Geometry and topology of smooth four-manifolds

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).The proposed research has two main parts. The first part concerns producing new smooth and symplectic closed four-manifolds so as to address a variety of problems that range from constructing non-diffeomorphic copies of standard four-manifolds with small Euler characteristics to the symplectic geography problem for four-manifolds with nontrivial fundamental groups, or from building new exotic families of four-manifolds that are distinguished by their stable cohomotopy Seiberg-Witten invariants to obtaining smoothly knotted but topologically unknotted embeddings of surfaces. The second part of the project deals with generalizations of Lefschetz fibrations and symplectic structures on smooth four-manifolds. In this research, singularity theory and handlebody techniques are combined to obtain new results on generalized fibrations on smooth four-manifolds, and to establish a useful description of smooth four-manifolds in terms of broken Lefschetz fibrations and moves between them, analogous to handlebodies and Kirby moves. Determining which broken Lefschetz fibrations can or cannot support a smooth four-manifold with nontrivial Seiberg-Witten invariant, and investigating the diffeomorphism types of certain four-manifolds using broken Lefschetz fibrations associated to them are two other problems contained in this research.Space and time combined, we live in a four dimensional world. The goal of this project is to better understand the intriguing nature of "four-manifolds", which are geometric objects locally modeled on space-time. There is an immense literature in theoretical physics related to the 'shape' of four-manifolds, and a great deal of mathematical research dedicated to this very subject. When considered with certain additional structures, four-manifolds exhibit numerous curious differences. For one, "symplectic structures", which appear in various equations in classical mechanics and string theory constitute a key theme of the proposed research. The "Seiberg-Witten invariants" that arise from differential equations in quantum field theory also play a key role. Using geometric and topological methods, along with new structures, the PI studies similarities and differences of four-manifolds.