Smooth 4-Manifolds

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，一个基本的问题是，是否任何允许光滑结构的拓扑单连通4-流形实际上允许无穷多个。这项提议提出了一种被称为“逆向工程”的技术，可以用来解决这个问题。许多，也许全部，成对的非微分纯但同胚单连通的4-流形的无限族的例子可以被重塑为逆向工程的例子。近年来，在产生b^=1的光滑单连通4-流形的新例子方面有了令人兴奋的发展，这个提议的一个主要目标是利用逆向工程来寻找更好的例子。这个项目的最终目标是开发更系统的光滑4-流形的结构，希望开始出现一个大致的图景，这将提出一个分类方案。这个建议的更广泛的影响将解决数学和理论物理之间的关系，研究生和本科生在拓扑学方面的机会，以及博士后研究员的职业发展。四维几何和拓扑学与物理学有着非常密切的联系。例如，提出者将研究辛流形的地理问题，这些辛流形已经被证明通过“超保角简单类型”的概念影响物理。任何关于Seiberg-Witten理论的一般性结果都有可能与物理学界产生互动，这将是一个基本的担忧。另一个基本目标是解决数学中的“流水线问题”。我们的方法是让学生和年轻的数学家研究有趣的问题。这项建议的一个关键方面是开发研究生和高级本科生可以接触到的问题。它提出的问题将是适合学生的论文问题和密歇根州立大学博士后研究员的研究项目。并讨论了适合本科生学习的计算问题。

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.