Family Floer Cohomology

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The development of geometry as an abstract discipline is based on shared human intuitions about dimension, direction, and distance. In the setting of non-commutative geometry where the notion of locality ceases to make sense, these intuitions usually offer little guidance, so that mathematicians are forced to fall back on algebraic and computational tools. The study of symplectic manifolds provides, via mirror symmetry, an approach to non-commutative geometry that starts from an honest geometric space and produces a non-commutative one formed by a class of subspaces called Lagrangians. For example, the algebras known to mathematicians as Clifford algebras and to physicists as Dirac matrices arise from the study of circles in the sphere. This research project will investigate such non-commutative spaces in higher dimensions, relating them to a program whose goal is to understand symplectic manifolds via the associated non-commutative spaces known as Fukaya categories. The project focuses on the setting of symplectic manifolds admitting Lagrangian torus fibrations. The investigator recently extracted a local-to-global description of the Fukaya category in this case. This research will pursue a series of projects aimed at extending the applicability of local-to-global approaches to general symplectic manifolds; this will require developing new tools for studying and computing Fukaya categories. The investigator aims to systematically develop these tools, starting from situations where they will provide alternate descriptions of Fukaya categories (e.g., for cotangent bundles). The project will also explore applications to the study of Lagrangian embeddings and to extensions of mirror symmetry.