Exotic 4- Manifolds, and geometric structures

这个项目将阐明光滑流形（广义空间）的几何和拓扑学中未解决的问题。这些流形与物理学家当前感兴趣的时空和弦理论有关。本计画的主要目标是研究和了解四维流形的奇异光滑结构和七维流形的镜像对偶。该项目将支持教育，为研究生提供学习机会，并为推进这些领域做出贡献。 PI将研究光滑4-流形，Stein流形，Lefschetz纤维化和G_2流形的拓扑。他计划通过将4-流形分解为基本的易于理解的片段，称为软木塞，塞子和PALF，来解决4-流形理论中许多未解决的问题（如光滑庞加莱猜想，s-协边问题和小奇异4-流形的存在）。PI将通过应用复杂和辛流形理论的技术以及体技术来研究这些作品。PI还计划通过Lefschetz纤维化的工具来研究高维Stein流形的拓扑，并研究它们与光滑4-流形的关系。此外，PI计划工作的几何和拓扑的某些类别的7维流形，称为G_2流形。他的目标是解释由G_2流形的结合子流形的变形引起的"镜像对偶"。G_2流形在弦理论中起着重要的作用，因而引起了物理学家的广泛兴趣。

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.