Higher-dimensional contact topology and quantum topology

Contact structures are central objects of study in modern geometry and topology. They appear naturally as the boundary of space-time in mathematical physics, and as such, play an essential role in the mathematical study of three- and four-dimensional spaces and the knotting of DNA. In dimension three, contact structures are comparatively well-understood due to dramatic advances in the previous decades. The goal of this research project is to study the relationship of contact topology to quantum physics and to quantum invariants of knots and links and further develop the study of contact structures in higher dimensions, which is still in its infancy despite significant advances in recent years. As part of this project, the Principal Investigator will also promote the training of future mathematicians. This research project on higher-dimensional contact topology, Floer theory, and their interactions with quantum topology has two parts: The first is to study the mathematics surrounding the recent discovery of the relationship between Hecke algebras --- essential ingredients in quantum knot invariants --- and the higher-dimensional Heegaard Floer homology of the cotangent bundle of a surface. One of the goals is to better understand the topological quantum field theory (TQFT) underlying this discovery, relate it to string topology, and define and analyze the 3- and 4-manifold invariants corresponding to this theory. The second is to continue the systematic study of convex hypersurface theory --- a technique to decompose a contact manifold into easier-to-analyze pieces --- following the works of the Principal Investigator in collaboration with Breen and Huang and to apply them in the analysis of codimension two contact submanifolds and higher homotopy groups of the space of contact structures. 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.