Algebraic structures in Floer theory

在辛拓扑中，Fukaya范畴的计算可以分为两个步骤：(1)确定一个拉格朗日集合，从这个集合中可以通过形式代数构造得到所有其他的拉格朗日函数；我们说这需要找到一个生成子范畴，然后(2)显式地确定这个生成子范畴的乘积(和更高的乘法)。PI建议研究一个新发展的生成准则对拉格朗日流形家族和与Landau-Ginzburg模型相关的Fukaya范畴的变体的推广。这将扩大大部分Floer理论可以明确描述的辛流形的类别。辛流形给出了一个抽象的设置，概括了经典力学中位置和动量之间的关系。最近，对这种流形的研究已经影响到从纽结(数学上理解为圆在空间中的嵌入)和理论物理的研究等各个领域。这项建议将研究出现在辛拓扑中的代数不变量，目的是加深我们对这一主题及其应用的理解。

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.