A conference on module theory and its relations to algebraic geometry,
category theory,
commutative algebra, homotopy theory, logic, and
representation theory.
Speaker  Abstract 
Ana Agore 
Extending structures for algebras
The talk is devoted to the dual of the Hochschild extension problem for associative algebras. Let $A$ be an algebra, $E$ a vector space containing $A$ as a subspace and $V$ a complement of $A$ in $E$. All algebra structures on $E$ containing $A$ as a subalgebra are described and classified by two nonabelian cohomological type objects which are explicitly constructed: ${\mathcal A}{\mathcal H}^{2}_{A} \, (V, \, A)$ will classify all such algebras up to an isomorphism that stabilizes $A$ and ${\mathcal A}{\mathcal H}^{2} \, (V, \, A)$ provides the classification up to an isomorphism of algebras that stabilizes $A$ and $V$. A new product, called the unified product, is introduced as a tool of our approach. Different types of split extensions of algebras are fully described in terms of special cases of unified products: in particular, the classical crossed product and its generalizations are special cases of the unified product. 
Silvana Bazzoni  Cotilting modules and homological epimorphisms We study the relation between 1cotilting classes and homological epimorphisms of rings.
We show that for a valuation domain $R$ there is a bijective correspondence between cotilting classes and equivalence classes of injective homological ring epimorphisms originating in $R$.
Partial results and some open problems will be stated for 1cotilting modules over arbitrary rings. 
Clinton Boys 
The graded representation theory of the alternating groups
When new results are proved for symmetric groups, a natural question arises: what do these results imply for the alternating group, the index2 subgroup of the symmetric group consisting of even permutations? In this talk we discuss the implications of the BrundanKleshchev graded isomorphism theorem for the alternating groups.

Simion Breaz 
$\Sigma$pure injective modules
We prove that a right $R$module $M$ is $\Sigma$pure injective if and only if $\Add(M)\subseteq \Prod(M)$. Consequently, the homotopy category $\Htp{\Modr R}$ satisfies the Brown Representability Theorem if and only if the dual category has the same property. We also apply this result to provide new characterizations for right puresemisimple rings or to give a partial positive answer to a question of G. Bergman. 
Martin Doubek  Generalized BV algebras and master equations in string theory via operads
Generalized BV algebras and master equations in string theory via operads
abstract: So called string vertices $S$ in closed string theory are required to satisfy the BV equation $d(S)+h\Delta(S)+1/2\{S,S\}=0$ in certain BV algebra with operations $d, \Delta, \{,\}$. We show how the solutions are equivalently described in terms of standard construction on the cyclic operad $Com$, thus revealing that the solutions are in fact homotopy algebras. The theory generalizes to operads other than $Com$ and yields master equations for open and openclosed string theories.

Mehmet Akif Erdal  The Equivariant James Spectral Sequence
The James spectral sequence, which has been introduced by Teichner in his
Ph.D. thesis, is a generalization of the AtiyahHirzebruch spectral
seqeunce. It converges to the generalized homology of the Thom spectra and
has many geometric applications. Let $G$ be a compact lie group, $U$ be a
complete $G$universe. Let $\xi:E \rightarrow BO_G(U)$ be a
$G$equivariant stable Gauss map. We construct the Thom $G$spectrum
$M\xi$, as a homotopy colimit of a certain functor. This construction
allow us to construct the equivariant $RO(G)$graded version of the James
Spectral Sequence, by using the homotopy colimit spectral sequence.

Sergio Estrada  On the absence of flat objects in finitely accessible categories
The category
$Qcoh(X)$ of quasicoherent sheaves on a quasicompact and quasiseparated scheme $X$ is finitely accessible. Therefore there is the usual categorical notion of flatness on finitely accessible categories. But there is also the standard
definition
of flatness in
$Qcoh(X)$ from the stalks. So it makes sense to wonder the relationship (if
any) between these two notions. In the talk we will discuss on the absence of nonzero categorical flat objects in many finitely accessible categories.
As particular instance, we show that $Qcoh(\mathbf{P}^n(R)))$ has no other
categorical flat objects than zero, where $R$ is any commutative ring.

Gabriella D'Este  
Pedro Antonio Guil Asensio  Hereditary pure torsion classes We define the notion of hereditary pure torsion classes over a ring and of pureinjective module relative to them and show that they generalize both the class of flat modules and of definable subcategories. We also prove that the work of Auslander on simple subfunctors of continuous functors can be extended to them to show that they are determined by sets of indecomposable relative pureinjective modules. The obtained results suggest that some type of topology may be associated to these classes of modules. This is a preliminary report of a join work with I. Herzog and P. Rothmaler. 
Daniel Herden  
Michal Hrbek 
Minimal generating sets of modules
This is a joint work with Pavel Růžička. Unless it is finitely generated, a module may not possess a generating set minimal with respect to inclusion (e.g. nonzero injective abelian groups). If a module has such generating set, it is called weakly based. We give a lucid characterization of weakly based modules over Dedekind domains and show some applications. We also address modules over simple, perfect, and semiartinian rings.

Peter Kalnai  Products of small modules
A module is said to be small if it is not a union of strictly increasing infinite countable chain of submodules. We show that the class of all small modules over selfinjective purely infinite ring is closed under direct products whenever there exists no strongly inaccessible cardinal. Joint work with Jan Zemlicka.

Derya Keskin Tutuncu  
Steffen Koenig  Quasihereditary algebras and boxes
Quasihereditary algebras arise in Lie theory for instance as Schur algebras and as blocks of the BernsteinGelfandGelfand category O. Algebras of global dimension at most two are quasihereditary, too. Such algebras come with socalled standard (Weyl, Verma) modules, and the category of modules with standard filtration is being studied. We describe this category by first considering its Ainfinity structure, then translating this into a differential graded setup and finally identifying it as a category of representations of a box (in the sense of Drozd, Ovsienko and Roiter). As an application we get that up to Morita equivalence, each quasihereditary algebra has an exact Borel subalgebra. In this context, Ringel duality becomes a special case of BurtButler duality.
This is joint work with Julian Kuelshammer and Sergiy Ovsienko. 
Wieslaw Kubis  Homogeneous objects via Katetov functors
We shall present the notion of a Katetov functor, leading to special objects with a strong homogeneity property.
Recall that an object $x$ is homogeneous with respect to the class of objects $K$ if every two embeddings of a fixed $K$object into $x$ are conjugate. The name `embedding' corresponds to a fixed subcategory whose arrows are supposed to be monic in the original categoryi.
Roughly speaking, a covariant functor is Katetov if, together with a fixed natural transformation from the identity, it `realizes' all primitive extensions of objects of a fixed category $S$ of `small' objects. Typical examples of $S$ are categories of finitely generated models of a some firstorder language. The range of a Katetov functor is then the category of all models that can be presented as colimits of directed systems in $S$. Some applications to categories of cell complexes will be given. 
Martin Markl  Natural operations and Koszul hierarchy The Koszul hierarchy (aka higher brackets or Koszul brackets) is an
explicit construction that, for any commutative associative algebra A
with a differential Delta (which is, very crucially, not necessary a
derivation), produces a sequence of multilinear maps
Phi_n : A x ... x A > A (n copies of A) such that (1) the operations Phi_n form a strongly homotopy Lie algebra, and (2) Phi_n = 0 implies Phi_{n+1} = 0 (heredity property) Koszul brackets are used for instance to define higherorder derivations: Delta is a degree n derivation if Phi_{n+1}(Delta) = 0. Higher order derivations play an important role e.g. in BRST approach to closed string field theory. Recently, a similar construction appeared also for associative (noncommutative) algebras. I was able to show that both brackets are given by the twisting by a specific unique automorphism and that they are essentially unique. Consequently, the notion of higherorder derivations is God given, not human invention. The proof is based on careful analysis of a space of natural operations. Here I employed the technique developed in my work with M. Batanin on the Deligne conjecture. As a matter of fact, the core of my theory is a vanilla version of Deligne's conjecture. I plan to focus my talk on this side of the story. 
Andrew Mathas  The graded representation theory of the symmetric groups
The representation theory of the symmetric groups is a classical subject which has its roots in the work of Frobenius and Young at the end of the 18th century. In the semisimple case almost everything is known, however, over finite fields the theory is much harder and many basic questions have yet to be answered. In 2008 there was a huge paradigm shift in this field when Brundan and Kleshchev used work of KhovanovLauda and Rouquier to show that the group algebras of the symmetric groups carry a natural Zgrading. This grading is intimately connected with canonical bases and the representation theory of the affine special linear groups and it gives new meaning to Ariki's Categorification Theorem, which proved the LLT conjecture.
In this talk I will survey these developments, with the aim being to give a taste for the main ideas whilst keeping the technicalities to a minimum. I will explain the KLR grading in terms of the wellunderstood semisimple representation theory of the symmetric groups and end by stating a conjecture for the dimensions of the irreducible modules of the symmetric groups which comes out of this new approach to this classical subject 
Gigel Militaru 
Quasitriangular algebras, braidings on bimodules and central simple algebras
We introduce and study a new class of algebras, we call it \emph{quasitriangular algebras}, as a generalization of the classical concepts of Azumaya algebras or central simple algebras. An algebra $A$ over a commutative ring $k$ is called a quasitriangular algebra if there exists an element $R \in A\ot A\ot A$ satisfying certain axioms. We prove that these elements $R \in A\ot A\ot A$, if exists, are in bijective correspondence to the set of all braidings on the monoidal category ${}_A{\mathcal M}_A$ of $A$bimodules; moreover we prove that any such braiding is a symmetry. The concept of quasitriangular algebra is related to three classical concepts in algebra as follows: (1) a commutative algebra is quasitriangular if and only if $k\to A$ is an \emph{epimorphism} in the category of rings; (2) if the invariants functor $G = ()^A:\ {}_A{\mathcal M}_A\to {\mathcal M}_k$ is a \emph{separable} functor, then $A$ is quasitriangluar; (3) any Azumaya algebra is a quasitria
ngular algebra. 
George Ciprian Modoi 
Some applications of deconstructibility in triangulated categories
We define deconstructible subcategories in triangulated categories in analogy with the abelian case. We show that a triangulated category with coproducts satisfies Brown representability, provided that itself is deconstructible. We use this result in order to indicate some triangulated categories which satisfy Brown representability, included the opposite of the derived category of a Grothendieck abelian category. 
Sinem Odabasi  
Flaviu Pop  On direct products and Ext^1 contravariant functors This is a joint work with Claudiu Valculescu. We show, using inequalities between infinite cardinals, that, if R is an hereditary ring, the contravariant derived functor Ext^1_R(−,G) commutes with direct products if and only if G is an injective Rmodule. 
David Pospisil  A classification of compactly generated cotstructures for commutative noetherian rings I will present some results from [1]. Namely the classification of compactly generated Homorthogonal pairs in a triangulated category and the classification of compactly generated cotstructures in a derived category of a commutative noetherian ring. [1] J. Stovicek, D. Pospisil, "On compactly generated torsion pairs and the classification of cotstructures for commutative noetherian rings", submitted to TAMS, 2013, arXiv: 1212.3122 
Mike Prest  Almost dual pairs and definable classes of modules
This is joint work with Akeel Ramadan Mehdi. Lenzing investigated categories of modules with support in (that is, which are in the $\varinjlim$closure of) categories of finitely presented left modules. Recently H. Holm considered the categories of right modules which arise as (character)duals of those with support in some category of finitely presented modules. By placing Holm's results in the context of elementary duality on definable subcategories we are able to extend them. In doing so we also prove that dual modules have enough indecomposable direct summands.
H. Holm, Modules with cosupport and injective functors, Algebr. Represent. Theor., 13 (2010), 543560. H. Lenzing, Homological transfer from finitely presented to infinite modules, pp. 734761 in Abelian Group Theory, Lecture Notes in Math., Vol. 1006, SpringerVerlag, 1983. A. R. Mehdi and M. Prest, Almost dual pairs and definable classes of modules, University of Manchester, preprint, 2013, arXiv:1304.4481 
Pavel Prihoda  Large projective modules over small symmetric groups I will recall several basic notions from
representation theory of finite groups that can be
applied to find idempotent ideals of integral group ring
of a finite group. Hopefully I will be able to show how
it works in some small examples. 
Oriol Raventos 
Obstruction theory for the representability of cohomological functors
We will describe an obstruction theory that enables us to decide if a cohomological functor defined from a subcategory of a triangulated category to the category of abelian groups is representable. We will also discuss the relation with the transfinite purity of rings and the transfinite Adams representability problem. 
Adam van Roosmalen  Serre functors and derived equivalences
This talk is based on joint work with Donald Stanley where we study the
role of the Serre functor in the theory of derived equivalences. Let A be
a klinear (k a field) abelian category such that the bounded derived
category Db A has a Serre functor, and let H be the heart of a
tstructure (D^{\leq 0}, D^{\geq 0}) on Db A. One can verify that if the
realization functor Db H > Db A is a triangle equivalence, then (1) the
tstructure is bounded and (2) D^{\leq 0} is closed under the Serre
functor. In this talk, we will consider some cases where the converse
holds as well.

Jiri Rosicky  A functorial form of Shelah's singular compactness theorem
We present a functorial form of Shelah's singular compactness theorem where
"free" objects are those belonging to the image of a functor. As a consequence,
we get a cellular form where "free" objects are cellular ones. This form covers
all known instances of Shelah's singular compactness theorem, i.e.,
Sfiltered modules, transversals and colourings of graphs.

Serap Sahinkaya  
Alexander Slavik  
Jan Stovicek 
Derived equivalences induced by big cotilting modules
Big cotilting modules have been originally studied as a well behaved
formal generalization of finite dimensional cotilting modules from
representation theory. Generalizing results of Happel, Reiten and Smalo,
and of Colpi, Gregorio and Mantese, I will show that big cotilting
modules are precisely images of injective cogenerators of Grothendieck
categories under certain tilting derived equivalences. If time permits,
I will discuss more aspects such as that the derived equivalence is
induced by a Quillen equivalence and that this induces an equivalence of
the corresponding Grothendieck derivators.

Carlo Toffalori 
Some model theory of modules over Bezout domains
This is a joint work with Gena Puninski. We develop the model theory of modules over commutative B\'ezout domains, in particular we characterize the domains whose lattice of ppformulae has no width and give some applictaions to the existence of superdecomposable pure injective modules. 
Jan Trlifaj 

Jan Zemlicka  Modules with restricted minimum condition
A module $M$ satisfies the restricted minimum condition if
$M/N$ is artinian for every essential submodule $N$ of $M$.
We discuss several properties of this class of modules.
