Combinatorial and Categorical methods in Representation Theory

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This is a project dealing with basic research in mathematics, in the field of algebra. The project is concerned with various aspects of the representation theory for finite dimensional algebras and related topics in algebra, geometry and combinatorics. The main topics of research in this project are: (1) We want to contribute to the theory of cluster structures on triangulated categories with Calabi-Yau properties. This is partially motivated by possible further applications to the theory of cluster a lgebras. (2) Cluster-tilting theory has provided generalizations of tilting theory, and we want to continue the research in this direction, with emphasis on understanding homological and combinatorial properties of the 2-CY tilted algebras. (3) We wan t to study support varieties determined by Hochschild cohomology. Especially we want to determine for which finite dimensional algebras, the Ext-groups between any two finitely generated modules is a finitely generated module over a Noetherian (graded) co mmutative ring defining the support variety. (4) We want to contribute to making software for computing projective resolutions. (5) We want to contribute to the interplay between representation theory and algebraic geometry given by degenerations of representations. We will study various concepts of partial orders on the set of isomorphism classes of modules of a given dimension related to degenerations. (5) We want to solve problems related to finding bounds for dimensions of triangulated categori es and representation dimension. In dealing with these topics, we will apply methods of homological algebra, combinatorics and geometry.