Advancing topology and group theory: division rings, L2-invariants and Dehn filling

The broad field of this project is topology, a branch of mathematics that studies geometric properties preserved under continuous transformations. The main tools used in this study are group theory and homology. A group is a collection of transformations of an object that preserve certain geometric properties. Groups have algebraic structures that make them easier to analyze than the geometric objects themselves. Homology counts the number of holes in various dimensions of a geometric object. Since continuous transformations cannot create or eliminate holes, homology is a powerful method for studying topology. In addition to advancing mathematical knowledge, the project promotes STEM education and broadens participation through outreach. Activities include a summer camp for high school students, a math night featuring games for elementary students, and a workshop for graduate students. These efforts aim to inspire students at multiple educational levels, build a stronger STEM pipeline, and support the development of future scholars. This project consists of three interconnected components. The first focuses on Simon’s Conjecture, which relates knot groups to knot genera. The PI proposes a new approach via division rings and aims to verify the conjecture for bi-orderable knot groups. The second component develops algebraic and topological characterizations of the Thurston norm on free-by-cyclic groups, which is currently accessible only through functional analytic techniques. The third component investigates how L2-Betti numbers behave under a group-theoretic construction known as Dehn filling, with applications including verifying Singer’s Conjecture for Dehn-filled manifolds. 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.