RUI: Knot Theory and the Four-Sphere

Understanding four-dimensional spaces holds significant scientific importance. It offers insights into fundamental theories like relativity and quantum mechanics, enhancing our conception of the structure and dynamics of the universe. Additionally, it enables exploration of complex phenomena such as higher-dimensional geometry, offering solutions to theoretical puzzles and practical challenges. Embracing the concept of four-dimensional spaces expands our intellectual horizons and has the potential to lead to advancements in technology and change how we interact with our surroundings. The focus of this research program is the study of knotted objects in dimension four and the connections that arise between the topology and geometry of the ambient four dimensional manifold and the knotted objects found within. This grant will allow the PI to carry out a robust program of undergraduate research, training, and mentoring; organize conferences, seminars, and an undergraduate math club; and implement student-centered teaching pedagogies. All of these activities help to create a new generation of researchers who will grapple with the challenging problem of understanding four-dimensional topology. The four-dimensional sphere is the simplest closed, four-dimensional manifold, yet it remains little understood in several important ways. First, it is unknown whether the four-sphere admits an exotic smooth structure, a question that has been settled in all other dimensions. Second, it is unknown whether a smoothly knotted two-sphere in the four-sphere with the same knot group as the unknotted two-sphere must itself be unknotted. Finally, it is unknown which four-manifolds arise as irregular three-fold covers of the four-sphere. This research program aims to make progress on these and important related problems through the study of knotted disks in homotopy four-balls; the diagrams of knotted surfaces in the four-sphere; and the topology, geometry, and symmetries of the four-manifolds obtained as branched covers over these surfaces. 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.