The Topology of Smooth 4-Manifolds

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This proposal concerns the interaction between low-dimensional topology and symplectic geometry. The first main theme of the proposed research is to study exotic smooth structures on small simply connected four-manifolds. The Principal Investigator and his collaborator B. Doug Park have shown that the complex projective plane blown up at two, three, and four points admit exotic irreducible smooth structures. The PI will continue his study with an aim of further understanding of smooth structures on small four-manifolds such as CP^2, S^2 x S^2, and CP^2#(-CP^2). The initial work suggests that it would be possible to construct an exotic smooth structure on some of these remaining small four-manifolds. The proposed research will also focus on the geography problem for smooth irreducible simply connected four-manifolds and the construction of surface bundles over surfaces with non-zero signature. This project may lead to new theorems on spin and non-spin symplectic geography, and provide new constructions relevant to attack the Symplectic Bogomolov-Miyaoka-Yau conjecture. A final theme in the proposed research is the construction of Stein and strong symplectic fillings of certain contact three-manifolds.This proposal studies the "exotic" smooth structures on small four-dimensional manifolds, i.e. the geometric objects which are locally modeled on space-time. Although it is known that many smooth four-dimensional manifolds admit the "exotic" smooth structures, such structures are very hard to construct if the manifold is small. The famous smooth four-dimensional Poincare conjecture illustrates this phenomenon. The PI recently developed a new and very effective technique that allows to tackle these small four-dimensional manifolds. The PI's approach shows a great promise in understanding the classification of smooth four-dimensional manifolds. This project uses the ideas and tools from several fields of mathematics, such as geometric topology, symplectic geometry, complex algebraic geometry, group theory and gauge theory. The problems involved in this research project also have interesting applications to physics, such as mirror symmetry and string theory.