In two dimensional topologically ordered matter, processes depend on gross topology rather than detailed geometry. Thinking in 2+1dimensions, particle world lines can be interpreted as knots or links, and the amplitude for certain processes becomes a topological invariant of that link. While sounding rather exotic, we believe that such phases of matter not only exist, but have actually been observed in quantum Hall experiments, and could provide a route to building a quantum computer.
Possibilities have also been proposed for creating similar physics in systems ranging from superfluid helium to
topological superconductors to semiconductor-superconductor junctions to quantum wires to spin systems to graphene to cold atoms.
Recordings of Condensed Matter Seminar events can be found on mediaspace: https://mediaspace.illinois.edu/channel/Condensed%2BMatter%2BSeminar%2BTalks%2BFall%2B2020/178724052