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Lagrangian Floer homology and the geometry of homological mirror symmetry

拉格朗日弗洛尔同调和同调镜像对称的几何

基本信息

批准号：
1406274
负责人：
Denis Auroux
金额：
$ 24.57万
依托单位：
University of California-Berkeley
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2014
资助国家：
美国
起止时间：
2014-07-01 至 2018-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1406274&HistoricalAwards=false
关键词：
Lagrangian Floer homology geometry homological

项目摘要

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.

理论物理学的新思想带来了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（研究力学系统的相空间）和纽结理论。这个项目的目的是调查和建立这些领域之间的一些固定的关系。例如，将研究复曲面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Aganagic和Vafa的猜想，根据该猜想，三维空间中的每个结都确定了6维辛可积系统，其几何形状反过来与最近发现的结不变量有关。通过测试理论物理学家最近的预测的有效性，这项工作将提高我们对现代几何各个领域之间正在出现的和仍然神秘的联系的理解。这个项目将使用拉格朗日弗洛尔同调作为工具来探索镜像对称的各个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）的超曲面和完全交叉仿射空间，有关包裹福谷类别和衍生类别的仿射品种和他们的镜像朗道-金兹伯格模型彼此。在另一个方向上，这个项目将开发新的结构的奇异拉格朗日环面辛流形，和他们的关系，环面退化和集群各种结构的镜像空间。最后，它将寻求从卡拉比-丘3-褶皱中拉格朗日环面的镜像对称性和跨壁性方面提供对新纽结不变量（例如物理学家Aganagic和Vafa最近引入的量子A-多项式）的解释。