Family Floer Cohomology

几何学作为一门抽象学科的发展是基于人类对尺寸、方向和距离的共同直觉。在非对易几何的背景下，局部性的概念不再有意义，这些直觉通常提供的指导很少，因此数学家被迫求助于代数和计算工具。对辛流形的研究通过镜像对称性提供了一种非交换几何的方法，它从一个诚实的几何空间出发，产生一个由一类称为拉格朗日的子空间形成的非交换几何空间。例如，数学家所熟知的Clifford代数和物理学家所熟知的狄拉克矩阵源于对球体中圆的研究。这个研究项目将在更高的维度上研究这种非交换空间，将它们与一个目标是通过相关的被称为Fukaya范畴的非交换空间来理解辛流形的程序联系起来。该项目的重点是设置允许拉格朗日环面纤颤的辛流形。在这种情况下，调查人员最近提取了Fukaya类别的局部到全球描述。这项研究将进行一系列旨在将局部到全局方法的适用性扩展到一般辛流形的项目；这将需要开发新的工具来研究和计算Fukaya范畴。调查员的目标是系统地开发这些工具，从它们将提供Fukaya类别的替代描述(例如，余切束)的情况开始。该项目还将探索在研究拉格朗日嵌入和镜像对称性扩展方面的应用。

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.