Floer theories in symplectic geometry and low dimensional topology

这个项目的一个主要目标是用Kaehler曲面镜像对称公式中的大结构极限的一个版本来完成Atiyah-Floer猜想的证明。这将把同调三球面的规范理论Floer同调与拉格朗日算子的Floer同调联系起来，后者源于与Heegaard分裂相关的平坦丛的模空间。该项目的另一大部分旨在将拉格朗日对应实现为精化Donaldson-Fukaya范畴上的可合成函子。通过使用规范理论模空间将拓扑态射(如3-余切或纠缠)表示为拉格朗日对应，这将导致拓扑不变量和TQFT。该项目属于辛几何和低维拓扑相互作用的一般领域。通过辛范畴构造拓扑不变量一直是这一领域的指导思想，尽管拉格朗日对应的几何组成只被部分定义。这个项目旨在实现这一愿景，基于组成的完整代数定义。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。更广泛地说，这个项目旨在进一步理解和阐述规范理论的分析基础，伪全纯曲线，以及一般的非线性偏微分方程模空间。

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.