues of triangulated categories, called n-angulated categories. For n=3, these are just triangulated categories. In this talk, we shall see that the Grothendieck group of such a category classifies its n-angulated subcategories; in the triangulated case, this is a classical result by Thomason. The talk is based on joint work with Marius Thaule.


Frauker Bleher, Closures of affine spaces in Grassmannians.

Abstract: This talk is about joint work with Ted Chinburg. Suppose k is an algebraically closed field. Let m<n be positive integers, and define Grass(m,n) to be the classical Grassmannian over k of all subspaces of dimension m in k^n. We consider embeddings of affine r-space A^r into Grass(m,n) given by taking the space spanned by the rows of an m x n matrix of linear polynomials in the r standard coordinates for A^r. We relate such embeddings and the closures of their images to degenerations of modules with simple top for a finite dimensional algebra over k. We then concentrate on the case r=2. We show that the generic embedding of A^2 into Grass(m,n) via such a matrix of linear polynomials has closure isomorphic to projective 2-space P^2. We also show that for each e>1 there is a positive dimensional family of embeddings for which the closure is the Hirzebruch surface X_e. While it is known, by work of the authors and Birge Huisgen-Zimmermann, that X_2 can arise from degenerations of modules as above, this is not known for the surfaces X_e with e>2.


Ilke Canakci, On extensions in the Jacobian algebra of a surface without punctures.

Abstract: Given an unpunctured surface (S,M), we study extensions in the Jacobian algebra J(Q,W) and in the cluster category C_{(S,M)}. We explicitly describe the middle terms of non-split short exact sequences in J(Q,W) and give a formula for dim Ext^1 (M_1, M_2) in terms of the intersection number of the arcs associated to indecomposable string modules M_1 and M_2. This is a joint work with Sibylle Schroll.


Calin Chindris, Moduli spaces for Schur-tame algebras.

Abstract: From the point of view of invariant theory, one is naturally led to think of an algebra based on the complexity of its Schur modules. In this talk, I will focus on the class of  Schur-tame algebras and their moduli spaces of modules. Along the way, I will describe a general reduction technique for dealing with moduli spaces for finite-dimensional algebras. This talk is based on joint work with Andrew Carroll, Ryan Kinser, and Jerzy Weyman.


Lars Christensen, Co-basechange of injective modules -- the other direction.

Abstract: Given a commutative ring R and an R-algebra S, it is a well-known fact that for every injective R-module E, the co-base changed module \mathrm{Hom}_R(S,E) is injective over S. In the talk I will discuss when the converse might be true, i.e. when does injectivity of \mathrm{Hom}_R(S,E) over S imply injectivity of E over R.


Jose Antonio De la Pena, On the Mahler measure of the Coxeter polynomial of algebras.

Abstract: click here


Ivon Dorado, Representations of p-equipped posets.

Abstract: This talk is based on joint work with Raymundo Bautista. For a prime number p, we define p-equipped posets, i. e. partially ordered sets with an order relation of p kinds, and its categories of representations and corepresentations over a normal field extension. The injective and projective objects are described. These categories are equivalent to a subcategory of modules over certain algebras, from which we obtain some properties, including the existence of almost split sequences. For a p-equipped poset, we establish a graph bijection between the preprojective components of the Auslander-Reiten quiver of its representations and its corepresentations.


Michael Gekhtman, Poisson-Lie groups and cluster structures.

Abstract: Coexistence of diverse mathematical structures supported on the same variety often leads to deeper understanding of its features. If the manifold is a Lie group, endowing it with a
Poisson structure that respects group multiplication (Poisson– Lie structure) is instrumental in a study of classical and quantum mechanical systems with symmetries. On the other hand, the  ring of regular functions on certain  Poisson varieties can have a structure of a cluster algebra. I will discuss results and conjectures on natural cluster structures in the rings of regular functions on simple complex Lie groups and Poisson–Lie structures compatible with these cluster structures. Much of this talk is based on an ongoing collaboration with M. Shapiro and A. Vainshtein.


Dolors Herbera, Semilocal rings, projective modules and decompositions of infinite direct sums of modules.

Abstract: click here


Lutz Hille, On the number of tilting modules for Dynkin quivers via polytopes.

Abstract: The number of tilting modules is classical known for type A and type D and recently attracted attention again in the work of Ringel and his coauthors. Moreover, certain closely related numbers also have been considered: the number of rigid modules, the number of exceptional sequences, the number of cluster tilting modules and the number of tilting complexes.

The aim of this talk is to relate all these numbers and to unify the computation, so that it is not case by case anymore. Moreover, the number of tilting modules does not depend on the orientation, however, so far, the computation depends on a choice of the orientation of the quiver. The principal idea is to define certain polytopes, so that the volume of these polytopes coincides with the number of tilting modules. Using this approach, we obtain several recursion formulas that relate the these numbers and allow to compute them in one strike for all Dynkin quivers and all orientations.


Alexander Ivanov, BV-algebra structure on Hochschild cohomology of the group algebra of quaternion group in characteristic 2 (joint with Guodong Zhou, Yury Volkov and Sergei Ivanov).

Abstract: We calculated Gerstanhaber and BV-algebra structures on Hochschild cohomology of the group algebra of quaternion group in characteristic 2. In the calculation we used an approach developed by Guodong Zhou and a description of the multiplicative structure previously obtained by A.I. Generalov.


Srikanth Iyengar, What annihilates Ext?

Abstract: Consider an algebra R that is finitely generated as a module over its center. This talk will be about elements in the center of R that annihilate Ext^n_R(M,N) for all finitely generated R-modules M and N, and for n large enough. I was lead to consider these elements in the course of an investigation on the finiteness of dimensions of derived categories. While the latter grew out of Auslander’s work on representation dimension, the study of annihilators of Ext has led us to a different set of ideas and techniques—namely, those involving separable algebras---pioneered by Auslander and his collaborators. This is part of an on-going collaboration with  Ryo Takahashi; see http://arxiv.org/abs/1404.1476.


Ellen Kirkman, Actions of finite dimensional Hopf algebras on AS regular algebras.

Abstract: Invariants A^H under Hopf algebra H actions on AS regular algebras A provide additional subrings of invariants of A.  We discuss the cases where A^H is AS Gorenstein, and where A^H is AS regular.


Liping Li, Homological dimensions of crossed products.

Abstract: Let A be a semiprimary Noetherian algebra over an algebraically closed field k with characteristic p \geqslant 0, and let G be a finite group whose elements acting on A as algebra automorphisms. In this talk we extend the usual induction and restriction functors from module categories to homotopy categories, and prove a criterion for the global dimension of the crossed product A#G to be finite. Moreover, we show that in this situation A#G and A have the same homological dimensions such as global dimension, finitistic dimension, and strong global dimension.


Daniel Lopez Aguayo, Potentials for some tensor algebras.

Abstract: This is joint work with Raymundo Bautista. Given a semisimple basic finite dimensional algebra S of finite dimension over a fixed field we define a theory of potentials for an S-bimodule with some additional assumptions. We introduce an ideal analogous to the Jacobian ideal introduced by H. Derksen, J. Weyman, and A. Zelevinsky. We prove that mutation is an involution on the set of right-equivalence classes of reduced potentials.


David Meyer, Universal deformation rings for representations of subgroups of \mathrm{GL}_2(\mathbb{F}_p)

Abstract: Let \Gamma be a finite group, and let V be an absolutely irreducible \mathbb{F}_{p}\Gamma-module.  By Mazur, V has a universal deformation ring R(\Gamma,V).  This ring is characterized by the property that the isomorphism class of every lift of V over a complete local commutative Noetherian ring R with residue field \mathbb{F}_p arises from a unique local ring homomorphism \alpha: R(\Gamma,V)\to R.We consider the case when \Gamma is an extension of a finite group G whose order is relatively prime to p, by an elementary abelian p-group N of rank 2.  We further suppose that  \mathbb{F}_{p} is a splitting field for G, and that G has a faithful, irreducible, two-dimensional \mathbb{F}_{p} representation.  Such groups G have been classified according to their images in the projective linear group  \mathrm{PGL}_2(\mathbb{F}_p).  We outline a strategy of how to use this classification to determine to what extent the knowledge of R(\Gamma,V) for all irreducible V can detect the fusion of N in \Gamma.


Van Nguyen, Finite generation of cohomology rings.

Abstract: People are interested in the cohomology rings for various reasons. One of the properties they look at is whether the cohomology ring of an object is finitely generated. In this talk, we show that some skew group algebras have Noetherian cohomology rings, a property inherited from their component parts. The proof is an adaptation of Evens' proof of finite generation of group cohomology. We apply the result to a series of examples of finite dimensional Hopf algebras in positive characteristic. This is joint work with S. Witherspoon.


Steffen Oppermann, A recollement approach to Geigle-Lenzing weighted projective varieties.

Abstract: This is a report on joint work in progress with Osamu Iyama and Boris Lerner. Motivated by Iyama's higher dimensional Auslander-Reiten theory, Iyama, Herschend, Minamoto and I generalized Geigle-Lenzing's weighted projective lines to higher dimensions. In work of Iyama-Lerner an alternate construction based on orders is given. The aim for this talk is to construct Geigle-Lenzing weighted projective varieties using recollements of abelian categories, starting from a usual projective variety with a tilting object. The advantage of this construction is that it works rather generally, so that we are able to cover natural examples which were not treated in the two previous constructions. In my talk I will first recall the classical constructions, and then try to explain how they led us naturally to our new one.


Daiva Pucinskaite, BGG-algebras of dominant dimension at least 2.

Abstract: Any finite-dimensional BGG-algebra A is related to a partial order of the set of isomorphism classes of simple A-modules. When (in addition) the dominant dimension of A is at least 2, then there exist an algebra B and a B-module G such that A and endomorphism algebra of G are isomorphic (Morita-Tachikawa Theorem). In this talk we consider BGG-algebras of dominant dimension at least 2 having indecomposable faithful module, and discuss the relationship between the partial order and the structure of the corresponding pair (B,G). Important examples of this algebras are block algebras of BGG category O.


Toni Rangachev, The epsilon multiplicity and polar varieties.

Abstract: The epsilon multiplicity is a generalization of the Buchsbaum-Rim multiplicity for submodules of free modules not necessarily of finite colength. In this talk we relate the epsilon multiplicity to the geometry of certain polar varieties of modules. We discuss applications of this relation to equisingularity theory.


Dylan Rupel, Greedy bases in rank 2 quantum cluster algebras.

Abstract: I will report on a joint work with Lee, Li, and Zelevinsky where we identify a quantum lift of the greedy basis of rank 2 cluster algebras.  In the quantum setting the beautiful combinatorics of compatible pairs is unfortunately not available, thus I will describe a purely algebraic approach to establishing the nice properties of the quantum greedy basis.


Ralf Schiffler, Cluster algebras and rings of snake graphs.

Abstract: Abstract snake graphs were introduced in [1] inspired by the labeled snake graphs appearing in the expansion formulas for cluster variables in cluster algebras of surface type [2]. While the labeled snake graphs are constructed from the crossing pattern of an arc in a surface with a fixed triangulation, the definition of abstract snake graphs is completely detached from triangulated surfaces and is simply given by describing the possible graphs in an elementary way. In analogy with the situation in cluster algebras, one can define a multiplication on the free abelian group generated by all abstract snake graphs by means of taking disjoint unions modulo certain relations. In this way, one obtains a ring of abstract snake graphs which has interesting relations to cluster algebras. This is joint work with Ilke Canakci.
[1] I. Canakci, R. Schiffler : Snake graph calculus and cluster algebras from surfaces, J. Algebra, 382, (2013) 240--281.
[2] G. Musiker, R. Schiffler, L. Williams : Positivity for cluster algebras from surfaces, Advances in Math. 227 (2011) 2241--2308.


Khrystyna Serhiyenko,  Induced and coinduced modules in cluster-tilted algebras.

Abstract: Let C be a tilted algebra and B the corresponding cluster-tilted algebra. We consider induction from mod C to mod B via the tensor product with B. It turns out that this functor has some interesting properties such as each projective B-module is induced by the corresponding projective C-module, but induction of any injective C-module results in the exact same module.  Similarly, we introduce a dual construction called coinduction functor.  Using both functors we construct an explicit injective resolution of each projective B-module.  This gives rise to another proof of the known result that cluster-tilted algebras are 1-Gorenstein.

Moreover, if B is representation finite then every module is both induced and coinduced from some tilted algebra C.  If B is not representation finite then every transjective module in B is either induced or coinduced from some C.  However, the situation with regular modules turns out to be more complicated.


Mufit Sezer, Modular invariants of the Klein four group.

Abstract: We study the rings of invariants for the indecomposable modular representations of the Klein four group. For each such representation  we give minimal generating sets for the Hilbert ideal and the field of fractions. We observe that, with the exception of the regular representation, the Hilbert ideal for each of these representations is a complete intersection. This is joint work with Jim Shank.


Luise Unger, On the Combinatorics of the Set of Tilting Modules.

Abstract: click here


Adam-Christiaan van Roosmalen, Numerically finite hereditary categories with Serre duality.

Abstract: In this talk, I will report on a classification (up to derived equivalence) of abelian hereditary categories with Serre duality, satisfying the additional condition that the numerical Grothendieck group (being the Grothendieck group modulo the radical of the Euler form) has
finite rank.  Categories satisfying this last property are called numerically finite, and this property is satisfied by the category of coherent sheaves on smooth projective varieties.


Yury Volkov, BV-differential on Hochschild cohomology of Frobenius algebra.

Abstract: click here


He Wang, Lie algebras of finitely generated groups and their formality properties.

Abstract: There are various Lie algebras associated to a finitely generated group G, including the associated graded Lie algebra, the Malcev Lie algebra and the holonomy Lie algebra. The natural homomorphisms between these Lie algebras are related to the formality properties of the group G. We study the formality properties of the group by exploring presentations of these Lie algebras in the case when the group G has a finite presentation. Using a generalization of the Magnus expansion and an explicit formula for cup products, we give an algorithm for finding a presentation for the holonomy Lie algebra. As an application of our methods, we investigate the formality properties of finitely generated torsion-free nipoltent groups and various generalizations of the pure braid groups. This is joint work with Professor Suciu.


Alexandra Zvonarëva, On the derived Picard group of a Brauer tree algebra.

Abstract: Brauer tree algebras arise naturally in modular representation theory to describe blocks of group algebras with cyclic defect group, also these algebras are precisely symmetric special biserial algebras of finite representation type. Derived equivalences of Brauer tree algebras were studied by many people. In my talk I will discuss how to use the technique of silting mutations developed by Aihara and Iyama to show that the derived Picard group of a Brauer tree algebra is generated by shift, Picard group and braid generators introduced by Schaps and Zakay-Illouz or by a slightly bigger set in the multiplicity free case. 

 
 
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