Floer homology, low-dimensional topology, and mirror symmetry

本课题研究辛几何，特别是拉格朗日花同调在镜像对称和低维拓扑中的应用。拉格朗日子流形和Fukaya范畴是Kontsevich同调镜像对称猜想和strominger - you - zaslow构造镜像对方法的核心。该项目将专注于几何现象，如瞬间校正，以便更好地理解镜像对称并扩大其范围。拉格朗日子流形在低维拓扑中也起着关键的作用，它们参与了3-流形和4-流形（封闭或有边界）的各种不变量的构造。这激发了从对称积的Fukaya范畴的角度研究3流形的有边heegaard - flower同调和4流形上破碎Lefschetz纤振的不变量，旨在提供更丰富的代数框架和揭示新的联系。从广义上讲，该项目旨在加强几何，拓扑和数学物理各个领域之间的现有联系。现代理论物理学对数学产生了巨大的影响，特别是对几何，其中由场论产生的方程导致了拓扑空间的新不变量和关于其几何的新猜想。该项目的一个目标是澄清由弦理论启发的预测的数学有效性和范围，将两个不同的数学领域相互联系起来（代数几何，研究多项式方程定义的集合，辛几何，研究经典力学的相空间）。另一方面，同样的数学思想也适用于研究三维和四维空间的拓扑结构。更具体地说，目的是探索如何沿着二维表面切割这样的空间可以导致各种拓扑不变量的新解释。

This project studies the applications of symplectic geometry, and in particular Lagrangian Floer homology, to mirror symmetry and to low-dimensional topology. Lagrangian submanifolds and Fukaya categories lie at the heart of Kontsevich's homological mirror symmetry conjecture and the Strominger-Yau-Zaslow approach to the construction of mirror pairs. The project will focus on geometric phenomena such as instanton corrections in order to gain a better understanding of mirror symmetry and broaden its scope. Lagrangian submanifolds also play a key role in low-dimensional topology, where they enter in the construction of various invariants of 3- and 4-manifolds (closed or with boundary). This motivates the study of bordered Heegaard-Floer homology of 3-manifolds and invariants of broken Lefschetz fibrations on 4-manifolds from the perspective of Fukaya categories of symmetric products, with the aim of providing a richer algebraic framework and revealing new connections.Broadly speaking, this project aims to reinforce the existing connections between various areas of geometry, topology and mathematical physics. Modern theoretical physics has had a tremendous impact on mathematics, and in particular on geometry, where equations arising from field theories have led to new invariants of topological spaces and new conjectures about their geometry. One goal of the project will be to clarify the mathematical validity and scope of predictions inspired by string theory, relating two different areas of mathematics to each other (algebraic geometry, which studies sets defined by polynomial equations, and symplectic geometry, which studies the phase spaces of classical mechanics). On the other hand, the same mathematical ideas have applications to the study of the topology of three and four-dimensional spaces. More specifically, the aim is to explore how slicing such spaces along two-dimensional surfaces can lead to new interpretations of various topological invariants.