Four-manifolds, surfaces, and mapping class groups

This project aims to develop a deeper understanding of the underlying mysteries of four-dimensional spaces, which are mathematical objects locally modeled on the space-time. These spaces also arise in physics, through classical mechanics, string theory, and quantum field theory. The project explores similarities and differences among such spaces when equipped with additional geometric structures, such as smooth, symplectic, and complex structures, using a mix of topological, geometric, analytical, combinatorial, and algebraic methods. Many of the central questions are translated into hands-on problems involving algebraic relations between curves on surfaces, offering accessible entry points for graduate and undergraduate students. The project includes broader community contributions through the creation of training opportunities for students and early-career mathematicians, such as organizing workshops and summer schools—and through work on new problems and updates to longstanding open problems in the field. The research lies in low-dimensional topology and geometry, particularly concerning the topology of smooth and symplectic 4-manifolds and contact 3-manifolds. Key goals include constructing exotic smooth structures on previously inaccessible families of 4-manifolds, developing new examples with definite or signature-zero intersection forms, and producing symplectic analogues of complex surfaces such as fake projective planes and ball quotients. The project also explores smooth mapping class groups and equivariant embeddings of surfaces in the 4-sphere. A variety of methods from gauge theory, geometric group theory, and symplectic and contact topology will be used to advance these investigations and uncover new structural results. 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.