Algebraic structures in Floer theory

在辛拓扑中，福谷范畴的计算可以分为两步：（1）确定一个拉格朗日集合，所有其他的拉格朗日集合都可以通过形式代数构造得到;我们说这需要找到一个生成子范畴，然后（2）明确地确定生成子范畴的乘积（和更高的乘法）。PI建议研究一个新开发的生成标准的拉格朗日流形的家庭，并与朗道-金兹伯格模型相关的福谷类别的变体的概括。这将扩大类辛流形的大部分弗洛尔理论可以明确地描述。辛流形给出了一个抽象的设置概括之间的关系的位置和动量在经典力学。对这种流形的研究最近已经影响了各种各样的领域，如纽结（数学上理解为空间中圆的嵌入）和理论物理学。这个建议将研究出现在辛拓扑中的代数不变量，目的是进一步理解这个主题及其应用。

In symplectic topology, computation of Fukaya categories can be broken into two steps: (1) identify a collection of Lagrangians from which all others can be obtained by formal algebraic constructions; we say that this entails finding a generating subcategory, then (2) explicitly determine the product (and higher multiplications) for the generating subcategory. The PI proposes to study generalisations of a newly developed generation criterion to families of Lagrangian manifolds, and to the variants of Fukaya categories associated to Landau-Ginzburg models. This will enlarge the class of symplectic manifolds for which large parts of Floer theory can be explicitly described.Symplectic manifolds give an abstract setting generalising the relationship between position and momentum in classical mechanics. The study of such manifolds has recently impacted areas as diverse as the study of knots (mathematically understood as embeddings of circles in space), and theoretical physics. This proposal will study algebraic invariants that appear in symplectic topology, with the goal of furthering our understanding of the subject and its applications.