CAREER: Configuration Spaces, Gauge Theory, and the Topology of 4-Manifolds

Low-dimensional topology is a branch of mathematics that studies the shapes and structures of three and four-dimensional objects. This project explores the properties of such objects, with a focus on the study of symmetries of geometric objects in dimension 4. The study of 4-dimensional shapes is significantly different from the other dimensions, because there exists a large number of exotic phenomena that are not possible in other dimensions. Since four-dimensional shapes provide the mathematical foundation for modeling the space-time of our physical universe, understanding these structures is of profound importance. This project seeks to promote the progress of science by developing innovative tools to understand these geometric objects via creative analogies between four-dimensional geometry and modern quantum physics. Beyond advancing fundamental mathematical research, the project contributes to society by supporting STEM education for the general audience. During the project, the Principal Investigator (PI) will train graduate and undergraduate students, organize high school educational activities, and participate in outreach programs designed to attract a broader group of students to the mathematical sciences. The goal of the project is to develop new tools and discover new phenomena in the study of 4 manifolds using gauge theory and the calculus of configuration spaces of points. There are three main directions of research within the project. The first direction studies the mapping class groups and the isotopy classes of embedded 2– and 3– dimensional manifolds in 4–manifolds using configuration spaces of points. The second direction studies the properties of 4–manifolds with bi-Lipschitz structures using tools from geometric analysis and gauge theory. This is also related to questions in C^0–symplectic topology. The third direction studies several problems in instanton Floer homology, including the structure theorem for singular Donaldson invariants and properties of instanton spectral invariants. 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.