Topics in Smooth and Symplectic 4-Manifolds

低维拓扑的中心问题之一是光滑单连通4流形的分类。新的结构使分类问题变得混乱，但也使分类理论充满活力，增强了分类的丰富性和多样性。还需要进一步的例子来确定一个合适的分类方案，并且提议者打算在这样的结构上工作。作者的最终目标是为光滑和辛4流形的新构造开发足够的技术，从而开始出现分类的总体图景。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。提议者将研究辛流形的地理问题，辛流形通过“超共形简单型”的概念已经被证明对物理学有影响。本提案的另一个基本目标是开发研究生和高级本科生可访问的问题。这个提案提出的问题，将是合适的论文问题，为提案者的未来学生。并讨论了拟给本科高年级学生讲授的计算问题。申请人还将在夏季支持研究生和本科生，并鼓励他们参加会议。本提案中提出的问题也将对密歇根州立大学的博士后研究员有用。四维流形理论在数学和物理上都很重要。在数学中，四维拓扑正处于一个十字路口，人们可以尝试应用发达的低维（3维或更少）拓扑技术，也可以希望利用强大的高维拓扑技术，如外科理论。在许多方面，最有趣的技术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学的输入，其中很明显，四维理论是中心。本建议采取后一种观点。它的关键技术围绕着量子场论中出现的Seiberg-Witten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.--