Smooth 4-Manifolds

低维拓扑学的中心问题之一是光滑单连通4-流形的分类。在这个方向上的一个基本问题是，是否任何拓扑单连通4-流形，承认一个光滑的结构，事实上承认无限多。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，也许是所有的，两两非同胚但同胚的单连通4-流形的无限族的例子都可以被改写为逆向工程的例子。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例子。该项目的最终目标是发展更系统的光滑4-流形的结构，希望能有一个总体的图像开始出现，这将建议一个分类scheme.The更广泛的影响，这一建议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“超共形简单型”的概念影响物理学。任何关于Seiberg-Witten理论的一般性结果都具有与物理学界产生相互作用的前景，这将是一个基本的问题。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的数学家研究有趣的问题。这项建议的一个关键方面是研究生和高级本科生可以访问的问题的发展。它提出的问题，这将是合适的论文问题的学生和博士后研究项目在密歇根州立大学。还讨论了适合本科高年级学生的计算问题。

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.