Christopher Brav: The Projective McKay correspondence

Kirillov has described a McKay correspondence for finite subgroups of PSL_{2}(C) that associates to each `height' function an affine Dynkin quiver together with a derived equivalence between equivariant sheaves on the projective line P^1 and representations of this quiver. The equivalences for different height functions are then related by reflection functors for quiver representations. Our main goal is to develop an analogous story for the cotangent bundle of P^1. We show that each height function gives rise to a derived equivalence between equivariant sheaves on the cotangent bundle T*P^1 and modules over the preprojective algebra of an affine Dynkin quiver. These different equivalences are related by spherical twists, which take the place of the reflection functors for P^1.

Ragnar-Olaf Buchweitz: The stable derived category of a Gorenstein ring

In this talk we define the category in the title and give various different description of it, as the homotopy category of totally acyclic complexes or as the stable module category of maximal Cohen-Macaulay modules, or, in case of a hypersurface, as the homotopy category of matrix factorisations. We explain the construction of (universal) complete resolutions as the analogue of projective resolutions in the stable case. We then exhibit some of the features of the associated Tate-extension groups, such as Serre duality and annihilation through Noether differents.

Ragnar-Olaf Buchweitz: The stable derived category of a graded Gorenstein ring after Orlov

We sketch the proof of Orlov's theorem that establishes a precise comparison between the derived category on coherent sheaves on the projective (virtual) space underlying a graded Gorenstein ring and the stable category of graded maximal Cohen-Macaulay modules over it, resulting in an equivalence when the ring is Calabi-Yau. We explain how to make the correspondence concrete in terms of "saturated Betti tables". We pay particular attention to the numerous questions -- and potential research projects -- that follow naturally from this correspondence. Time permitting, we will sketch a generalisation by Herbst, Hori, and Page of this result in the toric case, which vastly enriches our knowledge of derived equivalences between Calabi-Yau categories and produces families of realisations parametrised by the moment map of the given torus action.

Igor Burban: Cohen-Macaulay modules over surface singularities

Abstract: In this talk,  I am going to make a survey  of various results on Cohen-Macaulay modules over surface singularities. In particular, I shall explain  the classification of indecomposable Cohen-Macaulay modules  over simple, quotient and minimally elliptic singularities using  the geometric and algebraic McKay Correspondence.

Igor Burban: Cohen-Macaulay modules over non-isolated surface singularities

:  This talk is based on my joint work with Yuriy Drozd. Let (A,m) be a complete Noetherian local k-algebra of Krull dimension two. In works of Artin- Verdier, Auslander and Esnault it was proven that the category of maximal Cohen-Macaulay modules CM(A) over R has finite representation type if and only if A is a quotient singularity. Later  Dieterich, Kahn and Drozd-Greuel have shown that CM(A) is tame for A minimally elliptic. In my talk I am going to show that a wide  class of non-isolated singularities called degenerate cusps is Cohen-Macaulay tame. To prove this,  we introduce a new class of tame matrix problems called representations of decorated bunches of chains,  generalizing the classical bunches of (semi-)chains.

Wolfgang Ebeling: Singularities and Coxeter functors

To an isolated singularity of a two-dimensional complex analytic variety one can associate two different classes of graphs. The first class of graphs is only defined when the variety is a hypersurface and its definition uses a deformation of the singularity.  A graph of this kind is the Coxeter-Dynkin diagram with respect to a distinguished basis of vanishing cycles of the Milnor lattice. A categorification of this data is the directed Fukaya category of the singularity. On the other hand, one can consider the dual graph of a minimal resolution of the singularity or of its compactification. To a Coxeter-Dynkin diagram there is associated a Coxeter element which corresponds to the monodromy of the singularity. There is a mirror symmetry between some of these singularities. In the case of Kleinian and Fuchsian singularities, it was observed that the Coxeter element of one of these singularities is in a certain sense dual to a Coxeter element of some abstract lattice associated to the resolution graph of the mirror partner. We give a geometrical interpretation for these lattices and Coxeter elements, lifting them to triangulated categories of coherent sheaves. We also indicate relations of these Coxeter functors with the Poincaré series of the corresponding singularities. This is joint work with David Ploog.

Lutz Hille: Polynomial invariants for tilted algebras and a braid group action

If two algebras of finite global dimension are derived equivalent, then the entries in the Cartan matrix have to satisfy certain conditions. In fact, there exist polynomials in the entries of the Cartan matrix which have the same value on derived equivalent algebras. We call them polynomial invariants (for tilted algebras). The known polynomial invariants are obtained as the coefficients of the Coxeter polynomial. For algebras with upper triangular Cartan matrix it turns out that these polynomials are also invariant under a natural action of the braid group on the corresponding polynomial ring. We present the precise conjecture on the classification of the polynomial invariants and the invariants under the braid group action. Finally, we mention several applications.

Osamu Iyama: Cluster tilting in singularity theory, I

We recall work of Auslander and Reiten from the 1980s on the theory of almost split sequences in the category of Cohen-Macaulay modules. The stable categories of Cohen-Macaulay modules provide us a rich source of Calabi-Yau triangulated categories. Recently the concept of cluster tilting has been introduced for Cohen-Macaulay modules, which can be regarded as Calabi-Yau analogue of tilting theory for graded Cohen-Macaulay modules. We discuss higher theory of almost split sequences and Auslander algebras and their connection to Calabi-Yau algebras and Calabi-Yau tilted algebras. A typical example of cluster tilting is given by quotient singularities. They also provide an interesting relationship between stable categories of Cohen-Macaulay modules and cluster categories.

Bernard Keller: Cluster categories and Ginzburg algebras

This is a report on joint work with Dong Yang. I will recall how the mutation of quivers with potentials (as introduced by Derksen-Weyman-Zelevinsky) can be interpreted categorically using Ginzburg dg algebras and how this interpretation is linked to the one obtained via cluster categories. I will then present applications of this interpretation which yield new insight into recent results by Buan-Iyama-Reiten-Smith.

Sefi Ladkani:  On derived equivalences of triangles, rectangles and lines

I will present new results on derived equivalences of certain finite-dimensional algebras. These derived equivalences can be viewed as categorical interpretations of linear algebra statements about equivalences of bilinear forms given by explicit matrix shapes. These results imply unexpected derived equivalences among certain Auslander algebras or more generally endomorphism algebras of initial modules in the sense of Geiss-Leclerc-Schroer, incidence algebras of posets and other algebras generalizing the ADE-chain related to singularity theory. Among the quivers of these algebras one can find shapes of triangles, rectangles and lines.

Wendy Lowen: A Hochschild Cohomology Comparison Theorem for prestacks

We generalize and clarify Gerstenhaber and Schack's "Special Cohomology Comparison Theorem".  More specifically we obtain a fully faithful functor between the derived categories of bimodules over a prestack over a small category U and the derived category of bimodules over its corresponding fibered category. In contrast to Gerstenhaber and Schack we do not have to assume that U is a poset.

Daniel Murfet:  Residues and duality for singularity categories

Using Krause's unbounded singularity category and Grothendieck duality, we show how to obtain the Serre functor on the(bounded) singularity category of a projective Gorenstein variety. This leads to a description of the associated trace map in terms of residues.We will focus on the special case of hypersurface singularities of dimension one, where the trace formula can be written explicitly in terms of matrix factorisations.

Fernando Muro: On the existence and uniqueness of enhancements

Several classes of algebraic, geometric and topological objects are successfully studied via their derived category. Nevertheless derived categories are not good enough to encode some important invariants such as higher K-theory. This is why we need enhancements of triangulated categories. A triangulated category may have several non-equivalent enhancements or even no enhancement at all. In this talk we report on recent progress on the problem of existence and uniqueness of algebraic enhancements.

Amnon Neeman: From compactly generated to well generated categories

Abstract: We will give a brief reminder of compactly generated categories and their applications, and then review how well generated categories should be viewed as a natural large cardinal generalization.

Amnon Neeman: Some recent progress on Brown representability

We will give a review of the various known Brown representability theorems. Then we will discuss a major improvement that one can achieve using Rosicky functors.

Pedro Nicolas: The bar derived category of a curved dg algebra

Curved dg algebras, introduced by Positselski, are graded algebras endowed with a degree one derivation whose square is not necessarily zero but equals the commutator with a distinguished element of a degree 2 called curvature. They appear in nature as deformations of dg algebras, as duals of non-homogeneous quadratic algebras,... Associated to a curved dg algebra we construct a certain "derived" category which allows us to extend and explain some classical results of A-infinity theory.

Jose-Antonio de la Pena: Accessible algebras and singularities

Recently, we introduced in joint work with Helmut Lenzing the class of accessible algebras, which can be obtained by successive one-point extensions with exceptional modules, starting with the field k. Poset and tree algebras belong to this class, moreover the derived closure of the class of accessible algebras contains many well studied algebras. In particular, we consider the derived category of an extended canonical algebra related, via Orlov´s theorem, to the triangulated category of singularities of a graded singularity. Explicit formulas are given expressing the corresponding changes of the Coxeter polynomial.

David Ploog: Different categorical approaches to some surface singularities

Triangulated categories are valuable invariants of singularities, providing a link between geometry and representations of algebras. In this talk, we will compare two approaches for Kleinian and certain Fuchsian surface singularities: the algebraic one, using stable categories (or matrix factorisations) and a geometric one, utilising resolutions. These approaches yield genuinely different categories, each of them lifting the Coxeter element of the root lattice to an autoequivalence. There is a geometric functor showing that the two categories and their Coxeter functors are compatible in a natural sense. (Joint work with Chris Brav and Wolfgang Ebeling.)

Marco Porta:
DG enhancement of algebraic well generated triangulated categories

I will explain the main result of my PhD thesis, supervised by Prof. Bernhard Keller. Triangulated categories are more and more useful in several mathematical subjects. Nevertheless DG (differential graded) categories are a much more flexible tool. Thus, it is interesting to know when we can lift from the triangulated to DG world. We give an answer for the class of "algebraic" (well generated) triangulated categories, introduced by Keller in the 90's. This result is strongly reminiscent of a 1964 theorem of Gabriel and Popescu, which characterized the Grothendieck abelian categories as localizations of categories of modules over rings.

Idun Reiten: Cluster tilting in singularity theory, II

Based upon a paper with Burban, Iyama and Keller, we discuss Cohen-Macaulay modules over isolated hypersurface singularities. In particular we describe when there are cluster tilting objects in the corresponding stable categories of Cohen-Macaulay modules, and classify them when they exist. We also discuss connections between Cohen-Macaulay modules and the associated stable endomorphism algebras of cluster tilting objects, which are by definition 2-Calabi-Yau tilted algebras.

Claus Michael Ringel: Nakayama algebras of Loewy length 3.

Following Happel and Seidel, we consider the Nakayama algebras with directed Gabriel quiver such that all but 2 indecomposable projective modules have Loewy length 3. Of course, these algebras are representation-finite and one may wonder whether there are any problems when looking at its representation. But it turns out that this series of algebras forms an ADE-chain. We will consider the category of perfect complexes and show in which way the complexity of this category depends in the number n of simple modules. In particular, we will explain why the strong global dimension has to be infinite for n > 11 (as shown by Happel). In addition, we will outline some periodicity features.

Jiri Rosicky: Phantom accessible triangulated categories

Accessible categories are precisely free completions of small categories under k-filtered colimits for some regular cardinal k. We say that a category K is phantom acessible if it is equipped with a representation equivalence (= non-faithful equivalence) F to an accessible category. In the additive case, this means that K is accessible up to phantoms. We will discuss the relationship between phantom accessible and well generated triangulated categories in general and my proof that homotopy categories of stable model categories are phantom accessible in particular. Roughly saying, each phantom accessible triangulated category is well generated and has k-pure global dimension less or equal to 1 for some regular cardinal k. Phantom accessibility of well generated triangulated categories is unclear and is closely related to a recent conjecture of A. Neeman. Recall that he needed the fullness of F for proving Brown representability for the dual of K.

Stefan Schwede: Algebraic versus topological triangulated categories

The notion of a triangulated category is a conceptual language used in several areas of puremathematics. It has two historical origins, going back to the 1960s. In algebraic geometry,Verdier used triangulated categories as a convenient framework to describe dualityphenomena. Around the same time, Puppe introduced a very similar notion to extract the keyformal properties of the stable homotopy category of algebraic topology. In the first part of mytalk, I want to review the concept of a triangulated category, comment on its history and describe some key examples. Corresponding to the two roots, the constructions of the `algebraic' and `topological' examples oftriangulated categories have a very different flavor. The examples from algebra and algebraicgeometry typically arise from chain complexes by passing to chain homotopy classes or inverting homology isomorphisms. Such examples have underlying additive categories. Stable homotopy theory produces examples of triangulated categories by very different means, usingtopological spaces, and in this context the underlying categories are very non-additive before passing to homotopy classes. Still, all examples satisfy the same set of axioms, and one couldwonder if there is any trace left of how the examples were built. In the second part of the talk I want to explain some systematic differences between these two kindsof triangulated categories. These are certain subtle torsion phenomena, defined entirely in terms of the triangulated structure, which can happen in topological examples, but never in algebraic ones.

Stefan Schwede: On well generated, topological triangulated categories

This talk is on the thesis work of my PhD student Andreas Heider who showed that every  well generated, topological triangulated category is a localization of the derived category of a spectral category. This complements an analogous result of Porta in the realm of algebraic triangulated categories, where "spectral categories" are replaced by "differential graded categories".

Jan Stovicek:
On homotopy categories of complexes

If A is an additive category, the homotopy category of unbounded complexes over A, usually denoted by K(A), is well known to be a triangulated category. In this talk, I will discuss some properties of K(A), with a focus on the case when A is an additive subcategory of a module category. First, I will discuss when K(A) is well generated and show that this fails in many interesting cases. Second, in an attempt to get a remedy for this failure, I will define locally well generated triangulated categories and show that many categories of the form K(A) have this property. Unfortunately, locally well generated categories do not satisfy the Brown representability theorem in general. If time allows, I will indicate how to obtain triangulated adjoint functors without Brown representability. This talk is related to recent work of Neeman, Krause, Jorgensen and Holm.

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