Description du projet
Nouveaux liens entre la symétrie miroir, les algèbres de groupe et la théorie des représentations
Le domaine des mathématiques fournit un langage permettant de décrire notre univers. La théorie des cordes, qui postule que les particules sont en fait de minuscules cordes vibrantes, est l’un des cadres les plus prometteurs pour unifier les théories actuellement incompatibles de la mécanique quantique et de la relativité générale. La symétrie miroir, un phénomène naturel, est née de la théorie des cordes, et la symétrie miroir homologique en est l’une des versions. Avec le soutien du programme Actions Marie Skłodowska-Curie, le projet STABREP établit de nouveaux liens entre la symétrie miroir, les algèbres de groupe et la théorie des représentations par l’introduction de modèles géométriques.
Objectif
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.
Champ scientifique
Mots‑clés
Programme(s)
Régime de financement
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinateur
LE1 7RH Leicester
Royaume-Uni