Exotic 4- Manifolds, and geometric structures

奇异4-流形和几何结构

• 批准号: 1505364
• 负责人: Selman Akbulut
• 金额: $ 24.44万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-15 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1505364&HistoricalAwards=false
• 关键词: Exotic; Manifolds; geometric; structures

This project will shed light on unsolved problems in geometry and topology of smooth manifolds (generalized spaces). These manifolds are related to the space-time and string theories that are of current interest to physicists. The principal goal of this project is to study and understand exotic smooth structures of 4-dimensional manifolds and mirror dualities of 7-dimensional manifolds. This project will support education, by providing opportunities for graduate students to study and contribute to advancing these fields. The PI will investigate the topology of smooth 4-manifolds, Stein manifolds, Lefschetz fibrations, and G_2 manifolds. He plans to attack many of the unsolved problems in 4-manifold theory (such as the smooth Poincare Conjecture, the s-cobordism problem, and the existence of small exotic 4-manifolds) by breaking 4-manifolds into basic easy to understand pieces, which are called Corks, Plugs and PALFs. The PI will study these pieces by applying techniques from the complex and symplectic manifold theories along with the handlebody techniques. The PI also plans to study the topology of higher dimensional Stein manifolds by the tools of Lefschetz fibrations and to study their relation to the smooth 4-manifolds. In addition the PI plans to work on geometry and topology of certain classes of 7 dimensional manifolds, called G_2 manifolds. His goal is to explain "mirror dualities" arising from the deformations of associative submanifolds of G_2 manifolds. The G_2 manifolds are of current interest to physicists because they play important role in string theory.

该项目将阐明几何形状和光滑歧管拓扑（广义空间）的拓扑问题。这些流形与物理学家当前感兴趣的时空理论有关。该项目的主要目标是研究和理解7维流形的4维流形和镜子双重性的外来平滑结构。该项目将通过为研究生学习并为推进这些领域做出贡献的机会来支持教育。 PI将研究光滑的4个manifolds，Stein歧管，Lefschetz纤维和G_2歧管的拓扑拓扑。他计划通过将4个manifolds分解为基本的易于理解的作品，以4个manifold理论（例如平滑的庞康猜想，S-cobordism问题和存在的小型异国情调的4个manifolds）来攻击许多未解决的问题，这些问题称为易于理解的作品，这些作品被称为软木，塞子和Palfs。 PI将通过应用复杂和符号歧管理论的技术以及手柄技术来研究这些部分。 PI还计划通过Lefschetz纤维的工具来研究更高维的Stein歧管的拓扑，并研究它们与光滑的4个模型的关系。此外，PI计划在某些7维流形的某些类别（称为G_2歧管）的某些类别上进行几何和拓扑工作。他的目标是解释是由G_2歧管的关联子手机的变形引起的“镜子二元性”。 G_2歧管是物理学家目前感兴趣的，因为它们在弦理论中起着重要作用。