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)).