Descrizione del progetto
Nuove indagini sulla teoria della rappresentazione delle algebre e dell’algebra omologica
Le indagini su argomenti o processi complessi la cui misurazione o analisi risulta difficile vengono spesso effettuate mediante un proxy, ovvero un’osservazione o estrapolazione indiretta che rappresenta il segnale di interesse. Nel campo della matematica, la teoria della rappresentazione attinge a informazioni sulle strutture algebriche complesse mediante le azioni da loro svolte su strutture più semplici. Grazie al sostegno del programma di azioni Marie Skłodowska-Curie, il progetto COMBGEOREP sta sviluppando metodi omologici e geometrici per studiare le rappresentazioni dell’algebra, creando legami con la combinatoria, la teoria della rappresentazione dei gruppi e la geometria algebrica e simplettica.
Obiettivo
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.
Campo scientifico
Programma(i)
Argomento(i)
Meccanismo di finanziamento
MSCA-IF-EF-ST - Standard EFCoordinatore
LA1 4YW Lancaster
Regno Unito