Topics in Smooth and Symplectic 4-Manifolds

DMS-0305818 Ronald Fintushel低维拓扑的中心问题之一是光滑单连通4-流形的分类。新的结构混淆了分类问题，但也丰富了理论，加强了它的丰富性和多样性。 还需要更多的例子来确定一个适当的分类方案，提议者打算研究这种结构。提案人的最终目标是发展足够的技术，新的建设顺利和辛4-流形的一般图片分类将开始出现。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。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。这项建议的另一个基本目标是开发研究生和高级本科生可以访问的问题。这个建议提出的问题，这将是合适的论文问题，为提案人的未来的学生。同时也讨论了建议者计划给高年级本科生的计算问题。提议者还将在夏季支持研究生和本科生，并鼓励他们参加会议。在这个建议中提出的问题也将是有益的博士后研究员在密歇根州立大学。理论的4维流形是重要的数学和物理原因。在数学中，4维拓扑处于十字路口，人们可以尝试应用低维（3维或更少维）拓扑的成熟技术，也可以希望利用高维拓扑的强大技术，如手术理论。在许多方面，最有趣的技术都不是来自这两种方法，而是来自与复杂表面理论的类比和高能物理的输入，其中由于明显的原因，四维理论是中心。这一提议采用了后一种观点，其关键技术围绕着量子场论中的塞伯格-威滕方程。

DMS-0305818Ronald FintushelOne of the central problems of low-dimensional topology is the classification of smooth simply connected 4-manifolds. New constructions have confused the issue ofclassification, but also have invigorated the theory and reinforced its richness and diversity. Still further examples are needed to identify a suitable classification scheme, and the proposer intends to work on such constructions. The ultimate goal of the proposer is to develop enough techniques for new constructions of smooth and symplectic 4-manifolds that a general picture for classification will begin to emerge.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. The proposer will study geography problems for symplectic manifolds which have been shown to impact physics via the notion of 'superconformal simple type'. Another basic goal of this proposal is the development of problems which are accessible to graduate and advanced undergraduate students. This proposal presents problems which will be suitable thesis problems for the proposer's future students. It also discusses computational problems which the proposer plans to give to advanced undergraduate students. The proposer will also support graduate and undergraduate students during summers and encourage their participation in conferences. Problems posed in this proposal will also be useful for postdoctoral fellows at Michigan State.The theory of 4-dimensional manifolds is important for both mathematical and physical reasons. In mathematics, topology of 4 dimensions lies at a crossroad, where one can try to apply well-developed techniques of low-dimensional (3 and fewer dimensions) topology, and also one might hope to utilize powerful techniques of high dimensional topology, such as surgery theory. In many ways, the most interesting techniques come from neither of these approaches, but rather from analogies with complex surface theory and input from high energy physics, where for obvious reasons 4-dimensional theory is central. This proposal, takes this latter view.Its key techniques revolve around the Seiberg-Witten equations arising in quantum field theory.--