Wenzl's research on (unitary) braid representations was originally motivated by giving comparatively elementary constructions of important examples of subfactors. More recently, it has led to classification results of braided tensor categories whose Grothendieck semi-ring is the one of a representation category of a classical Lie group or one of the associated fusion categories. Wenzl plans to continue this study for the missing cases of spinor groups and exceptional groups. Such a classification provides a bridge between different approaches towards subfactors via quantum groups, loop groups and braid groups, which is useful for a number of constructions. In particular, this will be applied to constructing additional important examples of subfactors related to coproduct actions and twisted loop groups. Moreover, Wenzl's analysis of certain braid representations have already found applications in other approaches to subfactors, such as planar algebras or loop group constructions; so similar applications would seem likely for the current project. Classically, the study of symmetries in physical systems was closely linked to representation theory of groups. In recent years, however, results in areas such as conformal field theory, statistical mechanics and topology required more complicated mathematical tools. In particular, one has encountered tensor structures where the usual symmetries given via permutations of tensor factors are replaced by more complicated braiding symmetries. This has the advantage that one can read off much more information using the representation theory of the infinite braid groups than it would be possible with symmetric groups. So one can construct and classify such tensor categories in many cases by just using results about braid representations. This has been exploited in Wenzl's previous research, and he plans to apply this strategy for more complicated cases. This also has the potential for new results in the classical cases of group symmetries as limiting cases.
