Floer theories in symplectic geometry and low dimensional topology

该项目的主要目标是通过Kaehler曲面的镜像对称公式中的大结构极限来完成Atiyah-Floer猜想的证明。这将把同调三球的规范论花同调与由与heegard分裂相关的平束模空间产生的拉格朗日的花同调联系起来。该项目的另一个主要目标是在改进的Donaldson-Fukaya范畴上实现拉格朗日对应作为可组合函子。这将导致拓扑不变量和TQFT，通过使用规范理论模空间将拓扑态射（例如3-协态或缠结）表示为拉格朗日对应。该项目属于辛几何和低维拓扑之间相互作用的一般领域。通过辛范畴构造拓扑不变量一直是该领域的一个指导思想，尽管拉格朗日对应的几何组成只是部分定义。该项目旨在实现这一愿景，基于对组合的完整代数定义。此外，Atiyah-Floer猜想的证明将是理解3-流形不同不变量之间关系的重要一步。更一般地说，本项目旨在进一步理解和阐述一般非线性偏微分方程的规范理论、伪全纯曲线和模空间的解析基础。

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.