Last updated: April 21, 2012, 9:37 EST
Conference Titles and Abstracts
Vladimir Bavula, The largest left quotient ring of a ring.
Abstract: The left quotient ring (i.e. the left classical ring of
fractions) of a ring does not always exist and still, in general,
there is no good understanding of the reason why this happens.
Recently, for an arbitrary ring I proved existence of its largest left
quotient ring. A criterion is given of when the largest left
quotient ring of a ring is semi-simple (generalizing Goldie's
Theorem). We extend slightly Ore's method of localization to
localizable left Ore sets, give a criterion of when a left Ore set is
localizable, and prove that all left and right Ore sets of an
arbitrary ring are localizable (not just denominator sets as in Ore's
method of localization). Applications are given for certain classes of
rings (semi-prime Goldie rings, Noetherian commutative rings, the
algebra of polynomial integro-differential operators).
Yuri Berest, Derived representation schemes and cyclic homology.
Abstract: Many important varieties in algebra, geometry and physics can
be realized as moduli spaces of finite-dimensional representations of
associative algebras and groups. The simplest and most commonly used
are the classical representation scheme Rep_n(A) parametrizing the
n-dimensional representations of a f.g. algebra A over a field k and
its affine categorical quotient Rep_n(A)//GL_n(k) parametrizing the
isomorphism classes of semisimple representations. For a fixed n, each
of these schemes defines a (contravariant) functor on the category of
algebras. In this talk, I will explain how to construct the
corresponding derived functors in the category of DG schemes and compute their stable
homology as n goes to infinity. Time permitting, I will discuss some
applications of the derived representation schemes in knot theory and
low-dimensional topology. (The talk is based on joint work with A.
Ramadoss.)
Frauke Bleher, Brauer's generalized decomposition numbers and universal deformation rings.
Abstract: In this talk, I will study the problem of lifting to local
rings certain mod 2 representations V of a finite group G which belong
to a 2-modular tame block B of G having at least two isomorphism
classes of simple modules. Green's lifting theorem determines when such
a V may be lifted to the ring of infinite Witt vectors. I will
generalize this result by determining the full universal deformation
ring R(G,V) of such V using Brauer's generalized decomposition numbers.
The isomorphism type of R(G,V) will depend on whether the stable Auslander-Reiten quiver of B contains 3-tubes.
Thomas Bruestle, Poset structures on the cluster exchange graph.
Abstract: Cluster mutation is an involution, so the cluster exchange
graph comes a priori without any orientation. However, fixing an
initial cluster allows to turn it into an oriented graph with a unique
source and a unique sink. We discuss this order relation from various
points of view and indicate relations to other areas: it contains the
Happel-Unger poset of tilting modules. Moreover, maximal paths in the
oriented graph give rise to quantum dilogarithm identities, and they
allow to compute the complete spectrum of a BPS particle.
Aslak Buan, Rigid objects and torsion pairs in tubes.
Abstract: This is based on joint work with Baur and Marsh. We study
certain uniserial abelian categories, called tubes, and extensions of
such containing Prüfer and adic objects. We describe rigid objects in
the extended tubes, and the relationship to torsion pairs. We also
discuss a combinatorial model.
Giovanni Cerulli Irelli, On linear independence of cluster monomials and application to positivity in cluster algebras.
Abstract: In a recent preprint in collaboration with B. Keller,
D.Labardini-Fragoso and P.-G. Plamondon, we prove that the cluster
monomials of a skew-symmetric cluster algebra are linearly
independent over the ground ring. From the proof of this result we also
deduce that exchange graph of the custer algebra is independent on the
system of coefficients. I will present this results and if time permits
I will also provide some applications to positivity.
Claudia Chaio, Degrees of irreducible morphisms.
Abstract: Click here
Gabriella D'Este, Cancellations, additions and Lego-type constructions.
Abstract: the first part of my talk will deal with "non classical"
partial tilting modules, that is partial tilting modules T of
projective dimension at most 2, with the following property: if Hom (T, X) = 0 and Extm(T, X) = 0 for every m > 0, then X = 0 .
The second part of my talk will deal with partial tilting complexes T°
which are projective resolutions of partial tilting (but not tilting)
modules T with the
above property. As we shall see, there seems to be a weak relationship
between the complexes T° and the indecomposable complexes
“orthogonal” to them. For instance, some of these complexes look like
“mutations” of direct summands of the complexes T°, while other
are completely different.
Sachin Gautam, Geometry of quiver varieties.
Abstract: Click here
Ed Green, The symmetric special biserial algebras that are K2.
Abstract: Click here
Ivo Herzog, An introduction to ideal approximation theory.
Abstract: An introduction to the theory of ideal approximations in the
context of an exact category (joint with X. Fu, P.A. Guil Asensio and
B. Torrecillas). The main highlights are a 3-dimensional version of
Salce's Lemma, and the characterization of special precovering ideals
as ideals of phantom morphisms with respect to certain subfunctors of
Ext. Some applications to the powers of the phantom ideal will be given (joint with X. Fu).
Lutz Hille, Full exceptional sequences and tilting modules over the Auslander algebra of k[T]/Tt.
Abstract: We classify all exceptional and all spherical modules over the Auslander algebra of k[T]/Tt.
This module category contains many spherical objects defining
non-standard equivalences of the derived category. There is a close
connection between spherical and exceptional objects that we can
explain in detail for this example. Its importance came originally from
tilting bundles on rational surfaces. Any chain of (-2)-curves on a
rational surface defines a subcategory of coherent sheaves that is
equivalent to the category of modules over the Auslander algebra of k[T]/Tt. The main result in the talk gives a complete classification of all full exceptional sequences for the Auslander algebra of k[T]/Tt.
We use this construction and universal extensions to obtain tilting
modules. Tilting modules of projective dimension at most one have been
classified in a joint work with Brüstle, Ringel and Röhrle. We extend
this classification to all tilting modules. Moreover, the Auslander
algebra of k[T]/Tt seems to appear in various applications, we also give some overview. This is joint work with David Ploog.
Tom Howard, Upper bounds on complexity over representation-finite algebras.
Abstract: We will look into the question of whether there is some
exponential bound on the complexity of a module over a
representation-finite algebra, i.e. on the dimension growth of the
module's syzygies. In particular, we will show that for a truncated
path algebra, 2n is a sufficient upper bound. Since
these complexities are preserved under derived equivalences, our upper
bounds also apply to modules over derived equivalent algebras.
Birge Huisgen-Zimmermann, Fine and coarse moduli spaces in the representation theory of finite dimensional algebras.
Abstract: We will begin with the concepts of fine/coarse moduli spaces.
After a brief history and some classical examples, we will discuss two
fundamentally different lines of approach in applying these notions to
classification problems in the representation theory of finite
dimensional algebras. The juxtaposition will highlight differences in
goals and methodology. Both viewpoints will be illustrated with
prototypical results.
Miodrag Iovanov, Quiver and monomial algebras, coalgebras, and rational representations.
Abstract: Given a quiver Q, besides the quiver algebra K[Q], there is
another less well known but equally important associated object, the
quiver coalgebra KQ. A natural question is: what are the connections
between the two, is there any (bialgebra) compatibility between them,
and what kind of bialgebra structures can be found on a quiver
(co)algebra. We first give an account of algebras, coalgebras, rational
modules, and duality. We show that the path coalgebra can be obtained
from the quiver algebra as a certain type of graded finite dual; also,
it is the classical finite dual of the quiver algebra (i.e. the
coalgebra of representative functions) if and only if the quiver is
acyclic and only finitely many arrows exist between any two vertices.
We also give some results about recovering the quiver algebra from the
path coalgebra, and we mention the interpretation in categorical terms
of representations and rational modules of quivers. Time permitting, we
briefly discuss bialgebra structures on monomial (co)algebras, and new
categorical characterizations of monomial algebras.
Finally, we look at a particular class of quivers, which produce the
linear categories whose finite objects are all serial. We show how a
theory that parallels that of infinite abelian groups can be developed
for this situation; as a main application, we give an example of an
algebra all of whose rational (i.e. locally finite) left modules are
direct sums of indecomposables but not all right locally finite modules
decompose this way. This provides an interesting contrast to the finite
dimensional algebra case, where one of the main open questions asks
whether decomposability into indecomposables of all modules on one side
implies the analogue complete decomposability on the other.
David Jorgensen, On products in negative cohomology for n-Calabi-Yau categories.
Abstract: We investigate the structure of the Z-graded cohomology rings
of objects in n-Calabi-Yau triangulated categories. Almost by
definition these cohomology rings possess a natural duality. In
particular, the stable endomorphism ring of a finitely generated module
over a finite dimensional symmetric k-algebra is a Z-graded k-algebra
that possesses a natural duality between its positive and negative
sides. A consequence of this is that if the non-negative part of
the endomorphism ring has a regular sequence of central elements of
length 2, then all products between elements of negative degree are
trivial. As a corollary we show this holds for the Tate-Hochschild cohomology ring of a symmetric k-algebra.
We'll also show the same results hold over a commutative
zero-dimensional Gorenstein ring. This is based on joint work with
Petter Bergh and Steffen Oppermann.
Otto Kerner, Wild hereditary algebras.
Abstract: Click here
Ryan Kinser, Module varieties of finite-dimensional algebras.
Abstract: Given a finite-dimensional algebra A, the set of A-modules of
a fixed dimension d can be viewed as a variety. This variety carries a
group action whose orbits correspond to isomorphism classes of
A-modules. A natural problem is to characterize various properties of
an algebra A in terms of its module varieties.
For example, if A is assumed to have global
dimension one, then it is not difficult to show that A has finitely
many indecomposable modules (up to isomorphism) if and only if all of
its module varieties have a dense orbit, which is also if and only if
all weight spaces of semi-invariants in the coordinate rings of its
module varieties have dimension one. Our goal is to generalize these
statements (with modification) to higher global dimension. After
explaining the background, we present counterexamples to the naive
generalizations, along with plausible modifications and cases where
these modifications are correct. (Joint work with Calin Chindris and
Jerzy Weyman.)
Henning Krause, Koszul, Ringel, and Serre duality for strict polynomial functors.
Abstract: Strict polynomial functors were introduced by Friedlander and
Suslin in their work on the cohomology of finite group schemes. This
category admits a Koszul duality by recent work of Chalupnik and Touze.
In my talk I will give a gentle introduction to strict polynomial
functors (via representations of divided powers) and will explain the
Koszul duality, making explicit the underlying monoidal structure which
seems to be of independent interest. Then I connect this to Ringel
duality for Schur algebras and describe Serre duality for strict
polynomial functors.
Daniel Labardini-Fragoso, Tagged triangulations, potentials and cluster monomials.
Abstract: Linear independence of cluster monomials in skew-symmetric
cluster algebras with arbitrary coefficients has been proved recently
by Cerulli-Keller-LF-Plamondon. One of the main ideas in the proof had
been used before by Cerulli and myself to show the alluded linear
independence in the context of (possibly punctured) surfaces with
marked points and non-empty boundary. In this talk I will explain this
idea and some results on QPs of tagged triangulations that we showed in
order to apply it.
Helmut Lenzing, On Happel-Seidel symmetry.
Abstract: My talk deals with the class of Nakayama algebras given by
the path algebra of an equi-oriented finite quiver, equipped with all
nilpotency relations of a fixed degree. While these algebras are
representation finite, their derived categories turn out to be very
interesting. They have a close relationship to singularity theory and
the ADE-chain problem. Further, they enjoy a number of often-hidden
symmetries, the most prominent one was first investigated by Happel and
Seidel. While Happel-Seidel restricted mostly to the piecewise
hereditary situation, I will mostly deal with the complement, present a
number of these symmetries and, in particular, present a nice
particular instance of Orlov's theorem. The matter is related to recent
work with Kussin and Meltzer on triangle singularities f = xa + yb + zc.
Graham Leuschke, The MCM McKay Correspondence.
Abstract: The McKay correspondence is a collection of correspondences
and equivalences which encompasses representation theory, commutative
algebra, algebraic geometry, and mathematical physics. I will
give an informal introduction to the McKay correspondence from the
point of maximal Cohen-Macaulay modules over fixed rings of group
actions.
Liping Li, A generalized Koszul theory and its relation to the classical theory.
Abstract: Koszul theory plays an important role in the representation
theory of graded algebras. However, there are a lot of structures
(algebras, categories, etc) having natural gradings with non-semisimple
degree 0 parts. Particular examples include tensor algebras generated
by non-semisimple algebras A0 and (A0, A0)-bimodules A1,
and extension algebras of standard modules of standardly stratified
algebras. In this talk we introduce a generalized theory which
preserves many classical results and can be applied to the above
examples. We also describe its applications to directed k-linear
categories and a close relation to the classical theory.
Shiping Liu, Almost split sequences and approximations.
Abstract: Let A be an exact category, that is, an
extension-closed full subcategory of an abelian category. Firstly, we
give a new characterization of almost split sequences in A, which
yields some
necessary and sufficient conditions for A to have almost split
sequences. In case A has almost split sequences, we shall provide a
necessary and sufficient condition for an exact subcategory of A to
have almost split sequences. Finally, we show two applications of these
results.
Ivan Losev, Crystal for Rep(GLn) revisited.
Abstract: Consider the category Rep(GLn) of rational representations of GLn
over an algebraically closed field of positive characteristic. On this
category one has two distinguished endo-functors -- tensor
products with the tautological module and its dual. Each of them
decomposes into the direct sum of p endofunctors that leads to a
categorical action of the affine algebra \hat{sl}p. This gives rise to a canonical crystal structure on the set of simple GLn-modules.
In this talk we will describe this crystal combinatorially. This was
done previously by Brundan and Kleshchev. A new feature of our approach
is that it works for every categorical Kac-Moody action on a highest
weight cateogry modulo some compatibility conditions between the action
and the highest weight structure.
Frank Marko, Simple modules for Schur superalgebra S(2|2).
Abstract: Click here
Roberto Martínez Villa, Dualities in Koszul AS Gorenstein algebras.
Abstract: The main result in the talk is that for Koszul algebras Λ with Yoneda algebra Γ, such that both Λ and Γ
are graded AS Gorenstein noetherian of finite local cohomology
dimension on both sides, there are dualities of triangulated categories: grΛ[Ω-1] ≃ Db(QgrΓ) and grΓ[Ω-1] ≃ Db(QgrΛ).
Puiman Ng, Quotients of the finite derived category Db(mod kD5) as derived categories.
Abstract: Let D be the finite derived category Db(mod kD5)
and consider a certain quotient of D. This quotient category is
pretriangulated and is furthermore triangulated. Then by the extensive
use of the octahedral axiom of a triangulated category, we are able to
describe explicitly the action of the translation functor of the
quotient category, and to compare the distinguished triangles between
the two categories. Finally, we briefly describe the theorem which
shows that this quotient category can also be seen as a derived
category.
Steffen Oppermann, n-Representation infinite algebras.
Abstract: This talk is based on joint work with Martin Herschend and Osamu Iyama.
Iyama introduced a higher version of Auslander-Reiten theory, in which
(compared to classic AR-theory) all short exact sequences are replaced
by longer exact sequences. In joint work we introduced the notion of
n-representation finite algebras as a setup in which his higher
AR-theory applies as nicely as possible. It turns out that many
statements about representation finite hereditary algebras can be
generalized to this setup.
In my talk I will first recall the notion of n-representation
finiteness, and give some idea what the natural generalization of
AR-translation is in this setup. Then I will focus on the question what
an appropriate analog of representation infinite hereditary algebras
is. I will suggest a definition of n-representation infiniteness,
explain some basic properties of n-representation infinite algebras,
and finally illustrate how preprojective algebras can be used to get a
better picture at least in some ("tame") cases.
Charles Paquette, Semi-stable categories for Euclidean quivers.
Abstract: Let k be an algebraically closed field and Q a finite quiver
of Euclidean type. The semi-stable subcategories of rep(Q) are
important objects coming form geometric invariant theory. In the
Euclidean case, they are easy to describe. In this talk, I will show
what are the possible subcategories arising as intersection of
semi-stable categories of rep(Q). Contrary to the Dynkin case, these
categories are not necessarlily semi-stable and do not cover all the
thick subcategories of rep(Q). This is joint work with C. Ingalls and
H. Thomas.
Alice Pavarin, Recollements from partial tilting complexes.
Abstract: Click here
María Inés Platzeck, Fundamental domains of cluster categories inside module categories.
Abstract: Let H be a finite dimensional hereditary algebra over an algebraically closed field, and let CH be the corresponding cluster category. In this talk I will give a description of the (standard) fundamental domain of CH in the bounded derived category Db(H),
and of the cluster-tilting objects, in terms of the category mod B of
finitely generated modules over a suitable tilted algebra B.
Furthermore, I will explain how this description can be used to
obtain (the quiver of) an arbitrary cluster-tilted algebra. This talk
reports joint work with Juan A. Cappa and Idun Reiten.
Marju Purin, The generalized Auslander-Reiten condition.
Abstract: Click here
Jeremy Russell, A functorial approach to linkage.
Abstract: Martsinkovsky and Strooker generalized the notion of linkage
of algebraic varieties to linkage of finitely generated modules over
semiperfect noetherian rings. In an unpublished work,
Martsinkovsky generalized this definition even further to linkage on
the stable module category. This definition works over arbitrary
noetherian rings. Using the satellites and injective resolutions, I
will explain how to extend the definition of linkage on the stable
module category to linkage of finitely presented functors and discuss
the relationship between these two notions.
Peter Samuelson, Skein modules and the quantum torus.
Abstract: Skein modules are vector spaces associated to 3-manifolds
which are functorial with respect to embeddings of manifolds. We review
this construction and a result of Frohman and Gelca which shows that
the skein module of a knot complement is a module over the Z2-invariant subalgebra of the quantum torus (where Z2
acts by inverting x and y). We discuss some new results and conjectures
about the structure of these modules. If time permits, we may discuss
the question of deforming these modules to modules over the double
affine Hecke algebra of type A1.
Ralf Schiffler, On the first Hochschild cohomology of a cluster-tilted algebra.
Abstract: This is joint work with Ibrahim Assem and Maria Julia
Redondo. If B is a cluster-tilted algebra, then there exists a tilted
algebra C such that B is the relation extension of C. We study the
relation between the first Hochschild cohomology of B and C. If B is
tame or C is constrained we show that this relation is given by an
invariant n{B,C} which is defined in terms of the bound quivers of B and C.
Markus Schmidmeier, Operations on arc diagrams and degenerations for invariant subspaces of linear operators.
Abstract:
We study geometric properties of varieties associated with invariant
subspaces of nilpotent operators. There are reductive algebraic groups
acting on these varieties. We give dimensions of orbits of these
actions. Moreover, a combinatorial characterization of the partial
order given by degenerations is described. This is a report about
a joint project with Justyna Kosakowska from Torun.
Salvatore Stella, Polyhedral models for generalized associahedra via Coxeter elements.
Abstract: Motivated by the theory of cluster algebras, S. Fomin and A.
Zelevinsky have associated to each finite type root system a simple
convex polytope called generalized associahedron. It turns out that
this purely combinatorial gadget encodes many informations on the
associated cluster algebra making it an interesting object to study.
I will describe, after recalling the basic definitions, a family of
geometric realizations of these polytopes, parametrized by orientations
of the corresponding Dynkin diagram. I will also show that this
construction agrees with the one given by C. Hohlweg, C. Lange and H.
Thomas in the setup of Cambrian fans developed by N. Reading and D.
Speyer.
Hugh Thomas, Quotient-closed subcategories of the representations of a quiver.
Abstract: Let Q be a quiver without oriented cycles. We show that the
quotient-closed full subcategories of rep Q which are cofinite (i.e.
contain all but finitely many indecomposable representations of Q) are
naturally in bijection with the elements of the Weyl group W associated
to Q, and that the inclusion order on the subcategories corresponds to
the "sorting order" on W introduced by Armstrong. In the Dynkin case,
cofiniteness is trivially satisfied, and we obtain a bijection between
the quotient-closed subcategories and the elements of W. Even the fact
that these sets have the same cardinality seems to be new. The
representation
theory of preprojective algebras plays an important role in our
analysis. This is joint work with Steffen Oppermann and Idun Reiten.
Peter Tingley, Some musings on preprojective algebras.
Abstract: Preprojective algebras play an important role in geometric
representation theory of Kac-Moody algebras. However, the notion of a
preprojective algebra is older than this work. Here I will describe
preprojective algebras as they are typically defined in geometric
representation theory, I will explain why these are in fact
preprojective algebras according to older definitions, and then I will
discuss some interplay between these points of view. I am interested in
this because the early work of Dlab-Ringel included preprojective
algebras for non-symmetric types, which are usually not considered by
the geometric representation theory community. I will finish by
mentioning some very recent work with Vinoth Nandakumar showing that
some (but not all) of the geometric representation theory results
remain true in at least some non-symmetric cases.
Valerio Toledano-Laredo, Stability conditions and Stokes factors.
Abstract: This talk will be concerned with wall-crossing in an abelian
category A, that is the (dis)appearance of semistable objects when
varying a stability or slope condition on A.
D. Joyce gave a remarkable construction of a generating function which
counts semistable objects in A and, unlike the objects themselves, is
continuous (in fact, holomorphic) with respect to a change of the
stability condition.
I will explain how wall-crossing in A is governed by an ODE in the
complex plane with an irregular singularity at the origin, and the
jumps of semistable objects by the Stokes phenomena of that ODE, that
is the discontinuous changes in asymptotics as one crosses the Stokes
rays of that ODE. This gives in particular a
conceptual explanation of Joyce's construction, and realizes it as the
solution of an appropriate Riemann-Hilbert problem.
This is joint work with Tom Bridgeland.
Helene Tyler, The Auslander-Reiten components in the rhombic picture.
Abstract: For an indecomposable module M over a path algebra of a
quiver of type $\widetilde{\mathbb A}_n$, the Gabriel-Roiter measure
gives rise to four new numerical invariants; we call them the
multiplicity, and the initial, periodic and final parts. We describe
how these invariants for M and for its dual specify the position of M
in the Auslander-Reiten quiver of the algebra. This talk is based on
joint work with Markus Schmidmeier.
José Vélez-Marulanda, Universal deformation rings of modules over a certain symmetric special biserial algebra.
Abstract: Click here
Ben Webster, Category O for a quiver variety.
Abstract: One of the best-beloved playgrounds of infinite-dimensional
representation theory is the category O of Bernstein, Gelfand and
Gelfand. I'll show how this is actually a special case of a much
more general construction arising naturally in symplectic algebraic
geometry, and discuss recent advances of understanding in the quiver
variety case with connections to categorical representations of Lie
algebras.
Kunio Yamagata, Morita theory revisited.
Abstract: In his paper on dualities of module categories (1958), K.
Morita gave a characterization for a finite dimensional algebra to be
the endomorphism algebra of a faithful module over a finite dimensional
self-injective algebra. These endomorphism algebras form a class of
algebras properly containing all finite dimensional self-injective
algebras. From a joint work with O. Kerner I will introduce a new
characterization of the endomorphism algebra, and other related results.
Gufang Zhao, Noncommutative desingularization of orbit closures for some GLn-representations.
Abstract: A noncommutative desingularization of determinantal varieties
defined by maximal minors of generic matrices has been studied by
Buchweitz, Leuschke, and van den Bergh. In this talk, we generalize
their results to the orbit closures of other GLn-representations.
We describe a method to calculate the quiver with relations for any
non-commutative desingularizations coming from exceptional collections
over partial flag varieties. I will study noncommutative
desingularizations of generic determinantal varieties, determinantal
varieties defined by minors of generic symmetric matrices, and pfaffian
varieties defined by pfaffians of generic skew-symmetric matrices. We
introduce the language of equivariant quivers to describe these
noncommutative desingularizations more explicitly. For maximal minors
of square matrices and symmetric matrices, this gives a non-commutative
crepant resolution. This talk is based on my joint work with Jerzy
Weyman.