Family Floer Cohomology

族Floer上同调

基本信息

批准号：
1609148
负责人：
Mohammed Abouzaid
金额：
$ 32.77万
依托单位：
Columbia University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-07-01 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1609148&HistoricalAwards=false
关键词：
Family Floer Cohomology

项目摘要

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.

几何形状作为一门抽象学科的发展是基于关于维度，方向和距离的共同直觉。在非交通性几何形状的设置中，当地的概念不再有意义，这些直觉通常几乎没有指导，因此数学家被迫退回代数和计算工具。符号歧管的研究通过镜像对称性提供了一种非共同几何形状的方法，该方法始于诚实的几何空间，并产生由一类称为Lagrangians的子空间形成的非交通性的方法。例如，数学家称为Clifford代数的代数和物理学家，作为Dirac矩阵，是由对球体中的圆圈的研究产生的。该研究项目将在较高维度中调查此类非交流空间，将它们与一个程序有关，该程序的目标是通过称为福卡亚类别的相关非公开空间来理解符合歧管。该项目的重点是召集拉格朗日圆环纤维的符号歧管的设置。在这种情况下，研究人员最近提取了福卡亚类别的局部到全球描述。这项研究将追求一系列项目，旨在扩展局部到全球方法的适用性；这将需要开发用于研究和计算福卡亚类别的新工具。研究人员的目标是系统地开发这些工具，从他们将提供福卡亚类别（例如，对于cotangent束）的替代描述开始。该项目还将探讨研究拉格朗日嵌入的应用和镜像对称性的扩展。