In symplectic topology, computation of Fukaya categories can be broken into two steps: (1) identify a collection of Lagrangians from which all others can be obtained by formal algebraic constructions; we say that this entails finding a generating subcategory, then (2) explicitly determine the product (and higher multiplications) for the generating subcategory. The PI proposes to study generalisations of a newly developed generation criterion to families of Lagrangian manifolds, and to the variants of Fukaya categories associated to Landau-Ginzburg models. This will enlarge the class of symplectic manifolds for which large parts of Floer theory can be explicitly described.
Symplectic manifolds give an abstract setting generalising the relationship between position and momentum in classical mechanics. The study of such manifolds has recently impacted areas as diverse as the study of knots (mathematically understood as embeddings of circles in space), and theoretical physics. This proposal will study algebraic invariants that appear in symplectic topology, with the goal of furthering our understanding of the subject and its applications.
