Description du projet
Miroir, mon beau miroir: un regard plus attentif sur la géométrie complexe généralisée
La géométrie complexe généralisée englobe la géométrie complexe et la géométrie symplectique en tant que cas particuliers «extrêmes», toutefois, les structures complexes généralisées ne sont pas encore bien comprises dans toute leur généralité. La géométrie complexe et la géométrie symplectique sont liées l’une à l’autre par symétrie miroir, une relation spéciale entre les objets géométriques pertinents pour la théorie des cordes. Bien que certains résultats importants liés à la géométrie complexe ou symplectique aient été étendus aux structures complexes généralisées, la symétrie miroir n’a pas encore été étendue à ces structures. Le projet FuSeGC, financé par l’UE, prévoit de changer cela avec le premier résultat de ce type.
Objectif
Generalized complex geometry unifies complex and symplectic geometry, two important research areas in modern pure mathematics.
While generalized complex (GC) structures in full generality are not yet well-understood, a number of important results from complex or symplectic geometry have already been extended to these more general structures. Further, complex and symplectic geometry are intimately related to each other via mirror symmetry, a conjectured duality between certain complex and symplectic manifolds discovered in theoretical physics in the context of string theory. This duality has been proven in special cases.
For this project I propose an approach to extend homological mirror symmetry to certain subclasses and examples of GC manifolds, centred around three objectives:
(O1) Quantify the effect of stable GC compactifications of Landau-Ginzburg mirrors of del Pezzo surfaces on their Fukaya category.
(O2) Construct a Wrapped Fukaya category for oriented surfaces with log symplectic structures.
(O3) Develop and study a notion of 'holomorphic families of Fukaya categories'.
In particular in the case of (O1) and (O3), the construction of a Fukaya-type category would immediately suggest mirror partners for certain classes of examples, the first extension of mirror symmetry to the GC context.
During my PhD, I proved foundational results on Lagrangian-type submanifolds with boundary of stable GC manifolds, which naturally arise in examples and are candidates for objects of Fukaya-Seidel-type categories of stable GC manifolds.
As an MSC fellow, I would profit from world-leading expertise on symplectic geometry and Fukaya categories at my third-country host institution, while bringing in expertise on the novel research area of generalized geometry. I am looking forward to expanding my own skills in instruction and supervision through a mini course on generalized complex geometry and a Master's thesis project at my EU host KU Leuven.
Champ scientifique
Programme(s)
Régime de financement
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinateur
3000 Leuven
Belgique