The geometry of moduli spaces from low-dimensional topology and applications

Low Dimensional Topology is a branch of mathematics that studies shapes of three and four-dimensional objects. This project explores such objects using tools inspired by modern physics. One such tool is the Yang-Mills equation, used in quantum field theory to describe electro-weak interactions. Solutions to the Yang-Mills equation on a manifold can reveal deep insights into the underlying topology. The construction of configuration spaces on manifolds is inspired by Feynman diagrams; it has recently been used to answer long-standing open problems in low-dimensional topology. These ideas also interact closely with many other areas of mathematics, such as non-linear partial differential equations, algebraic topology, and algebraic geometry. The PI will use the existing tools in gauge theory and configuration space theory to study questions in three- and four-dimensional topology and develop new tools in this field by working on fundamental analytical questions about gauge-theoretic equations. During the project, the PI will train graduate and undergraduate students, organize high school educational activities, and participate in outreach programs to attract more students to mathematics. The research activities of this project will focus on the following three major directions. The first direction studies the higher algebraic structures in instanton Floer homology and its relations with gluings of 3-manifolds. The second direction studies the analytic properties of generalized Seiberg–Witten equations. The third direction explores the applications of Kontsevich configuration space integrals in 4-dimensional topology and uses it to study smooth mapping class groups in dimension 4. 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.