Mathematical Sciences: 4-Manifolds and Symplectic Topology

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The purpose of this project is better to understand smooth and symplectic 4-manifolds. Smooth 4-manifolds are spaces which are locally diffeomorphic to Euclidean 4-space. Symplectic manifolds are endowed with additional structure which keeps track of areas and volumes. For both types of 4-manifolds, the most basic questions of existence and uniqueness are still poorly understood; it is these questions which the project addresses. Traditionally, these two types of objects have been studied by rather different methods, but the principal investigator's research to date shows that much can be gained from a unified approach. In fact, he has solved major problems in both fields by simple "cut-and-paste" constructions (log transforms and connected sums along surfaces). He intends to apply these and related techniques to produce even better examples (for example, simply connected, irreducible symplectic 4-manifolds with small second betti numbers). These, and other examples which the principal investigator has already produced, should be detectable by means of gauge theory. Other examples which might be produced (such as a simply connected, symplectic, non-Kaehler 6-manifold, or a symplectic 4-manifold which splits as a connected sum of indefinite pieces) could be detected by elementary means. Smooth manifolds of dimension 4 are presently the least understood manifolds of any dimension. Only in the last decade has substantial progress been made in understanding the possible shapes of these objects. This is particularly ironic in that our own universe is an example of a 4-dimensional manifold. Symplectic manifolds have also come under intense study recently. These objects first appeared in connection with classical mechanics, the physics of macroscopic objects such as mechanical systems and orbiting satellites. Subsequently, they have turned out to play a deep role in quantum physics, so they are of great interest to physicists who are attempting to understand the fundamental forces of nature. Symplectic manifolds are also of interest to pure mathematicians because of their appearance in such diverse fields as algebraic geometry, gauge theory and Lie group theory. It is the principal investigator's belief that there are deep and largely unexplored connections between the theories of symplectic manifolds and smooth 4-manifolds. The instant research project is to exploit these connections to illuminate both theories.