Topology of Smooth 4-Manifolds

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DMS-0204579Selman AkbulutProposer plans to investigate the topology of smooth 4-manifolds by decomposing them into Stein pieces which are easy to understand (since they turn out to be Lefschetz fibrations over the 2-disk). He would like use the techniques of complex and symplectic manifolds to understand the restriction this decomposition imposes to the topology of the original manifold. Proposer hopes to apply these techniques to attack some unsolved problems of 4-dimensional topology, such as finding 4-dimensional fake s-cobordisms, and to the problem of constructing a fake S^3 x S^1. Proposer also plans to work on the topological problems arising from complex algebraic geometry, such as problems coming from the complex conjugation, branched covers and the curve counting problem in real algebraic geometry.Proposer plans to investigate structure of 4-dimensional manifolds by decomposing them into pieces that are topologically easier to understand (4-manifolds are spaces which locally look like 4-dimensional Euclidean space-time we live in). These smaller pieces carry certain `natural complex structures' which analytical techniques apply. By these techniques Principal investigator hopes to construct certain 4-manifolds believed to exists but not yet have been found. One of the reasons 4-manifolds are of interest is because of the physicists model of the 10-dimensional universe which consists of a 4-dimensional space-time plus 6-dimensional complex piece.