4-Manifolds, Calibrated Manifolds, Real Algebraic Varieties

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Proposer plans to investigate the topology of smooth 4-manifolds by attacking some unsolved problems in 4-manifolds by decomposing them into basic easy to understand pieces (PALF's), and studying the pieces by applying techniques of complex and symplectic manifold theory. He also plans to work on calibrated manifolds, and on real algebraic varieties. In particular he plans to study 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) Proposer hopes to get a global understanding of the gauge theories of low dimensional manifolds, and construct a counting theory for these submanifolds (similar to Gromov-Witten counting theory of holomorphic curves in symplectic manifolds). Also, Proposer wants to continue to work on the project of topological characterization of real algebraic sets.Three and four dimensional manifolds, and certain classes of seven and eight dimensional manifolds (so called G2 and Spin(7) manifolds) are current interest of physicist because they play central role in understanding of space-time and the String theory physics. Also, algebraic sets are a nice way to describe topological spaces in equations, but not all the topological spaces can be described this way. Proposal plans to characterize all the topological spaces that can be described as real algebraic sets.