Smooth 4-Manifolds

平滑 4 歧管

• 批准号: 0704091
• 负责人: Ronald Fintushel
• 金额: $ 23.34万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0704091&HistoricalAwards=false
• 关键词: Smooth; Manifolds

One of the central problems of low-dimensional topology is the classification of smooth simply connected 4-manifolds. In this direction a basic question is whether any topological simply connected 4-manifold which admits a smooth structure in fact admits infinitely many. This proposal suggests a technique called 'reverse engineering' which can be used to attack this problem. Many, perhaps all, of the examples of infinite families of pairwise nondiffeomorphic but homeomorphic simply connected 4-manifolds can be recast as examples of reverse engineering. In recent years there have been exciting developments in producing new examples of smooth simply connected 4-manifolds with b^+=1, and a main goal of this proposal is to use reverse engineering to find even better examples. The ultimate goal of this project is to develop more systematic constructions of smooth 4-manifolds with the hope that a general picture begins to emerge that will suggest a classification scheme.The broader impact of this proposal will address the relationship between mathematics and theoretical physics, opportunities for graduate and undergraduate students in topology, and career development of postdoctoral fellows. Four-dimensional geometry and topology has very close ties to physics. For example, the proposer will study geography problems for symplectic manifolds which have been shown to impact physics via the notion of 'superconformal simple type'. Any general results concerning Seiberg-Witten theory hold the prospect of engendering interaction with the physics community, and this will be a basic concern. Another basic goal of this is to address 'pipeline issues' in mathematics. Our approach is to to get students and young mathematicians working on interesting problems. A key aspect of this proposal is the development of problems which are accessible to graduate and advanced undergraduate students. It presents problems which will be suitable thesis problems for students and research projects for postdoctoral fellows at Michigan State. It also discusses computational problems which are suitable for advanced undergraduate students.

低维拓扑的核心问题之一是光滑单连通4流形的分类。在这个方向上，一个基本问题是任何允许光滑结构的拓扑简单连接的4流形实际上是否允许无限多个。该提案提出了一种称为“逆向工程”的技术，可以用来解决这个问题。许多（也许是全部）成对非微分同胚但同胚单连通 4 流形的无限族示例都可以被改写为逆向工程的示例。近年来，在生成 b^+=1 的平滑单连接 4 流形的新示例方面取得了令人兴奋的进展，该提案的主要目标是使用逆向工程来找到更好的示例。该项目的最终目标是开发更系统的平滑 4 流形结构，希望开始出现一个总体图景，从而提出分类方案。该提案的更广泛影响将解决数学和理论物理之间的关系、拓扑学研究生和本科生的机会以及博士后研究员的职业发展。四维几何和拓扑与物理学有着非常密切的联系。例如，提议者将研究辛流形的地理问题，这些问题已被证明通过“超共形简单类型”的概念影响物理学。任何有关塞伯格-维滕理论的一般结果都有望与物理学界产生互动，这将是一个基本问题。另一个基本目标是解决数学中的“管道问题”。我们的方法是让学生和年轻数学家研究有趣的问题。该提案的一个关键方面是提出研究生和高年级本科生可以理解的问题。它提出的问题将适合密歇根州立大学学生的论文问题和博士后研究员的研究项目。它还讨论了适合高年级本科生的计算问题。