Triangulated categories and associative algebras in the geometry of singular varieties
Overview
The project belongs to algebraic geometry, homological algebra and representation theory in its interaction with theoretical physics. In the given research projects we can underline three guiding lines: - study of derived categories of coherent sheaves on singular varieties and their behavior in families, in particular in the case of degeneracies of elliptic curves, applications to the theory of the Yang-Baxter equation; - study of derived categories of certain associative algebras, their group of exact auto-equivalences, in particular effects of the Zopf group; combinatorial description of indecomposable complexes, derived categories of coherent sheaves on noncommutative curves; - stable category of Cohen-Macaulay modules over possibly noncommutative Gorenstein rings and their homological properties. There is interest in the study of these topics from the side of algebraic geometry and representation theory, as well as from other areas of mathematics (Yang-Baxter equation) and theoretical physics (mirror symmetry, D-Branes). The interactions between these areas make the topic particularly appealing.
DFG-Procedure Material Grants
(individual postdoctoral research grant, 24 months of a postdoctoral position)
More Information
Results
Together with Bernd Kreußler I introduced the notion of geometric r-matrices. Together with Olivier Schiffmann I studied the Drinfeld double of the Hall algebra of a tubular weighted projective straight line and showed that on this algebra one has an action of SX2(Z). Together with lyama, Keller and Reiten, I have described all cluster-tilt objects in the stable category of Cohen-Macaulay moduli of a reduced one-dimensional hypersurface singularity. L. Bodnarchuk, I. Burban, Yu. Drozd and G.-M. Greuel, Vector bundles and torsion free sheaves on degenerations of elliptic curves, Global Aspects of Complex Geometry, 83-129, Springer, (2006).Project-related publications (selection)