Project description
Mirror, mirror on the wall: A closer look at generalised complex geometry
Generalised complex geometry includes complex and symplectic geometry as its 'extremal' special cases, but generalised complex structures in full generality are not yet well-understood. Complex and symplectic geometry are related to each other through mirror symmetry, a special relationship between geometric objects of relevance to string theory. Although some important results related to complex or symplectic geometry have been extended to generalised complex structures, there has yet been no extension of mirror symmetry to these structures. The EU-funded FuSeGC project plans to change that with the first such result.
Objective
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.
Fields of science
Programme(s)
Funding Scheme
MSCA-IF - Marie Skłodowska-Curie Individual Fellowships (IF)Coordinator
3000 Leuven
Belgium