Last updated: April 25, 2017, 18:44 EDT

Javad Asadollahi, Categorical resolutions of bounded derived categories of artin algebras.

Abstract: Recently, Pu Zhang has proved that if an artin algebra $\Lambda$ has a module $T$ of finite injective dimension such that its left orthogonal ${}^\perp T$ is of finite type, then the bounded derived category of $\Lambda$ admits a categorical resolution. We generalize this result and show that bounded derived category of every artin algebra admits a categorical resolution. This, in particular, implies that bounded derived categories of artin algebras of finite global dimension determine bounded derived categories of all artin algebras. This in a sense provides a categorical level of Auslander's result stating that artin algebras of finite global dimension determine all artin algebras.


Frauke Bleher, Non-commutative deformation rings.

Abstract: In the 1980's Mazur, using work of Schlessinger, introduced techniques of deformation theory to systematically study lifts of mod p representations of Galois groups to commutative p-adic rings. In this talk we will discuss how to generalize this deformation theory to study lifts of mod p representations to suitable non-commutative local rings. In particular, we will present some examples that will highlight the differences between the commutative and the non-commutative deformation theory of Galois representations. This is joint work with Ben Margolin.


Benjamin Briggs, The characteristic action of Hochschild cohomology on the derived category.

Abstract: This is joint work with Vincent Gelinas. If A is a dg algebra over a field k, one has the well-known characteristic homomorphism \chi from HH^*(A,A) to the graded centre of the derived category D(A). While many authors have exploited this map, not a lot is known about its behaviour in general. We are interested in the images of the induced morphisms \chi_M : HH^*(A,A) \to Ext_A(M,M). We prove that \chi can also be computed as the edge map in a spectral sequence computing the Hochschild cohomology of the full (enhanced) derived category D^{dg}(A). Thus, \chi is actually independent of A (i.e. it is derived Morita invariant). This description allows us to constrain the image of \chi_M in terms of the natural A_\infty structure on Ext_A(M,M). Lastly, when A is augmented over k, we interpret this in terms of Koszul duality. We describe the image of chi_k precisely, generalising a theorem of Buchweitz, Green, Snashall and Solberg from the case of Koszul algebras. This is partly aimed at support theory, where it is important to understand when Ext_A(k,k) is finitely generated over the image of \chi_k.


Thomas Brüstle, On kissing numbers and a basis for Ext$^1$ over gentle algebras.

Abstract: This is a report on joint work with Guillaume Douville, Kaveh Mousavand, Hugh Thomas and Emine Y\i ld\i r\i m. We like to point out that Yann Palu, Vincent Pilaud, and Pierre-Guy Plamondon informed us that they are working on similar ideas. Furthermore, we ought to mention that Ilke Canakci, David Pauksztello and Sibylle Schroll have obtained similar results in arXiv:1609.09688, however based on different techniques.
The basic question is to find a nice combinatorial description of a basis for Ext$_A^1(X,Y)$ when $A$ is a gentle algebra. So why are so many people working on that subject at this moment? In fact, one might wonder: Why do we not yet have such a nice description, given that the work of Crawley-Boevey provides a nice combinatorial basis for Hom$_A(X,Y)$, and already thirty years ago Butler and Ringel gave a nice combinatorial description of the Auslander- Reiten translate, both results valid for string algebras? Well, it turns out that all experts found insurmountable difficulties when trying to put things together for the case of string algebras. And only recently, progress has been made when restricting to gentle algebras: Canakci, Pauksztello and Schroll manage to describe the Hom spaces between indecomposables in the derived category, which yields the desired description of Ext$^1$. Another approach follows the idea of adding some arrows to the quiver to make what we call a "fringed" quiver. This new, enlarged algebra is still gentle, but any vertex of the original algebra has two incoming and two outgoing arrows: this makes the combinatorics of Butler and Ringel a lot more pleasant. Another pleasant surprise is that Crawley-Boevey’s basis elements re-appear in the work of Thomas McConville as “kisses” in a certain situation. The idea is to count kisses to get the dimension of Hom$(X, \tau Y)$, and show which of those kisses provide a basis for Ext$^1(Y,X).$


Jon Carlson, Separable ring objects in the stable category.

Abstract: This is joint work with Paul Balmer. I will discuss the classification of separable ring objects in the stable category of the group algebra of a cyclic p-group. Along the way we characterize the Kelly radical of the module and stable categories.


Man Wai Cheung, Quiver representations and theta functions.

Abstract: Scattering diagrams theta functions and broken lines were developed in order to describe toric degenerations of Calabi-Yau varieties and construct mirror pairs. Later, Gross-Hacking-Keel-Kontsevich unravel the relation of those objects with cluster algebras. In the talk, we will discuss how we can combine the representation theory with these objects. We will also see how the broken lines on scattering diagram give a stratification of quiver Grassmannians using this setting.


Calin Chindris, Decomposing moduli of representations of finite- dimensional algebras

Abstract: This talk is based on joint work with Ryan Kinser. It is about studying representations of finite-dimensional algebras within the general framework of Geometric Invariant Theory (GIT). This interaction between representations of algebras and GIT leads to the construction of moduli spaces of representations as solutions to the classification problem of semi-stable representations, up to S-equivalence. In this talk, I will present a Krull-Schmidt type decomposition for moduli spaces for arbitrary finite-dimensional algebras. If time permits, I will discuss some applications of this decomposition result.


Raquel Coelho Simoes, Negative Calabi-Yau triangulated categories.

Abstract: Calabi-Yau (CY) triangulated categories are those satisfying a useful and important duality, characterised by a number called the CY dimension. Much work has been carried out on understanding positive CY triangulated categories, especially in the context of cluster-tilting theory. Even though CY dimension is usually considered to be a positive (or fractional) number, there are natural examples of CY triangulated categories where this “dimension” or parameter is negative, for example, stable module categories of self-injective algebras. Therefore, negative CY triangulated categories constitute a class of categories that warrant further systematic study. In this talk, we will consider an important class of generating objects of negative CY triangulated categories, namely simple-minded systems and study their mutation behaviour. We will focus on an example given by triangulated categories generated by spherical objects, whose combinatorics plays a useful role in the study of its representation theory.


Samuel Dean, Recollements of functor categories.

Abstract: I will give a theorem for constructing recollements of abelian categories. Its hypotheses are commonly satisfied, so I will give many examples. One of the examples shows that the Auslander-Gruson-Jensen duality can be extended to a recollement. I will explain the similarities between this and a recollement which is due to Krause. This gives a structural explanation for why we have two equivalent notions of purity for modules (since the kernel in both of these recollements is the category of pure exact sequences).


Georgios Dalezios, Quillen equivalences for stable categories of Gorenstein projective and injective objects.

Abstract: For an abelian category, we investigate when the stable category of Gorenstein projective objects and its injective analogue are triangulated equivalent. To this end, we realize these stable categories as homotopy categories of certain (non-trivial) model categories and give conditions that ensure the existence of a Quillen equivalence between the model categories in question. This work is joint with S.Estrada (Murcia) and H.Holm (Copenhagen).


Harm Derksen, Invariant theory for quiver representations.

Abstract: Given a quiver and a dimension vector, we have an action of a product of general linear groups on the representation space. We can study the ring of (semi-)invariants for this action. I will give an overview of what is known about these invariant rings and the geometric meaning. I will also discuss a recent joint result with Visu Makam which gives a polynomial degree bound for the degrees of generating semi-invariants for quivers.


Ivon Dorado, AR-quivers of some p-equipped posets,

Abstract: In a joint work with Raymundo Bautista, we show that the categories of representations and corepresentations of p-equipped posets are equivalent to subcategories of modules over certain algebras. In fact, the subcategory of their finitely generated objects has almost split sequences. Then, we describe some properties of a component of the Auslander-Reiten quiver of p-equipped posets. These results allow us to construct, in the usual way, the AR-quiver of some p-equipped posets.


Karin Erdmann, Support varieties for modules of selfinjective algebras.

Abstract: This is an expository talk.


Sergio Estrada, Periodic modules and acyclic complexes.

Abstract: We study the behavior of modules $M$ that fit into a short exact sequence $0\to M\to C\to M\to 0$, where $C$ belongs to a class of modules $\mathcal C$, the so-called $\mathcal C$-periodic modules. We give a rather general framework to improve and generalize some well-known results of Benson and Goodearl and Simson. We will combine techniques of hereditary complete cotorsion pairs and presentation of direct limits, to conclude, among other applications, that if $M$ is any module and $C$ is cotorsion, then $M$ will be also cotorsion. This will lead to some meaningful consequences in the category $\textrm{Ch}(R)$ of unbounded chain complexes. This is based on a joint work with Silvana Bazzoni and Manuel Cort\'es Izurdiaga.


Eleonore Faber, Endomorphism rings of conic modules over toric algebras.

Abstract: In this talk we consider normal toric algebras R over a field of characteristic p >0. The module M of p^e-th roots of R is then the direct sum of so-called conic modules. With a combinatorial method we construct certain complexes of conic modules over R and explain how these yield projective resolutions of modules over the endomorphism ring End_R(M). Thus we obtain a bound on the global dimension of End_R(M), which shows that this endomorphism ring is a so-called noncommutative resolution of singularities (NCR) of R (or Spec(R)). This is joint work with Greg Muller and Karen E. Smith.


Pedro Fernando Fernández Espinosa, Categorification of some integer sequences.

Abstract: The term categorification of an integer sequence was introduced by Fahr and Ringel to the process which consists of considering numbers in the sequence as invariants of objects of a given category in such a way that identities between numbers in the sequence can be considered as functional relations between objects of the category. Ringel and Fahr gave a categorification of Fibonacci numbers by using preprojective and regular components of the 3-Kronecker In this talk we describe how Kronecker modules can be used to categorize the integer sequence A052558 in the OEIS. We recall that such sequences count the number of ways of connecting n+1 equally spaced points on a circle with a path of n line segments ignoring reflections.


Alexander Garver, Minimal length maximal green sequences.

Abstract: Maximal green sequences are important objects in representation theory, cluster algebras, and string theory. It is an open problem to determine the set of lengths of the maximal green sequences of a quiver. We use the combinatorics of surface triangulations and the basics of scattering diagrams to address this problem. Our main result is a formula for the length of minimal length maximal green sequences of quivers defined by triangulations of a punctured disk or an unpunctured annulus. We will also discuss how minimal length maximal green sequences are related to the derived equivalence classification of the Jacobian algebras associated with these surfaces. This is joint work with Thomas McConville and Khrystyna Serhiyenko.


Vincent Gelinas, Tilting for MCM modules over homogeneous complete intersections.

Abstract: Stable categories of graded Maximal Cohen-Macaulay modules over a (sufficiently nice) graded Gorenstein ring are known to admit tilting objects in dimension zero, due to Yamaura, and in dimension one, due to recent work of Buchweitz-Iyama-Yamaura. This talk will discuss a natural obstruction in higher dimension for Gorenstein rings of geometric origin, exemplified with the classification of homogeneous complete intersection k-algebras of dimension $\geq 2$ admitting a tilting module.


Edward Green, Modules via Gr\"obner bases.

Abstract: I will introduce a new approach to the study of modules over a ring KQ/I, I admissible.


Ivo Herzog, Coordinatization & quantum logic via coherent functors.

Abstract: The implicit context for a large portion of the work of Maurice Auslander is the category of coherent functors. I will give a brief introduction to this deep influential idea and I will attempt draw a link with 1) the classical work of Artin and von Neumann on coordinatization of lattices and 2) quantum logic.


Lutz Hille, Stable representations of quivers.

Abstract: Stable representations have first appeared in the work of King on moduli spaces. Later they became important also in connection with cluster algebras. There are essentially two approaches, stability with respect to a weight (an element in the dual of the Grothendieck group) and stability with respect to a slope. The second approach seems not to be considered, even it has much better properties. We give an overview to both stability notions and some known results and also mention some new developments.


Alina Iacob, Totally acyclic complexes.

Abstract: It is known that over an Iwanaga-Gorenstein ring the Gorenstein projective (Gorenstein injective, Gorenstein flat) modules are simply the cycles of the exact complexes of projective (injective, flat) modules. We consider the question: are these characterizations working only over Iwanaga-Gorenstein rings? Among other results we prove the following theorem:
Let R be a two sided noetherian ring of finite finitistic flat dimension that satisfies the Auslander’s condition. The following are equivalent:
1. R is Iwanaga-Gorenstein.
2. Every acyclic complex of injective left (right) R-modules is totally acyclic.


David Jorgensen, Totally acyclic approximations.

Abstract: We give approximations of totally acyclic complexes by simpler ones via an adjoint pair of triangle functors. We also discuss various applications.


Ryan Kinser, Geometry of type D quiver representations.

Abstract: In previous joint work with Jenna Rajchgot, we found a close connection between type A quiver representation spaces and Schubert varieties in partial flag varieties. In this talk, I will discuss our recent generalization of this work to type D quivers, where partial flag varieties are replaced with symmetric spaces GL(a+b)/GL(a)xGL(b).


Mark Kleiner, Preprojective roots of Coxeter groups.

Abstract: Certain results on representations of quivers have analogs in the structure theory of general Coxeter groups. A fixed Coxeter element turns the Coxeter graph into an acyclic quiver, allowing for the definition of a preprojective root. A positive root is an analog of an indecomposable representation of the quiver. The Coxeter group is finite if and only if every positive root is preprojective, which is analogous to the well-known result that a quiver is of finite representation type if and only if every indecomposable representation is preprojective. Combinatorics of orientation-admissible words in the graph monoid of the Coxeter graph relates strongly to reduced words and the weak order of the group.


Maitreyee Kulkarni, Cylinders over Dynkin diagram and cluster algebras.

Abstract: Let G be a Lie group of type ADE and P be a parabolic subgroup. It is known that there exists a cluster structure on the coordinate ring of the partial flag variety G/P (see the work of Geiss, Leclerc, and Schroer). Since then there has been a great deal of activity towards categorifying these cluster algebras. Jensen, King, and Su gave a direct categorification of the cluster structure on the homogeneous coordinate ring for Grassmannians (that is, when G is of type A and P is a maximal parabolic subgroup). In this setting, Baur, King, and Marsh gave an interpretation of this categorification in terms of dimer models. In this talk, I will give an analog of dimer models for groups in other types by introducing a technique called “constructing cylinders over Dynkin diagrams”, which can (conjecturally) be used to generalize the result of Baur, King, and Marsh.


Haydee Lindo, Trace ideals and rigidity.

Abstract: I will present some new results regarding trace ideals of modules and algebras over commutative rings. This continues the project begun in arXiv:1603.08576 relating the center of the endomorphism ring of a module M, over a commutative noetherian ring, to the endomorphism ring of the trace ideal of M.


Isaias David Marín Gaviria, Representations of equipped posets and its applications.

Abstract: The theory of representation of equipped posets was introduced by A.G. Zavadskij and his students. This theory arises as a natural generalization of the classical theory of representation of posets introduced and developed by Nazarova, Roiter and their students in the 1970's. Now we know that according to Bautista and Dorado the theory of representation of equipped posets is a particular case of the classifcation of the so called algebraically equipped posets. In this talk, we describe a categorifcation (in the sense of Ringel and Fahr) of Delannoy arrays by using suitable lattice paths and some ideas arising from the theory of representation of equipped posets. Aplications of these results in visual cryptography are described as well.
Joint with; Pedro Fernandez (pfernandez@unal.edu.co), and Julian Serna (rjsernav@unal.edu.co).
References
[1] R. Bautista, I. Dorado, and l, Algebraically equipped posets, arXiv 1501.03074v1 (2015).
[2] P. Fahr and C. M. Ringel, A partition formula for Fibonacci numbers, Journal of integer sequences 11 (2008), no. 08.14.
[3] P. Fahr and C.M. Ringel, Categorification of the Fibonacci numbers, arXiv 1107.1858v2 (2011), 1-12.
[4] H. Koga and H. Yamamoto, Proposal of a Lattice-based Visual Secret Sharing Scheme for Color and Gray-scale Images, IEICE TRANS. FUNDAMENTALS E81-A (1998), no. 6, 1262-1269.
[5] A.G. Zavadskij, Representations of generalized equipped posets and posets with automor- phisms over Galois field extensions, Journal of Algebra 332 (2011), no. 1, 386-413.


Frantisek Marko, Cryptosystem based on invariants of groups.

Abstract: We describe a public-key cryptosystem based on invariants of groups and formulate a problem in invariant theory related to the security of such cryptosystem.


Jacob Matherne, Combinatorial Fourier transform for type A quiver representation varieties.

Abstract: For a given dimension vector d, we consider the space of representations of the linearly-oriented type A quiver. This affine space has a stratification by orbits for a product of general linear groups, where the orbits are isomorphism classes of representations with dimension vector d. The Fourier--Sato transform, a geometric version of the Fourier transform that we meet in analysis, is a functor which matches up orbits for this quiver with orbits for the reversed quiver in an interesting way. We introduce certain triangular arrays of nonnegative integers and, with them, give a combinatorial algorithm for computing the Fourier--Sato transform in this setting. This is joint (in progress) work with Pramod N. Achar and Maitreyee Kulkarni.


Agustín Moreno Cañadas, Representation of equipped posets and its applications.

Abstract: The theory of representation of equipped posets was introduced by A.G. Zavadskij and his students. This theory arises as a natural generalization of the classical theory of representation of posets introduced and developed by Nazarova, Roiter and their students in the 1970's. Now we know that according to Bautista and Dorado the theory of representation of equipped posets is a particular case of the classi cation of the so called algebraically equipped posets. In this talk, we describe a categori cation (in the sense of Ringel and Fahr) of Delannoy arrays by using suitable lattice paths and some ideas arising from the theory of representation of equipped posets. Aplications of these results in visual cryptography are described as well. Joint with: Pedro Fernand ez (pfernandez@unal.edu.co), Isaias Mar  in (imaringa@unal.edu.co) and Julian Serna (rjsernav@unal.edu.co).
References
[1] R. Bautista, I. Dorado, Algebraically equipped posets, arXiv 1501.03074v1 (2015).
[2] P. Fahr and C. M. Ringel, A partition formula for Fibonacci numbers, Journal of integer sequences 11 (2008), no. 08.14.
[3] H. Koga and H. Yamamoto, Proposal of a Lattice-based Visual Secret Sharing Scheme for Color and Gray-scale Images, IEICE TRANS. FUNDAMENTALS E81-A (1998), no. 6, 1262-1269.
[4] A.G. Zavadskij, Representations of generalized equipped posets and posets with automorphisms over Galois field extensions, Journal of Algebra 332 (2011), no. 1, 386-413.


Van Nguyen, Preprojective algebras of tree-type quivers.

Abstract: In this talk, we recollect several descriptions of preprojective algebras and show that these descriptions are indeed equivalent for any tree-type quiver $Q$. In particular, we construct irreducible morphisms, in the Auslander-Reiten quiver of the transjective component of the bounded derived category of its path algebra $kQ$, that satisfy what we call the $\lambda$-relations, where $\lambda$ a nonzero element in the field $k$. When $\lambda = 1$, the relations are known as mesh relations. When $\lambda = −1$, they are known as commutativity relations. Using this technique together with the results given by Baer-Geigle-Lenzing, Crawley-Boevey, Ringel, and others, we show that for any tree- type quiver, several descriptions of its preprojective algebra are equivalent.


Charles Paquette, Group actions on cluster categories and cluster algebras.

Abstract: We introduce the notion of admissible action of a group G on a quiver with potential (Q,W). This induces an action of G on the corresponding cluster category C(Q,W) and on the corresponding cluster algebra A(Q). At the level of C(Q,W), this yields a G-precovering functor F: C(Q,W) \to C(Q_G, W_G) where Q_G is the orbit quiver and W_G is the orbit potential. This functor is compatible with the Iyama-Yoshino mutation of a G-orbit of a summand of a cluster-tilting object of C(Q,W). At the level of the cluster algebra A(Q), the action of G yields an algebra A_G of G-orbits of the cluster variables of the G-stable clusters. This gives generalized cluster algebras that are different from the ones introduced by Lam and Pylyavskyy. For cluster algebras arising from surfaces, those algebras are associated to some triangulated orbifolds. As in the classical case of an oriented Riemann surface with marked points, the algebra can be obtained by mutations that are specified by exchange polynomials. These algebras are different from the ones defined by Felikson- Shapiro-Tumarkin. A cluster character can also be defined to relate the category C(Q_G, W_G) to the algebra A_G.


David Pauksztello, Discrete triangulated categories.

Abstract: The notion of a discrete derived category was first introduced by Vossieck, who classified the algebras admitting such a derived category. Due to their tangible nature, discrete derived categories provide a natural laboratory in which to study concretely many aspects of homological algebra. Unfortunately, Vossieck's definition hinges on the existence of a bounded t-structure, which some triangulated categories do not possess. Examples include triangulated categories generated by 'negative spherical objects', which occur in the context of higher cluster categories of type A infinity. In this talk, we compare and contrast different aspects of discrete triangulated categories with a view toward a good working definition of such a category. This is a report on joint work with Nathan Broomhead and David Ploog.


Mike Prest, Additive model theory.

Abstract: Model theory and algebra fit together rather well in the context of modules.  This is partly explained by the pp-elimination of quantifiers theorem in the model theory of modules and by the very large overlap between model-theoretic and functor-category-theoretic methods in this context.  Still, model theory offers a particular perspective and techniques which can be useful in both discovery and proof.  I will try to convey something of that perspective and describe some key ideas from model theory.


Gena Puninski, The almost split sequences in definable categories.

Abstract: There are two classical settings, where the existence of almost split sequences was established by Auslander: the category of finite dimensional modules over finite dimensional algebras, and the category of finitely generated Cohen- Macaulay modules over an isolated singularity. There is another classical result, also due to Auslander: if M is a finitely presented module with a local endomorphism ring over a ring R, then M is a sink of an almost split sequence in the category of all R-modules. We will show that an analogue of this Auslander result holds true for any definable category closed with respect to extensions. The proof uses a model theoretic approach developed by Ivo Herzog.


Michelle Rabideau, F-polynomial formula from continued fractions.

Abstract: We work in the setting of cluster algebras from surfaces with principal coefficients. There is a correspondence between arcs on the surface, cluster variables in the cluster algebra and snake graphs. In fact, there is a formula for cluster variables and thus F-polynomials in terms of the perfect matchings of snake graphs. It is also known that the number of perfect matchings of a snake graph is equal to the numerator of the positive continued fraction describing that snake graph. Moreover, there is a continued fraction equation for cluster variables with trivial coefficients. A natural question is to investigate the relationship between the positive continued fraction and the F-polynomial of a cluster variable. In this talk we describe how the F-polynomial is determined by a continued fraction of Laurent polynomials.


Dylan Rupel, On generalized minors and quiver representations.

Abstract: A fundamental result in the theory of cluster algebras due to Berenstein, Fomin, and Zelevinsky is the existence of (upper) cluster algebra structures on the coordinate rings of the double Bruhat cells in semi-simple algebraic groups. In every case, the initial cluster consists of a collection of generalized minors with generalizations of the classic Jacobi- Desnanot identity for minors of a matrix providing many of the exchange relations. However, most non-initial cluster variables are yet undocumented functions on the group and, in particular, they are usually expected not to be generalized minors. In the case of Coxeter double Bruhat cells, the cluster algebra is acyclic and via categorification cluster variables can be understood in terms of the representation theory of an acyclic quiver. In this talk I will describe how this categorification can be used to show that all cluster variables for Coxeter double Bruhat cells in affine type A are given by generalized minors and then discuss new identities among generalized minors resulting from these observations.


Jeremy Russell, (Co-)torsion via stable functors.

Abstract: The injective and projective stabilizations of additive functors will be used to define the torsion submodule of a module and the cotorsion quotient module of a module. This is done in complete generality, without any restrictions on rings or modules.  Over commutative domains the newly defined torsion coincides with the classical torsion, and for finitely presented modules over arbitrary rings it coincides with the 1-torsion (also known as the Bass torsion). The notion of the cotorsion module of a module does not seem to have a classical analog. On the level of functor categories, the Auslander-Gruson-Jensen functor sends cotorsion to torsion. If the injective envelope of the ring, viewed as a module over itself, is finitely presented, then the right adjoint of the Auslander-Gruson-Jensen functor sends torsion to cotorsion. In particular, over an artin algebra this gives a duality between torsion and cotorsion. This is joint work with Alex Martsinkovsky.


Julia Sauter, Shifted modules and desingularizations.

Abstract: We prove that finite-dimensional algebras of positive dominant dimension possess uniquely determined tilting (and cotilting) modules containing a higher cosyzygy of the algebra (resp. a higher syzygy of the injective generator). We refer to these modules as (co)shifted modules and to their endomorphism rings to (co)shifted algebras. If the dominant dimension is at least 2, we prove that shifted algebras come with a recollement from which we obtain the dual of the shifted module as an intermediate extension. In that case, we also show that shifted algebras are isomorphic to endomorphism rings in certain homotopy categories. As geometric applications we have
(1.) a realizations of rank varieties (i.e. closed subvarieties of the representation space given by Hom-inequalities) as affine quotient varieties,
(2.) a desingularization of orbit closures in the representation space provided the module is quotient-finite (i.e. there are only finitely-many isomorphism classes of quotients).
(3.) in quiver Grassmannians we look at the closure of a locus of all submodules isomorphic to a given module N. We find a desingularization of these subvarieties provided that the module N is quotient-finite.


Ralf Schiffler, Cluster algebras and continued fractions.

Abstract: This talk is on a combinatorial realization of continued fractions in terms of perfect matchings of the so-called snake graphs, which are planar graphs that have first appeared in expansion formulas for the cluster variables in cluster algebras from triangulated marked surfaces. I will also explain applications to cluster algebras, as well as to elementary number theory. This is  a  joint work with Ilke Canakci.


Sibylle Schroll, Varieties and algebras.

Abstract: This is joint work with Ed Green and Lutz Hille. Let K be a field. We construct an affine algebraic variety V whose points are K-algebras. In particular, every algebra of the form KQ/I is in such a variety. By construction, V contains a distinguished point corresponding to a monomial algebra. We show that this monomial algebra governs many of the properties of any algebra in V.


Emre Sen, Phi dimension of cyclic Nakayama algebras.

Abstract: Igusa and Todorov introduced phi function which generalizes the notion of projective dimension. We study the behavior of the phi functions for cyclic Nakayama algebras of infinite global dimension. In particular we show that phi dimension is 2 if and only if algebra satisfies certain symmetry conditions.


Alexander Slávik, Very flat modules and quasi-coherent sheaves.

Abstract: The class of very flat modules over a commutative ring has many nice properties; for instance, the notion of very flatness is Zariski-local, therefore it makes perfect sense to define very flat quasi-coherent sheaves. The main idea is to use this class as a remedy for the lack of projective quasi-coherent sheaves, replacing the usual approach (due to Murfet) via flat sheaves. We show that the derived categories of the exact categories of very flat and flat sheaves are equivalent for any quasi-compact semi-separated scheme. Joint work with S. Estrada.


Oeyvind Solberg, Going relative with Maurice - a survey.

Abstract: In four joint research papers and a survey paper with Maurice Auslander we studied relative homological algebra over artin algebras.  In this talk I will give the foundation of this theory, review some of the problems we studied and look at some applications of relative homological theory.


Greg Stevenson, Bounded derived categories, normalisations, and duality.

Abstract: Given a ring, one is frequently led to consider the bounded derived category of suitably finite modules; this category, from some points of view, has much better properties than the perfect complexes and admits interesting Verdier quotients such as the singularity category. I'll discuss joint work with John Greenlees proposing an analogue of the bounded derived category for certain differential graded rings, in terms of a kind of Noether normalisation. In particular, I'll highlight the way in which, given a sufficiently nice normalisation, one obtains a pleasingly symmetric situation upon passing to Koszul duals, together with examples demonstrating that this applies in many cases of interest.


Hipolito Treffinger, Stability conditions, tau-tilting theory and maximal green sequences.

Abstract: This is joint work with Thomas Brüstle and David Smith. In this talk we will study how the tau-tilting modules introduced by Adachi, Iyama and Reiten induce stability conditions as introduced by King. This will allow us to tag parts of the wall and chamber structure of a given algebra via its tau-tilting modules. Later, we will introduce the green paths in the wall and chamber structure of an algebra and we will use them to give a characterization of the maximal green sequences in the module category of the algebra. Note that some of the results contained in this talk were found independently by Thomas and Speyer.


Jan Trlifaj, Locality for sheaves associated with tilting.

Abstract: A classic result by Raynaud and Gruson says that the notion of a vector bundle is Zariski local. Vector bundles are particular instances, for n = 0, of the more general notions of quasicoherent sheaves induced by n-tilting modules. Using a recent classification of tilting classes over commutative rings, we show that also these general notions are Zariski local, for all $n$ and all schemes (joint work with Michal Hrbek and Jan Stovicek).


Jose Velez-Marulanda, On universal deformation rings for Gorenstein algebras.

Abstract: Let k be an algebraically closed field, and let Λ be a finite dimensional k-algebra. We prove that if Λ is a Gorenstein algebra, then every (maximal) Cohen-Macaulay Λ-module V whose stable endomorphism ring is isomorphic to k has a universal deformation ring R(Λ,V), which is a complete local commutative Noetherian k-algebra with residue field k, and which is also stable under taking syzygies. We investigate a particular non-self-injective Gorenstein algebra Λ_0, which is of infinite global dimension and which has exactly three isomorphism classes of finitely generated indecomposable Cohen-Macaulay Λ_0-modules V whose stable endomorphism ring is isomorphic to k. We prove that in this situation, R(Λ_0,V) is isomorphic either to k or to k[[t]]/(t^2).


Shijie Zhu, Dominant dimension and tilting modules.

Abstract: We study which algebras have a tilting module which is both generated and cogenerated by projective-injective modules. Auslander algebras have such a tilting module and for algebras of global dimension 2, Auslander algebras are classified by the existence of such a tilting module. We show that, independently of global dimension, the existence of such a tilting module is equivalent to the algebra having dominant dimension at least 2. Furthermore, such a tilting module is also a cotilting module if and only if the algebra is 1-Auslander-Gorenstein. As a special family, we show that algebras extended from Auslander algebras by certain injective modules admit a tilting module which is generated and cogenerated by projective-injective modules. This is joint work with V. Nguyen, I. Reiten, and G. Todorov.
 
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