This collaborative project will study the topology of smooth 4-dimensional manifolds, in connection with well-known problems in low-dimensional topology. We will focus on the construction of new smooth manifolds with symplectic structures, including Stein manifolds and symplectic fillings of certain contact 3-manifolds. Recent advances in techniques based on knot surgery and Luttinger surgery for creating exotic manifolds with small Euler characteristic will be coupled with computations of gauge-theoretic and symplectic invariants. We will make use of 4-dimensional handlebody techniques in these constructions, with an organizing principle being the search for 'corks' and 'plugs' as a technique for changing the smooth structure. Techniques of gauge theory and symplectic geometry will be used to investigate the classification of symplectic 4-manifolds and their symmetry groups.
The physical world of space and time is a 4-dimensional space whose local structure is well understood but whose large-scale (or topological) properties remain mysterious. This Focused Research Group will explore the global topology of 4-dimensional spaces, with a goal of understanding what kinds of spaces (called 4-dimensional manifolds) can exist as mathematical objects, and what the properties of such manifolds are. Of particular interest will be the problem of existence and uniqueness of symplectic structures, as well as that of determining the symmetries of a given manifold. The group will investigate how subtle changes in the smooth structure of a manifold can be achieved by gluing together pieces of different manifolds. Such changes will be detected by combining expertise from several disciplines, including powerful techniques derived from gauge theories of mathematical physics.
The Project did concern the interaction between low-dimensional topology and symplectic geometry. The main themes of the proposed research were: 1) to study the exotic smooth structures on small simply connected four-manifolds, 2) construction of exotic Stein and strong symplectic fillings of certain contact 3-manifolds, and 3) to study the geography problem for smooth irreducible simply connected four-manifolds This project used the ideas and tools from several fields of mathematics, such as geometric topology, symplectic geometry, complex algebraic geometry, group theory and gauge theory. Major Results: The PI have published four research papers related to this project. 1. "Lantern substitution and new symplectic 4-manifolds with b_{2}^{+} = 3", with Jun-Yong Park, Math. Research. Letters, 1 (2014), pages 1-17. 2. "Singularity links with exotic Stein fillings", with Burak Ozbagci, Journal of Singularities, volume 8, (2014), pages 39-49. 3. "Reducible smooth structures on 4-manifolds with zero signature", with M. Ishida, and D. Park, Journal of Topology, no 3, 7(2014), pages 607-616. 4. Dissolving 4-manifolds, covering spaces, and Yamabe invariant, with M. Ishida, and D. Park, Annals of Global Analysis and Geometry, (2014). FRG Workshop and Conferences: In the summer of 2012, the PI co-organized, with Selman Akbulut (MSU), Denise Auroux (UC Berkeley), Yasha Eliashberg (Stanford), Ko Honda (USC), Cagri Karakurt (UT Austin), and P. Ozsvath (Princeton), NSF Sponsored FRG Graduate Summer School and Conference on ``Holomorphic Curves and Low Dimensional Topology''. The program involved over 60 graduate students, 30 post-docs, and over 100 more senior researchers in the fields of Low Dimensional Topology and Symplectic Geometry. A similar NSF FRG Graduate Summer School and Conference on ``Topology and Invariants of Smooth 4-Manifolds'' and "Topology and Invariants of $-Manifolds" were held at the University of Minnesota from July 31 to August 10, 2013 and at the Simons Center for Geometry and Physics, Stony Brook University from August 18-27, 2014. The PI was one of the main organizers of these events.
