The Fukaya category for conical singularities

The Fukaya category is a powerful invariant of exact symplectic manifolds in the form of an A-infinity category, the objects of which consist of (a suitable class of) its Lagrangian submanifolds, while morphisms are Floer cochains.

This category exhibits a very rich an interesting algebraic structure which also is a crucial ingredient in Kontsevich´s homological mirror symmetry.

Together with collaborators, we have previously constructed Floer complexes for Lagrangians that have cylindrical ends inside certain exact symplectic manifolds with concave conical ends of a special type.

Our goal is to, first, use modern techniques from symplectic field theory for extending this construction to the completely general geometric setting.

Second, we plan to extend the construction to a Fukaya category of exact symplectic manifolds with concave conical ends, and then establish generation results in this setting.The envisioned theory has several applications.

Any exact Lagrangian with conical singularities become an object in such a category, after removing the singular locus from the ambient space.

For instance, the isotropic skeleton of a Weinstein manifold can be presented as Lagrangian with conical singularities, and by this work it becomes realised as an object of the Fukaya category.

On a different note, contact manifolds give rise to cylindrical Liouville cobordisms by passing to their symplectisations. In this manner we obtain a Fukaya category associated to any contact manifold.