Exotic 4- Manifolds, and geometric structures

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This project will shed light on unsolved problems in geometry and topology of smooth manifolds (generalized spaces). These manifolds are related to the space-time and string theories that are of current interest to physicists. The principal goal of this project is to study and understand exotic smooth structures of 4-dimensional manifolds and mirror dualities of 7-dimensional manifolds. This project will support education, by providing opportunities for graduate students to study and contribute to advancing these fields. The PI will investigate the topology of smooth 4-manifolds, Stein manifolds, Lefschetz fibrations, and G_2 manifolds. He plans to attack many of the unsolved problems in 4-manifold theory (such as the smooth Poincare Conjecture, the s-cobordism problem, and the existence of small exotic 4-manifolds) by breaking 4-manifolds into basic easy to understand pieces, which are called Corks, Plugs and PALFs. The PI will study these pieces by applying techniques from the complex and symplectic manifold theories along with the handlebody techniques. The PI also plans to study the topology of higher dimensional Stein manifolds by the tools of Lefschetz fibrations and to study their relation to the smooth 4-manifolds. In addition the PI plans to work on geometry and topology of certain classes of 7 dimensional manifolds, called G_2 manifolds. His goal is to explain "mirror dualities" arising from the deformations of associative submanifolds of G_2 manifolds. The G_2 manifolds are of current interest to physicists because they play important role in string theory.