Topics in Smooth and Symplectic 4-Manifolds

Michigan State University, East Lansing MI

Investigators

Abstract

DMS-0305818 Ronald Fintushel One of the central problems of low-dimensional topology is the classification of smooth simply connected 4-manifolds. New constructions have confused the issue of classification, 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. --