The proposer plans to study topology of smooth 4- manifolds, calibrated manifolds, and real algebraic varieties. The plan to attack to unsolved problems in 4-manifold theory is to decompose 4-manifolds into basic, easy to understand, pieces, which are called Corks, Plugs and Palfs, and study these pieces by applying techniques from complex and symplectic manifold theory. Questions about knottedness and uniqueness of Corks are particularly important in this context. An immediate consequence of these techniques is the construction of exotic Stein manifolds. The P.I. also plans to work on certain classes of 7 and 8 dimensional manifolds (so called G2 and Spin(7) manifolds). By studying the certain families of 3 and 4 dimensional submanifolds in them (so called associative and Cayley submanifolds) the P.I. hopes to get a global understanding of the gauge theories of low dimensional manifolds, and to construct a counting theory for these submanifolds (similar to Gromov-Witten counting theory of holomorphic curves in symplectic manifolds). Also intended is to explain mirror duality in terms of G2 manifolds. Finally the P.I. wants to continue working on the project of topological characterization of real algebraic sets.
4-dimensional manifolds (spaces) appear to decompose into small basic pieces (particles) which we call "Corks" and "Plugs". These pieces determine the exotic structures of the underlying 4-manifold. One can think of corks and plugs as freely moving particles in 4-manifolds like Fermions and Bosons in physics, or little knobs on a wall to turn on and off; the ambient exotic lights in a room. The P.I. plans to study the structure of these Corks and Plugs. The P.I. also plans to study G2 and Spin(7) manifolds (certain 7 and 8 dimensional spaces), which are of current interest in physicists, because they play important role in the String theory and M-theory. The P.I. also plans to work on the problem of determining which topological spaces are algebraic sets. Making a topological space algebraic helps us to understand many of the properties of the space.
