Smooth 4-Manifolds

低维拓扑的核心问题之一是光滑单连通4流形的分类问题。在这个方向上的一个基本问题是，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。这个建议提出了一种叫做“逆向工程”的技术，可以用来解决这个问题。许多，也许是所有的，无限族的对非同构但同胚的单连通4流形的例子可以被重新塑造为逆向工程的例子。近年来，在生成具有b^+=1的光滑单连通4流形的新例子方面有了令人兴奋的进展，本提案的主要目标是使用逆向工程来找到更好的例子。这个项目的最终目标是开发光滑4流形的更系统的结构，并希望一个总体的图景开始浮现，从而提出一个分类方案。该提案的广泛影响将涉及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，以及博士后的职业发展。四维几何和拓扑学与物理学有着密切的联系。例如，申请者将研究辛流形的地理问题，辛流形已经通过“超共形简单型”的概念对物理学产生了影响。关于Seiberg-Witten理论的任何一般结果都有与物理界产生相互作用的前景，这将是一个基本问题。这样做的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开发研究生和高级本科生可以访问的问题。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的研究项目。还讨论了适合高年级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.