Floer theories in symplectic geometry and low dimensional topology

这个项目的一个主要目标是完成Atiyah-Floer猜想的证明，通过Kaehler曲面镜像对称公式中的大结构极限。这将涉及规范理论的弗洛尔同调的同调三球的弗洛尔同调拉格朗日产生的模空间的平丛相关联的Heegaard分裂。该项目的另一个重要部分是将拉格朗日对应实现为精化Donaldson-Rumeaya范畴上的可组合函子。通过使用规范理论模空间将拓扑态射（例如3-协边或缠结）表示为拉格朗日对应，这应该会导致拓扑不变量和TQFT。该项目属于辛几何和低维拓扑之间相互作用的一般领域。通过辛范畴的拓扑不变量的建设一直是在这一领域的指导性视野，虽然拉格朗日对应的几何组成只是部分定义。这个项目的目的是实现这一愿景，基于完整的代数定义的组成。此外，Atiyah-Floer猜想的证明将是理解三维流形不同不变量之间关系的重要一步。更一般地说，这个项目的目的是进一步理解和规范理论，伪全纯曲线和一般非线性偏微分方程的模空间的分析基础的阐述。

A primary goal of this project is to finish the proof of the Atiyah-Floer conjecture by a version of the large structure limit in a formulation of mirror symmetry for Kaehler surfaces. This would relate the gauge theoretic Floer homology of a homology three-sphere to a Floer homology of Lagrangians which arise from moduli spaces of flat bundles associated to a Heegaard splitting. Another large part of the project aims to realize Lagrangian correspondences as composable functors on refined Donaldson-Fukaya categories. This should lead to topological invariants and TQFT's by using gauge theoretic moduli spaces to represent topological morphisms ( e.g. 3-cobordisms or tangles) as Lagrangian correspondences.The project belongs into the general realm of interaction between symplectic geometry and low dimensional topology. The construction of topological invariants via a symplectic category has been a guiding vision in this field although the geometric composition of Lagrangian correspondences is only partially defined. This project aims to realize this vision, based on a full algebraic definition of compositions. Moreover, a proof of the Atiyah-Floer conjecture would be an important step towards understanding the relations between different invariants of 3-manifolds. More generally, this project aims to further the understanding and exposition of the analytic foundations of gauge theory, pseudoholomorphic curves, and moduli spaces of nonlinear PDE's in general.