Algebraic structures in Floer theory

在辛拓扑中，Fukaya范畴的计算可分为两个步骤：(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类别，然后(2)显式地确定生成子类别的乘积（以及更高的乘法）。PI建议将新开发的一代准则推广到拉格朗日流形族，以及与朗道-金兹堡模型相关的深谷范畴的变体。这将扩大辛流形的种类，因为它可以显式地描述花理论的大部分。辛流形给出了一个抽象的集合，概括了经典力学中位置和动量之间的关系。对这种流形的研究最近影响了许多不同的领域，如对结（数学上理解为空间中圆的嵌入）和理论物理的研究。本提案将研究出现在辛拓扑中的代数不变量，目的是进一步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.