The Structure of Smooth 4-Manifolds

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DMS-0204041Ronald SternFor the last 25 years it has been the goal of exciting and deep mathematics to classify smooth 4-dimensional manifolds. An arsenal of techniques has been thrown at this problem; it is the focus of dozens of research groups. The most successful attempts have associated to each 4-dimensional manifold the solution space to complex systems of equations that arise in particle physics: the Yang-Mills equations and the monopole equations of Seiberg and Witten. These solution spaces are useful in distinguishing cunningly constructed 4-dimensional manifolds. The result of this assault is that 4-dimensional manifolds are more complicated than we ever expected. As a result, it is impossible to predict a classification scheme. It is the goal of this project to more systematically approach the existence and uniqueness framework for a classification scheme. The first goal is to understand why smooth structures on 4-manifolds are sensitive to local topological change. This project describes the underpinnings of how the smooth structures change in the known constructions. Log transformations on null-homologous tori are shown to play a significant role. The first step is to determine if two smooth structures on a fixed homeomorphism type of simply-connected smooth 4-manifold are related by a sequence of log transformations on null-homologous tori. The second step is to determine the characteristic numbers of irreducible smooth 4-manifolds and to determine how these topological invariants affect their Seiberg-Witten invariants. A notion of general-type smooth 4-manifolds is given and a conjectured restriction on their Seiberg-Witten invariants is proposed. This project outlines new constructions that show that there are general-type manifolds that fill out the regions determined by these restrictions. Careful investigation of these constructions should indicate why they are best possible. Other questions related to potential restrictions on the characteristic classes of irreducible simply-connected smooth 4-manifolds will be investigated.Excitement has been generated by the idea that the puniest of all forces, gravity, may in fact be a strong as nature's other three fundamental forces: the strong force which binds protons and neutrons together in atomic nuclei; the weak force which governs radioactive decay; and the forces that govern electricity and magnetism. The perceived mismatch between these three forces and gravity creates a theoretical nightmare; it's the principle reason we have yet to find a grand unified theory. However, it has recently been hypothesized that this weakness is a mirage; the force of gravity only appears weak because its force is diluted in our own universe and most of gravity's force radiates out into extra dimensions. All other forces remain trapped in our 3-dimensional world, while gravity is free to roam other dimensions. With this hypothesis, there could be other worlds that are parallel to our own; they all neatly stack up, each oblivious of the other, with gravity the only force that moves between them. This would also account for the missing dark matter of our universe; it actually resides in other parallel universes. New mathematics will be generated in this project to further explore these ideas. Much of the relevant mathematics has already exposed the special nature of dimensions three and four. These parallel universes may be explained by the theory of (singular) foliations. The proposed study of singular foliations may structure the way in which we view our own universe -how we stack up with possible parallel universes. These singular foliations will also provide new insight into the classification of 4- dimensional manifolds. At bottom, the goal of this project is to develop more systematic constructions of smooth 4-dimensional manifolds with the hope that a general picture begins to emerge that will at least suggest a classification scheme.