Projektbeschreibung
Neuartige Untersuchungen zur Darstellungstheorie von Algebren und homologischer Algebra
Bei der Untersuchung komplexer Themen oder Prozesse, die sich nur schwer messen und analysieren lassen, wird häufig ein Proxy verwendet, also eine indirekte Beobachtung oder Extrapolation, die das relevante Signal repräsentiert. Auf dem Gebiet der Mathematik gewinnt die Repräsentationstheorie durch ihre Wirkung auf einfachere Strukturen auch Erkenntnisse über komplexe algebraische Strukturen. Mit Unterstützung des Marie-Skłodowska-Curie-Maßnahmenprogramms entwickelt das Projekt COMBGEOREP homologische und geometrische Methoden zur Untersuchung der Darstellungen von Algebren. Dabei stellt es Verbindungen zur Kombinatorik, zur Darstellungstheorie von Gruppen, sowie zur algebraischen und symplektischen Geometrie her.
Ziel
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.
Wissenschaftliches Gebiet
Programm/Programme
Thema/Themen
Aufforderung zur Vorschlagseinreichung
Andere Projekte für diesen Aufruf anzeigenFinanzierungsplan
MSCA-IF-EF-ST - Standard EFKoordinator
LA1 4YW Lancaster
Vereinigtes Königreich