On July 14 2012 the following meeting will take place at the Mathematical Institute of Cologne.
|9:30-10:30|| Lennart Galinat
Matrix Factorizations and Autoequivalences of Calabi-Yau Hypersurfaces
|11:00-12:00|| Andreas Hochenegger
Spherelike Twist Functors
|14:00-15:00|| Hanno Becker
Models for Singularity Categories
|15:40-16:40|| Wassilij Gnedin
Maximal Cohen-Macaulay modules over Curve Singularities of type P
|17:00-18:00|| Martin Kalck
Frobenius Categories and Gorenstein rings for rational surface singularities
Guests who need a support to get an accomodation in Cologne are asked to write me an e-mail
Orlov has constructed an equivalence between the derived category of a hypersurface of degree n+1 in projective n-space and the homotopy category of graded matrix factorisations. We explain a method to construct the matrix factorisation corresponding to the structure sheaf under this equivalence. Along the way we will review Orlov's theorem and a recent result of Ballard, Favero and Katzarkov relating certain autoequivalences on either side of the equivalence.
I will report on a ongoing joint work with Martin Kalck and David Ploog.
We call an object F in a triangulated category d-spherelike, if its endomorphism algebra is isomorphic to the cohomology of a d-sphere. If we would also ask for F to be a d-Calabi-Yau object, we would arrive at the well-known notion of a d-spherical object. For such an object, P. Seidel and R. Thomas defined a functor, the so-called spherical twist, which is an auto-equivalence.
Without this CY-condition, we define analogously a spherelike twist functor which still incorporates interesting properties. The aim of this talk is to demonstrate these, by presenting examples from algebraic geometry and representation theory.
This talk is about the construction of various Quillen models for categories of singularities.
I will begin by recalling the connection between abelian model structures, cotorsion pairs and deconstructible classes and then describe the construction of the singular models. Next, I will explain how Krause's recollement for the stable derived category can be obtained model categorically.
Finally, as an example I will show that Positselski's contraderived model for the homotopy category of matrix factorizations is Quillen equivalent to a particular singular model structure on the category of curved mixed complexes.
Curve singularities of type P are certain complete intersections built from two copies of simple curve singularities of type A. Drozd and Greuel have proved that the classification problem of indecomposable maximal Cohen-Macaulay modules over reduced curve singularities of type P is tame.
In this talk, we show three things. First, we observe that non-reduced curve singularities of type P are tame. Second, we demonstrate how to glue the indecomposable maximal Cohen-Macaulay modules over (possibly non-reduced) curve singularities of type P from maximal Cohen-Macaulay over simple curve singularities of type A. Third, we translate them into matrix factorizations over hypersurface singularities of type T.
In recent work, Iyama and Wemyss associated a triangulated category T to any rational surface singularity R. More precisely, it arises as the stable category of the category of so called special Cohen-Macaulay R-modules, for which they give a Frobenius exact structure.
We show that T is a direct sum of well known categories. Namely, it splits into blocks of stable categories of maximal Cohen-Macaulay modules over ADE-surface singularities. On the way, we develop some techniques to express a certain class of Frobenius categories as categories of Gorenstein-projective modules over an Iwanaga-Gorenstein ring. This is of independent interest. This is joint work with Osamu Iyama, Michael Wemyss and Dong Yang.