Project description
Novel investigations into the representation theory of algebras and homological algebra
Investigations of complex topics or processes that are difficult to measure and analyse are often done by means of a proxy, an indirect observation or extrapolation representative of the signal of interest. In the field of mathematics, representation theory draws insight about complex algebraic structures via their actions on simpler ones. With the support of the Marie Skłodowska-Curie Actions programme, the COMBGEOREP project is developing homological and geometric methods to study representations of algebras, creating links with combinatorics, group representation theory, and algebraic and symplectic geometry.
Objective
Representation theory is the study of complex algebraic structures such as groups and rings via their actions on simpler algebraic structures, such as vector spaces. The naturality of this idea of studying complex problems by ‘linearisation’ means that representation theory has strong interactions with many areas of mathematics. This project lies in the area of representation theory of algebras and homological algebra. The overall goal is to develop homological and geometric methods to study representations of algebras creating links with combinatorics, group representation theory, algebraic and symplectic geometry. The principal research objectives are: 1) Use the geometry of Riemann surfaces to study skewed-gentle algebras and their tau-tilting theory. 2) Develop cluster-theoretic techniques in negative Calabi-Yau (CY) triangulated categories by: a) constructing negative CY cluster categories; b) developing the theory of simple-minded systems in stable module categories. The geometry of surfaces provides equivalences between derived categories of gentle algebras and Fukaya categories in symplectic and algebraic geometry. The extension of these methods to skewed-gentle algebras should significantly broaden the scope of this interaction between algebra and geometry. The theory of negative CY categories is considerably underdeveloped despite their occurrence in important contexts such as stable module categories in group representation theory. Cluster theory provides powerful combinatorial methods for positive CY categories which initial work by Coelho Simões suggests is amenable to development in the negative CY setting. The project will be carried out by Raquel Coelho Simões under the supervision of Jan Grabowski at Lancaster University. It will serve to establish Coelho Simões as a research leader in her field through work in a highly active research area at an institution sitting in a broad network of universities with major strength in the field.
Fields of science
Programme(s)
Funding Scheme
MSCA-IF-EF-ST - Standard EFCoordinator
LA1 4YW Lancaster
United Kingdom