Geometry and topology of 4-manifolds

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The overarching goal of this project is to develop a deeper understanding of the structure of four dimensional manifolds, which are geometric objects locally modeled on the four-dimensional space-time we live in. Smooth, symplectic and complex structures on these objects are broadly featured in theoretical physics: for example, in classical mechanics, particle physics, string theory, and quantum field theory. This project combines modern mathematical tools to explore similarities and differences of four dimensional manifolds equipped with such structures. The research program will invoke various new ideas and techniques that create leverage to address a number of important problems through a study of algebraic relations between curves on surfaces, which is fairly accessible to graduate and advanced undergraduate students. The broader impacts of this endeavor are through mentoring and outreach. This is a research project in low dimensional geometry and topology, aiming to provide new insight to a variety of profound questions on smooth and symplectic four-manifolds, contact three-manifolds and their fillings. A central goal of the project is to explore the existence of exotic smooth and symplectic structures in dimension four, with an eye towards deriving sensible classification schemes. Some of the projects are contriving novel constructions of exotic rational and ruled surfaces, fake symplectic projective planes, ball quotients and Calabi-Yau surfaces, an in-depth study of the geography of surface bundles over surfaces, and investigations of various stable classification schemes in dimension four. The research program will draw from the fields of topology, geometry, analysis and algebra, where gauge theory, new mapping class group techniques and symplectic surgeries recently discovered by the PI and his collaborators will play vital roles.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.