Floer homology, low-dimensional topology, and mirror symmetry

Florer 同调、低维拓扑和镜像对称

• 批准号: 1007177
• 负责人: Denis Auroux
• 金额: $ 43.64万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1007177&HistoricalAwards=false
• 关键词: Floer; homology; low; dimensional; topology

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.

这个项目研究辛几何的应用，特别是拉格朗日弗洛尔同调，镜像对称和低维拓扑。 拉格朗日子流形和福谷范畴是孔采维奇同调镜像对称猜想和斯特罗明格-丘-扎斯洛构造镜像对的方法的核心。 该项目将侧重于几何现象，如瞬子校正，以更好地了解镜像对称性并扩大其范围。 拉格朗日子流形在低维拓扑中也起着关键作用，它们参与了3-和4-流形（封闭或有边界）的各种不变量的构造。 从对称积的福谷范畴出发，研究3-流形的有边Heegaard-Floer同调和4-流形上破缺Lefschetz纤维化的不变量，旨在提供更丰富的代数框架，揭示新的联系。广义地说，本项目旨在加强几何、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，特别是对几何学的影响，在几何学中，由场论产生的方程导致了拓扑空间的新的不变量和关于它们的几何的新的不变量。 该项目的一个目标是澄清数学的有效性和弦理论启发的预测范围，将两个不同的数学领域相互联系起来（代数几何，研究由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三维和四维空间的拓扑学研究。 更具体地说，目的是探索如何切片这样的空间沿着二维表面可以导致各种拓扑不变量的新的解释。