Descrizione del progetto
Specchio, servo delle mie brame... Uno sguardo più attento alla geometria complessa generalizzata
La geometria complessa generalizzata include la geometria complessa e quella simplettica quali casi speciali «estremi»; tuttavia, le strutture complesse generalizzate in piena generalità non sono ancora ben comprese. La geometria complessa e la geometria simplettica sono legate l’una all’altra attraverso la simmetria speculare, una relazione speciale tra oggetti geometrici di rilevanza per la teoria delle stringhe. Anche se alcuni importanti risultati relativi alla geometria complessa o simplettica sono stati estesi a strutture complesse generalizzate, la simmetria speculare non è stata ancora estesa a queste strutture. Il progetto FuSeGC, finanziato dall’UE, prevede di cambiare questa situazione con il primo di questi risultati.
Obiettivo
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.
Campo scientifico
Parole chiave
Programma(i)
Argomento(i)
Meccanismo di finanziamento
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinatore
3000 Leuven
Belgio