Smooth 4-Manifolds

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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.