Quantum symmetries in diagram algebras and topological quantum field theory

In recent years TQFT has been driven by the combinatorics of braid relations, Temperly-Lieb algebras, and unitary representations of the Thompson group, following the seminal work of Vaughn Jones and others. The key observation is that unitary representation of the Thompson groups are better at producing links than braid relations. This project seeks to combine the group theory, quantum groups, representation theory, and operator algebras to understand and extend Jones' construction. Moreover, we hope to use recent developments in higher order Thompson groups to build new links and their invariants through systematic study.