Descripción del proyecto
Espejito, espejito: un análisis más detallado de la geometría compleja generalizada
La geometría compleja generalizada incluye la geometría compleja y la simpléctica como sus casos especiales «extremos», pero las estructuras complejas generalizadas aún no se comprenden bien del todo. La geometría compleja y la simpléctica están relacionadas entre sí a través de la simetría especular, una relación especial entre objetos geométricos relevante para la teoría de cuerdas. Aunque algunos resultados importantes relacionados con la geometría compleja o simpléctica se han extendido a las estructuras complejas generalizadas, la simetría especular todavía no se extendido a estas estructuras. El proyecto FuSeGC, financiado con fondos europeos, tiene por objeto cambiar esta situación con el primer resultado en esta línea.
Objetivo
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.
Ámbito científico
Palabras clave
Programa(s)
Régimen de financiación
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinador
3000 Leuven
Bélgica