Descrizione del progetto
Nuove connessioni tra la simmetria speculare, le algebre dei cluster e la teoria della rappresentazione
Il campo della matematica fornisce un linguaggio con cui descrivere il nostro universo. La teoria delle stringhe, secondo cui le particelle sono in effetti piccole stringhe vibranti, è uno dei quadri più promettenti per unificare le teorie attualmente incompatibili della meccanica quantistica e della relatività generale. La simmetria speculare, un fenomeno naturale, ha avuto origine dalla teoria delle stringhe e la simmetria speculare omologica ne è una versione specifica. Grazie al sostegno del programma di azioni Marie Skłodowska-Curie, il progetto STABREP sta istituendo nuove connessioni tra la simmetria speculare, le algebre dei cluster e la teoria della rappresentazione mediante l’introduzione di modelli geometrici.
Obiettivo
In this project, I will establish new connections between cluster algebras, mirror symmetry and representation theory through the introduction of geometric models.
Mirror symmetry is a natural phenomenon, first observed in superstring theory, consisting of two main approaches: the A-model, focused on the symplectic side of a Calabi-Yau manifold X, and the B-model, focused on the complex side of the manifold. Mirror symmetry is a duality between the two models. Based on this, Kontsevich formulated his famous homological mirror symmetry conjecture for categories. In this conjecture, the A-model is the Fukaya category of X, and the B-model is the derived category of coherent sheaves of X^. Cluster algebras were introduced in the early 2000s to provide a combinatorial framework for dual canonical bases. Many new ideas in representation theory have their origin in cluster algebras, bringing together category theory, particularly Calabi-Yau categories, combinatorics and the geometry of Riemann surfaces. In exciting recent developments cluster theory and homological mirror symmetry have been linked through scattering diagrams, opening up both theories.
In this project, I will study the connections between cluster combinatorics and scattering diagrams through Calabi-Yau algebras, which appear in a natural way in cluster theory and mirror symmetry. I will develop geometric models for the representation theory of Calabi-Yau algebras encoding, in particular, their (co)homology. This will lead to a complete understanding of these algebras and their role in the mirror symmetry program.
Dimer models are intrinsically linked to both cluster algebras and mirror symmetry. As part of my project, I will generalize dimer models to the general setting of special multiserial algebras. Both Calabi-Yau algebras and special multiserial algebras are of wild representation type and my geometric models will lead the way to an understanding of stability conditions for wild algebras.
Campo scientifico
Parole chiave
Programma(i)
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Meccanismo di finanziamento
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinatore
LE1 7RH Leicester
Regno Unito