Last updated: April 8, 2011, 18:08 EST

Conference Titles and Abstracts

Seidon Alsaody, On morphisms of finite dimensional absolute valued algebras.

Abstract: Click here

Silvana Bazzoni,
On the abelianization of derived categories and a negative solution to Rosicky's problem.

Abstract: Joint work with Jan Stovicek. We prove for a large family of rings R that their λ-pure global dimension is greater than one for each infinite regular cardinal 
λ. This answers in negative a problem posed by Rosicky. The derived categories of such rings then do not satisfy the Adams λ-representability for morphisms for any λ. Equivalently, they are examples of well generated triangulated categories whose λ-abelianization in the sense of Neeman is not a full functor for any λ. In particular we show that given a compactly generated triangulated category, one may not be able to find a Rosicky functor among the λ-abelianization functors.

Frauke Bleher, Connection of universal deformation rings to defect groups.

Abstract: In the eighties, Mazur, using work of Schlessinger, introduced techniques of deformation theory to the study of p-adic lifts of mod p representations of Galois groups. In this talk we will consider a question posed by Bleher and Chinburg about the connection of the universal deformation ring of a mod p representation of a finite group whose stable endomorphisms are all given by scalars to the defect groups of the p-block of G to which the representation belongs. We will discuss some positive and negative answers to this question.

Thomas Brustle, Cluster structures from surfaces without punctures.

Abstract: Any Riemann surface with marked points gives rise to a cluster category. We review this construction and discuss various properties, mostly restricting to the case where all marked points lie
on the boundary. We also explain the connection to earlier results of Fock and Goncharov, and Fomin, Shapiro and Thurston, as well as to some more recent developments.

Andrew Carroll, Geometry of representation spaces for gentle string algebras.

Abstract: It is well-known that if Q is a quiver with neither loops nor oriented cycles, then the coordinate ring of the space of representations of dimension vector d contains no non-trivial invariant functions relative to the action by GL(d), the product of general linear groups.  On the other hand, the ring of invariant functions with respect to the subgroup SL(d) admits a number of beautiful descriptions. 
    I will describe a procedure for determining the rings of semi-invariants for kQ/I when the preceding is a gentle string algebra.  This approach utilizes the GL(d)-decomposition of the coordinate rings for the aforementioned representation spaces.  I will give requisite background including the definition of string algebras.  Time-permitting, I will also describe the construction of generic modules in these representation spaces.

Leonid Chekhov, Groupoids of upper-block-triangular matrices: Poisson algebras and their affine extensions.

Abstract: (joint work with M.Mazzocco, Loughborough Univ.,UK). We generalize Bondal's construction of groupoid of upper-triangular matrices to the case of matrices composed from blocks (and having a nonstrictly upper-triangular form). We find the Poisson structure for entries of these matrices, construct the braid-group action, extend these algebras to semiclassical twisted Yangian algebras, and find their central elements in the both affine and non-affine cases.

Calin Chindris,
On the invariant theory for tame tilted algebras.

Abstract: I will present several characterizations of the tameness of a tilted (more generally, quasitilted) algebra in terms of the invariant theory of the algebra in question. Along the way, I will explain how moduli spaces for finite-dimensional algebras behave with respect to tilting functors, and to theta-stable decompositions.

Lars Christensen,
Brauer - Thrall for totally reflexive modules.

Abstract: For a commutative noetherian local ring that is not Gorenstein, it is known that the category of totally reflexive modules is representation infinite, provided that it contains a non-free module.

Over short local rings it will be shown how, starting from a non-free cyclic totally reflexive module, one can construct a family of indecomposable totally reflexive modules that contains, for every natural number n, a module that is minimally generated by n elements. Moreover, if the residue field is algebraically closed, then one can construct for every n an infinite family of indecomposable and pairwise non-isomorphic totally reflexive modules, each of which is minimally generated by n elements. The modules in both families have periodic minimal free resolutions of period at most 2.

The talk is based on joint work with Dave Jorgensen, Hamid Rahmati, Janet Striuli, and Roger Wiegand.

Lucas David-Roesler, On algebras from surfaces without punctures.

Abstract: We introduce a new class of finite dimensional gentle algebras, the surface algebras, which are constructed from an unpunctured Riemann surface with boundary and marked points by introducing cuts in internal triangles of an arbitrary triangulation of the surface. We show that surface algebras are endomorphism algebras of partial cluster-tilting objects in generalized cluster categories, we compute the invariant of Avella-Alaminos and Geiss for surface algebras and we provide a geometric model for the module category of surface algebras.

Gabriella D'Este, Indecomposable complexes and beyond.

Abstract: We show that bounded complexes (of projective modules with morphisms up to homotopy) and not only right bonded ones) are very often complicated enough to distinguish partial tilting complexes from tilting complexes in the sense of Rickard.  We will see that also rather short indecomposable complexes play a big role.

Ernst Dieterich, On finite dimensional division algebras.

Abstract: Click here

Yuriy Drozd, Representations of linear groups over Euclidean algebras.

Abstract: Let A be a finite dimensional algebra over the field of complex numbers, which is derived equivalent to the path algebra of a Euclidean quiver, G be the group of automorphisms of a finitely
generated projective A-module. We study the space G* of irreducible unitary representations of G and prove that it contains an open dense subset isomorphic to the product of several spaces GL(n)* and,
perhaps, the factorspace U/S, where U is the set of vectors having all different coordinates and S is the permutation group naturally acting on U.

Grégoire Dupont, Positivity in cluster algebras of Dynkin type A and affine type A.

Abstract: This is a preliminary report on a joint work with Hugh Thomas (U. New Brunswick).
We study the positive cone of a cluster algebra of Dynkin (resp. affine) type A. In particular, we prove that the extremal elements of this positive cone form a linear basis of the cluster algebra.

Federico Galetto, Orbit closures for the representations associated to graded Lie algebras: an interactive approach.

Abstract: The representations of simple groups with finitely many orbits are parametrized by graded simple Lie algebras. Many properties of the orbit closures of these representations are encoded by the minimal free resolution of their coordinate rings. I will describe an interactive method to construct such resolutions using Macaulay2.

Ed Green, Generalized matrix artin algebras.

Abstract: Click here

Tom Howard, When is the complexity of a module translation invariant?

Abstract: The complexity of a module measures the growth rate of the terms in a minimal projective resolution.  Complexity was first introduced by Alperin, who noted that for representations of finite groups, the resulting growth is always polynomial.  Avramov and others noticed exponential growth rates over certain commutative noetherian rings, and in a recent paper I have detailed classes of algebra for which these growth rates are products of exponential and polynomial functions.  This raises the question of which growth rates are possible.  In particular, one wonders whether the growth rates are always translation invariant, meaning that the rate does not depend on where along projective resolution we start.  This question of translation invariance is equivalent to several other important questions regarding complexity and its role as a homological invariant.  I will discuss these connections, and give conditions which ensure that the complexity of a module is translation invariant.

Colin Ingalls, The rationality of the Brauer-Severi variety of Sklyanin algebras.

Abstract: Iskovskih's conjecture states that a conic bundle over a surface is rational if and only if the surface has a pencil of rational curves which meet the discriminant in 3 or fewer points, (with one exceptional case).  We generalize Iskovskih's proof that such conic bundles are rational, to the case of projective space bundles of higher dimension.  The proof involves maximal orders and toric geometry.  As a corollary we show that the Brauer-Severi variety of a Sklyanin algebra is rational.

Dawid Kędzierski, Schofield induction for sheaves on weighted projective lines.

Abstract: Click here

Leila Khatami, Pairs of commuting nilpotent matrices.

Abstract: Click here

Ryan Kinser, Representation rings of quivers.

Abstract: Under the operations of direct sum and pointwise tensor product, the isoclasses of representations of a fixed quiver Q generate a (commutative) ring known as the representation ring R(Q) of the quiver.  The analogous construction for modular representations of finite groups has been studied more extensively.  By Gabriel's theorem, R(Q) is a finitely generated abelian group if and only if Q is Dynkin.  Our goal is to determine for which quivers is R(Q) modulo its ideal of nilpotents finitely generated as an abelian group (which we call the MFT property).  This classification problem is known to
admit an answer in terms of a finite list of "forbidden minors" (smallest quivers not having this property).

In this talk we will discuss some tools which are useful to study this problem.  In particular, we will focus on a new construction (work in progress) which we call "quiver trace functions" that give surjective
homomorphisms from R(Q) to the underlying field of the representations.  The existence of such a function implies that Q is not MFT; we will give plenty of examples and the list of 12 currently known forbidden minors, and a general discussion of how to construct a quiver trace function via a closed walk in Q.

Patrick Le Meur, Interactions between Hopf algebras and Calabi-Yau algebras.

Abstract: Click here

Helmut Lenzing, On an algebraic analysis of singularities.

Abstract: There are two competing methods for the algebraic analysis of a (graded, noetherian, isolated, Gorenstein) singularity R. One method is to apply Serre's construction (also called category of tails) to R by forming the quotient category of all finitely generated graded R-modules modulo its Serre subcategory of modules of finite length. Typically, this yields the category of coherent sheaves coh(X) on a weighted projective, usually non-commutative, variety X. The upshot is, that X is simpler than R by
dimensional reasons. The second construction is the singularity category Sing(R) of R which is due to Buchweitz and Orlov. The singularity category comes in various incarnations; a quite convenient one is the stable category CM(R)/[proj(R)] of graded Cohen-Macaulay R-modules modulo all morphisms factoring through projectives. Sing(R) is a good measure for the complexity of the singularity, since Sing(R)=0 if and only if R is regular. The main aim is to explain and illustrate Orlov's theorem (2005) dealing with the relationship of Sing(R) and the bounded derived category D(coh(X)) of coherent sheaves on X. There are three cases, depending on a numerical invariant a, the Gorenstein parameter of R. The theorem states that the two categories are equivalent for a=0 while otherwise, and
depending on the sign of a, one of the two triangulated categories sits in the other one as a perpendicular category to an exceptional sequence of size |a|. We illustrate the power of the theorem by a variety of interesting examples.

Hagen Meltzer,  Exceptional modules over canonical algebra.

Abstract: We study indecomposable and, in particular, exceptional modules over canonical algebras in the sense of Ringel. In joint work with Dirk Kussin we have described all indecomposable modules over domestic canonical algebras by vector spaces and matrices. The problem is much more complicated for tubular and wild canonical algebras. For tubular canonical algebras, we have shown how all exceptional modules can be obtained from those of rank zero and rank one, which again gives a description by vector spaces and matrices. In joint work with Piotr Dowbor and Andrzej Mróz we have developed an  algorithm and a computer program for those modules. Concerning exceptional modules over wild canonical modules, we refer to the talk of Dawid Kędzierski. Finally, we report on recent work with Piotr Dowbor and Andrzej 
Mróz about non-exceptional modules over tubular canonical algebras, where we can compute the relevant matrices using telescopic functors developed in joint work with Helmut Lenzing.

Charles Paquette, A proof of the strong no loop conjecture.

Abstract: This is a joint work with K. Igusa and S. Liu.  In this talk, I will give a proof (at least a sketch) of the strong no loop conjecture for finite dimensional elementary algebras, and in particular, for finite dimensional algebras over an algebraically closed field.  Recall that the strong no loop conjecture states that a simple module of finite projective dimension over an artin algebra has no non-zero self-extension. I will also give some generalizations of this result.

Marju Purin,  τ-Complexity of cluster tilted algebras.

Abstract: Click here

Jeremy Russell, Asymptotic stabilization of the tensor product.

Abstract: Three equivalent definitions are known for Vogel cohomology that work over arbitrary rings.  For Vogel homology, only 1.5 definitions are known. Completely missing is an analog of a construction of Buchweitz and Benson-Carslon via the inversion of the syzygy endofunctor.  In this talk we provide the missing link.

Kavita Sutar, Resolutions of defining ideals of orbit closures.

Abstract: Click here

Ralf Schiffler, On cluster algebras from surfaces.

Abstract: Fomin, Shapiro and Thurston have associated cluster algebras to Riemann surfaces with boundaries and marked points. For these cluster algebras, an explicit formula is known for the cluster variables and hence for the cluster monomials. In this talk, we recall this formula and describe how one can use it toward constructing bases for the cluster algebras.

Markus Schmidmeier, Gabriel-Roiter families occurring in tubes.

Abstract: he Gabriel-Roiter measure was first introduced by Roiter in his 1968 proof of the first Brauer-Thrall conjecture.  For a finite length module, the pair consisting of the GR-measure and the GR-comeasure defines the position of the module in the rhombic picture, as defined by Ringel. 

It turns out that modules in the same vicinity in the rhombic picture display similar behaviour with respect to Auslander-Reiten translation.  In particular, the set of modules, which is given by intersecting a ray with a coray in a stable tube in the Auslander-Reiten quiver, corresponds to a limit point in the rhombic picture.  We show that in the special case of quivers of type $\widetilde{A_n}$ with suitable orientation, the system of limit points in the rhombic picture provides a tiling of the corresponding tube.

This is a talk about a joint project with Helene Tyler (Manhattan College)..

Peter Tingley, Towards affine MV polytopes.

Abstract: MV polytopes give a useful realization of finite type crystals (combinatorial objects related to representations of complex simple Lie algebras). There has been some effort to generalize MV polytopes
to other Kac-Moody algebras. Recent work of Baumann and Kamnitzer constructs MV polytopes from Nakajima quiver varieties, which do generalize beyond finite type. I will describe this construction in
finite type. I will then discuss the symmetric affine case, which gives rise to affine analogues of MV polytopes. This is joint work with Pierre Baumann and Joel Kamnitzer.

Alberto Tonolo, When an abelian category with a tilting object is a module category?

Abstract: An abelian category with arbitrary coproducts and a small projective generator is equivalent to a module category. A tilting object in an abelian category is a natural generalization of  a small projective generator. Moreover, any abelian category with a tilting object admits arbitrary coproducts. It naturally arises the question when an abelian category with a tilting object is equivalent to a module category. By a result of Colpi, Gregorio and Mantese the problem simplifies in understanding when, given  an associative ring R and  a faithful torsion pair (X,Y)  in the category of right R-modules,  the heart of the t-structure  H(X,Y) associated with (X,Y) is equivalent to a category of modules.

Anatoly Vershik, Totally nonfree actions of the groups and factor-representations.

Abstract: The action of the group G on the space X called totally nonfree if for each two points x,y in X
their stabilizers are different Stab_x=\{g:gx=x\} \ne Stab_y=\{g:gy=y\}. For the measure preserving actions of the countable groups this is equivalent to a simple property of the family of the sets of fixed points fix(g)=\{x\in X:gx=x\}; g\in G. The main appication of this notion is the method of description of the characters of the group in terms of such kind of action, namely \chi(g)=meas( fix(g)).