New ideas from theoretical physics have brought forth a convergence between areas of mathematics as diverse as algebraic geometry (which studies spaces defined by polynomial equations), symplectic geometry (which studies the phase spaces of mechanical systems), and knot theory. This project aims to investigate and establish some of the conjectured relationships between these fields. For example, a deep connection between the symplectic geometry of monomial functions on toric spaces and the algebraic geometry of spaces defined by a single polynomial equation will be studied. The project will also explore a conjecture of physicists Aganagic and Vafa according to which every knot in 3-dimensional space determines a symplectic integrable system in 6 dimensions, whose geometry is in turn related to a recently discovered knot invariant. By testing the validity of recent predictions made by theoretical physicists, this work will enhance our understanding of the emerging and still mysterious connections between various areas of modern geometry.
This project will use Lagrangian Floer homology as a tool to explore various geometric aspects of mirror symmetry and its connections to classical questions in symplectic topology and low-dimensional topology. One main goal of the project will be to establish Kontsevich's homological mirror symmetry conjecture (in both directions) for hypersurfaces and complete intersections in affine space, relating the wrapped Fukaya categories and derived categories of affine varieties and their mirror Landau-Ginzburg models to each other. In another direction, this project will develop new constructions of exotic Lagrangian tori in symplectic manifolds, and their relations to toric degenerations and to cluster variety structures on the mirror spaces. Finally, it will seek to provide an interpretation of new knot invariants (such as the quantum A-polynomial recently introduced by physicists Aganagic and Vafa) in terms of mirror symmetry and wall-crossing for Lagrangian tori in Calabi-Yau 3-folds.
