BIREP – Representations of finite dimensional algebras at Bielefeld

See here for the schedule.

**Ragnar Buchweitz: **Maximal Cohen-Macaulay Modules over Complete Intersections

Maximal Cohen-Macaulay modules over a hypersurface ring have
a beautifully succinct description in terms of matrix factorizations,
and the question addressed here is how this generalizes over
complete intersection singularities.
We will show that there is indeed equally well such a succinct
description that can be viewed as a graded matrix factorization
of the generic hypersurface containing the complete intersection.
This has immediate consequences for the structure of the homomorphism
groups in the stable derived category of such singularities and,
potentially, allows for generalizations of all those recent results
that explore the stable derived category as a "geometric" object.

**Ragnar Buchweitz: **The Intersection Pairing on Matrix Factorizations

One of the baffling recent results on matrix factorizations is
a (partly) affirmative answer to a conjecture due to Dao:
For an isolated hypersurface singularity in an odd number of variables,
thus, of even dimension, even and odd Ext groups between
maximal Cohen-Macaulay modules should always have equal length.
There are by now three, rather different affirmative answers,
one due to Moore-Piepmeyer-Spiroff-Walker in the graded case
over an arbitrary field, one to Polyshchuk-Vaintrob in the setting of
dg categories, and another one due to van Straten and the
speaker in the analytic case.
We will report on the last of these that exploits the topology
of the hypersurface singularity to obtain the result.
It would be very interesting to learn whether Physics can shed any
light on why this kind of symmetry should hold, and what should
replace it for general singularity categories?

**Igor Burban: ** Representation Theory of Matrix Factorizations

In this talk I am going to recall some basic results on matrix factorizations
and maximal Cohen-Macaulay modules. In particular, I shall review various
methods to
classify indecomposable Cohen-Macaulay modules in the
representation-finite case, which includes a description of matrix
factorizations over ADE singularities.

**Igor Burban: ** Decorated bunches of chains and Cohen-Macaulay modules over
non-isolated surface singularities

In this talk (based on a joint work with Yuriy Drozd) I shall introduce a
new class of matrix problems called "representations of bunches of
chains", outline a proof of their tameness and explain a classification
of indecomposable canonical forms. This technique will be applied to
describe Cohen-Macaulay modules over a certain class of non-isolated
surface singularities called degenerate cusps.
In particular, I shall deduce a classification of matrix factorizations of
the potential w = xyz.

**Nils Carqueville: ** Matrix factorisations in field and string theory

In this introductory talk I shall review the notions of topological field theory and topological string theory in their algebraic formulation. Then I will indicate how matrix factorisations constitute the boundary sector of such theories in the case of affine B-twisted Landau-Ginzburg models. If time permits I will close with some rudimentary comments on the CFT/LG correspondence.

**Nils Carqueville: ** Topological defects in Landau-Ginzburg models

Matrix factorisations of W(x) - W'(y) also describe topological defects between Landau-Ginzburg models with potentials W and W'. This motivates the study and construction of the rigid monoidal structure of the category MF(Wx1-1xW), which is joint work together with Ingo Runkel. In particular, the "defect action on bulk fields" gives rise to a map from the Grothendieck ring of MF(Wx1-1xW) to endomorphisms of the Hochschild homology of MF(W).

**Wolfgang Ebeling: ** Geometric analysis of singularities

Let f : (C^{n+1},0) → (C,0) be the germ of a holomorphic function with an isolated singularity at the origin. The talk will be an introduction to two concepts which are important for the analysis of such a singularity.

First, let Bε ⊂ C^{n+1} be the ball in C^{n+1} of radius ε > 0 around the origin and denote by Dδ ⊂ C the disc of radius δ > 0 around 0. Let Dδ′ be the disc with the origin removed. A well-known result by J. Milnor states that for sufficiently small ε and δ ≪ ε the restriction of f to X := f^{−1}(Dδ′ ) ∩ Bε is a smooth locally trivial fibration. The fibre X_{t} = X ∩f^{−1}(t), t ∈ Dδ′, is called the Milnor fibre of f. It has the homotopy type of a bouquet of n-spheres.

On the other hand, we consider the algebra Q_{f} = O_{n+1}/Jf where O_{n+1} is the ring of germs at 0 of holomorphic functions and Jf the ideal generated by the partial derivatives ∂f/∂zi (0 ≤ i ≤ n). This is called the Milnor algebra of f. We discuss properties of this algebra. We relate it to the de Rham cohomology of the Milnor fibre.

**Alexander Eifmov: ** Reconstruction of hypersurface singularity from its triangulated category of singularities

We will show that a polynomial with singularity at origin can be reconstructed, up to a formal change of variables, from the A_{∞}-endomorphism algebra of the structure sheaf of origin
in the triangulated category of singularities. This result is contained in arXiv:0907.3903.

**David Favero: ** Graded matrix factorizations, functor categories, and orbit categories

The category of dg functors between categories of graded matrix factorizations, can be described by a different category of matrix factorizations whose grading is a bit more subtle. This gives rise to some nice "geometric" behavior between various categories, perhaps more easily seen through the lens of homological mirror symmetry. I will give an overview of recent results obtained in joint work with Matthew Ballard and Ludmil Katzarkov on this subject and provide examples of some of the geometry alluded to. If time permits I will very briefly describe some applications to Rouquier dimension for derived categories of coherent sheaves.

**Lutz Hille: ** Hochshild cohomology and tilting objects

Tilting sheaves allow to compare the derived category of coherent sheaves on a variety with the derived category of finite dimensional modules over a finite dimensional algebra. In this talk we construct a resolution of the diagonal for any variety with a tilting bundle. This construction allows to compare Hochschild homology and Hochschild cohomology for the variety and the corresponding module category. Our results are valid for schemes that are projective over an affine scheme. Finally we discuss rational surfaces in detail and obtain consequences on the level of the corresponding Grothendieck groups.

**Osamu Iyama: ** Tilting and cluster tilting for stable categories of Cohen-Macaulay
modules

Representation theory of CM (=Cohen-Macaulay) modules are initiated
by Auslander-Reiten. I will discuss properties of the stable categories
of CM modules over Gorenstein singularities in the context of tilting
theory as well as cluster tilting theory. In particular we will see that
the stable categories of graded (resp. ungraded) CM modules are often
realized as derived (resp. cluster) categories of certain algebras by
using tilting (resp. cluster tilting) theory.

**Dirk Kussin: ** Triangle singularities II

In this continuation of Lenzing's talk on triangle singularities X we
treat the following topics about the stable category of vector bundles
on X: exceptionality of vector bundles of rank two, in particular of
Auslander bundles; the link to subspace problems of nilpotent
operators (studied by Ringel-Schmidmeier, and Simson); two (classes
of) "natural" tilting objects and their corresponding finite
dimensional endomorphism algebras; resulting so-called ADE-chains; a
proof of an "duality" phenomenon observed by Happel-Seidel.

**Helmut Lenzing: ** Matrix factorizations for triangle singularities

My talk reports on joint investigations with D. Kussin and H. Meltzer on the category of matrix factorizations
for the triangle singularity f=x^{a} + y^{b} + z^{c} for a given weight triple of integers >1. The mathematics of f
is encapsulated in the associated weighted projective line X with three weighted points of order (a,b,c). The category coh X
of coherent sheaves on X is obtained by Serre construction from the suitably graded (by a rank-one abelian group L) coordinate algebra S=k[x,y,z]/(f). The category vect X of vector bundles on X happens to be equivalent to the
category of (maximal) L-graded Cohen-Macaulay modules over S, which induces on vect X the structure of a Frobenius
category such that the indecomposable projective-injective objects are just the line bundles on X. The associated stable
category vect X is then triangulated and equivalent to the homotopy category of L-graded matrix factorizations of f. We are
going to discuss various categorical properties of vect X, in particular its Calabi-Yau properties and the existence of a tilting
object. A central aspect of the talk concerns the comparison of coh X and vect X in the spirit of Orlov's theorem.

**Daniel Murfet: ** Stable derived category

**Daniel Murfet: ** Link homology and convolution

Using functors between categories of matrix factorisations, Khovanov and Rozansky have defined an interesting family of homological link invariants. I will present their construction and some related joint work of myself with Nils Carqueville.

**David Ploog: ** Kleinian and Fuchsian singularities

We will report on work of Kajiura, Saito, Takahashi and Ueda who describe the categories of graded matrix factorisations for Kleinian and (certain) Fuchsian singularities of genus 0. In all cases covered, the categories possess full, exceptional sequences.

**Ian Shipman: ** A geometric approach to Orlov's theorem

Orlov proved a comparison theorem between the derived bounded category of coherent sheaves on a nonsingular projective hypersurface, and the category of graded matrix factorizations for its defining equation. I will describe what graded matrix factorizations are when the ambient space is an arbitrary scheme or stack. Then I will explain how to see Orlov's theorem by stringing together some very natural geometrically defined functors.

**Atsushi Takahashi: ** Mirror symmetry between orbifold curves and cusp singularities with group
action

We consider an orbifold Landau-Ginzburg model (f,G), where f is an
invertible polynomial in three variables and G a finite group of
symmetries of f containing the exponential grading operator, and its
Berglund-Hübsch transpose (f^{T}, G^{T}). We show that this defines a
mirror symmetry between orbifold curves and cusp singularities with group
action. We define Dolgachev numbers for the orbifold curves and Gabrielov
numbers for the cusp singularities with group action. We show that these
numbers are the same and that the stringy Euler number of the orbifold
curve coincides with the G^{T}-equivariant Milnor number of the mirror
cusp singularity.

**Michael Wemyss: ** Maximal Modification Algebras and non-isolated AR duality

I will explain the concept of a maximal modification algebra, which is a simultaneous generalization of cluster tilting modules (actually maximal rigid modules) and Van den Bergh's noncommutative crepant resolutions. I will largely give an overview of what these are supposed to do for us, then link to the stable category of CM modules and explain, from our perspective, why this category has both good and bad properties. One of these is that for non-isolated singularities, the stable category has some form of AR duality. This is joint work with Iyama.