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.
I do research in topology, which is an area of pure mathematics. In recent decades, research in topology and research in theoretical physics have greatly benefited each other through a vigorous exchange of ideas. This has led to the solution of many outstanding problems in topology. Even more problems remain open. Their solution requires new and more sophisticated methods. Development of such methods is an essential part of this project. To be specific, the methods I developed in my research generalize the celebrated Atiyah--Singer Index Theorem, which in its modern form goes back to the 1960's and is considered by many the crowning achievement of the 20th century mathematics. This theorem has an overarching nature in that it cuts across several mathematical disciplines, such as analysis, differential equations, topology, algebra, and number theory, and is nowadays an invaluable tool in both mathematics and physics. Together with my collaborators, Tomasz Mrowka and Daniel Ruberman, I generalized this theorem to a new class of manifolds, called manifolds with periodic ends, and applied it to the study of several outstanding problems in manifold topology. In addition, our research was recently related to a new development in physics having to do with topological insulators. This is a new class of materials with potential applications to building cost-effective, super fast computers. These materials are based on a certain quantum effect similar to superconductivity, and it is the mathematical model explaining this effect that our research is relevant to. Throughout the duration of the grant I contributed to graduate education in mathematics via organizing graduate workshops and teaching in some of them, and via advising graduate students. Two of my graduate students at the University of Miami will be defending their Ph.D. theses in the spring of 2015.
