Family Floer Cohomology

几何学作为一门抽象学科的发展是基于人类对维度、方向和距离的共同直觉。在非对易几何中，定域性的概念不再有意义，这些直觉通常提供不了什么指导，因此数学家被迫求助于代数和计算工具。辛流形的研究通过镜像对称提供了一种非交换几何的方法，该方法从一个诚实的几何空间开始，并产生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，数学家称为克利福德代数和物理学家称为狄拉克矩阵的代数都是从对球体中的圆的研究中产生的。这个研究项目将调查这种非交换空间在更高的维度，将它们与一个程序，其目标是了解辛流形通过相关的非交换空间称为福谷类别。该项目的重点是设置辛流形承认拉格朗日环面纤维化。研究者最近提取了该病例中福谷类别的局部到全局描述。这项研究将追求一系列的项目，旨在扩大适用性的局部到全球的方法，一般辛流形，这将需要开发新的工具，研究和计算福谷类别。研究者的目标是系统地开发这些工具，从它们将提供福谷类别的替代描述的情况开始（例如，对于余切丛）。该项目还将探索应用程序的拉格朗日嵌入和镜像对称的扩展的研究。

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.