Description du projet
Nouvelles recherches sur la théorie des représentations des algèbres et de l’algèbre homologique
L’étude de sujets ou de processus complexes, difficiles à mesurer et à analyser, se fait souvent à l’aide d’une valeur approchée, une observation indirecte ou une extrapolation représentative du signal d’intérêt. Dans le domaine des mathématiques, la théorie des représentations permet de comprendre les structures algébriques complexes par le biais de leurs actions sur des structures plus simples. Avec le soutien du programme Actions Marie Skłodowska-Curie, le projet COMBGEOREP développe des méthodes homologiques et géométriques pour étudier les représentations des algèbres, en créant des liens avec la combinatoire, la théorie des représentations des groupes et la géométrie algébrique et symplectique.
Objectif
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.
Champ scientifique
Programme(s)
Régime de financement
MSCA-IF-EF-ST - Standard EFCoordinateur
LA1 4YW Lancaster
Royaume-Uni