Lagrangian Floer homology and the geometry of homological mirror symmetry

来自理论物理的新思想带来了不同数学领域的融合，如代数几何(研究由多项式方程定义的空间)、辛几何(研究机械系统的相空间)和纽结理论。这个项目旨在调查和建立这些领域之间的一些猜想关系。例如，将研究环形空间上单项函数的辛几何与由单个多项式方程定义的空间的代数几何之间的深层联系。该项目还将探索物理学家阿加纳奇和瓦法的一个猜想，根据这个猜想，3维空间中的每个纽结都决定了6维的辛可积系统，而该系统的几何又与最近发现的纽结不变量有关。通过检验理论物理学家最近预测的正确性，这项工作将加深我们对现代几何各个领域之间正在出现的、仍然神秘的联系的理解。本项目将使用拉格朗日Floer同调作为工具，探索镜像对称的各个几何方面，以及它与辛拓扑和低维拓扑中的经典问题的联系。该项目的一个主要目标将是建立仿射空间中超曲面和完全交的Kontsevich同调镜像对称猜想，将仿射簇的包裹Fukaya范畴和派生范畴及其镜像Landau-Ginzburg模型相互联系。在另一个方向上，这个项目将在辛流形上发展奇异拉格朗日环面的新构造，以及它们与环面退化的关系，以及在镜像空间上聚集各种结构。最后，它将试图从镜像对称性和Calabi-Yau 3-折叠中拉格朗日环面的交叉性的角度来解释新的纽结不变量(例如物理学家阿加尼奇和瓦法最近引入的量子A多项式)。

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.