Lagrangian Floer homology and the geometry of homological mirror symmetry

理论物理学的新思想带来了不同数学领域之间的融合，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）和结理论。本项目旨在调查和建立这些领域之间的一些推测关系。例如，我们将研究环空间上单项式函数的辛几何与由单多项式方程定义的空间的代数几何之间的深刻联系。该项目还将探索物理学家Aganagic和Vafa的一个猜想，根据该猜想，三维空间中的每个结决定了6维空间中的辛可积系统，其几何形状反过来与最近发现的结不变量相关。通过测试理论物理学家最近的预测的有效性，这项工作将增强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花同调作为工具来探索镜像对称的各种几何方面及其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Kontsevich的同调镜像对称猜想（在两个方向上），用于超曲面和仿射空间的完全相交，将仿射变体的包裹的Fukaya类别和衍生类别及其镜像Landau-Ginzburg模型相互联系起来。在另一个方向上，本项目将发展辛流形中奇异拉格朗日环面的新构造，以及它们与镜像空间上的环面退化和聚类变化结构的关系。最后，它将寻求提供新的结不变量的解释（如量子a -多项式最近由物理学家Aganagic和Vafa介绍）在镜面对称和跨壁方面的拉格朗日环面在Calabi-Yau 3折叠。

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.