This project concerns the mathematical underpinnings of topological quantum computation; and some practical aspects of `higher' representation theory.
The scientific project predates the funded project,
and perhaps the most intriguing output so far is
a complete classification of braid representations.
(A braid representation is a strict monoidal fuctor from the category of braids to the category of matrices - also known as a solution to the constant Yang-Baxter Equation.)
This is a major open question from the 20th century, the solution of which
would have many applications in computational physics,
topological quantum computation, topological quantum field theory,
and elsewhere.
But the general problem is considered impossible.
However we were able to solve it, subject only to a natural
`charge conserving' restriction.
This has lead to an extensive research programme using the method and results,
including several publications
(for example one line generalises the charge-conserving condition;
and another classifies loop-braid representations).
And this was the heart of the proposal that
was funded for the ABC project.
The braid representation classification has its origins, in turn, in work of Damiani-Martin-Rowell introducing loop Hecke algebras (as well as much earlier work of Deguchi, Saleur, Rittenberg and others). See for example this paper of Janssens et al for a recent review of loop Hecke.
And in general for now see our works on the arXiv for more information.
* page under construction