Quantum symmetry is a broad term used to describe higher categorical symmetries which appear in field theories.  These symmetries generalise group actions and are often encoded by the action of a more general mathematical  object which weakly resembles a group. 

Quantum symmetries make a surprise appearance in the subfactor theory of Vaughan Jones.  Jones studied inclusions of simple von Neumann algebras N inside M.  It is now understood that under the right assumptions, this inclusion is encoded by a group like object acting on N and a crossed product type construction yielding M. This group like object is called a unitary tensor category, it is the mathematical object that encodes the quantum symmetry on N.

My work is on the existence, classification and structure of quantum symmetries on those C*-algebras classified by the Elliott programme. The Elliott programme is a large scale research programme with the objective of classifying C*-algebras through simpler data. To a C*-algebra one can associate a pair of abelian groups (its K-theory) and a Choquet simplex (its space of traces), the Elliott programme studies in which case this data is enough to tell C*-algebras apart. In the last years, there has been a lot of interest in a dynamical Elliott programme, which aims to classify actions of groups on C*-algebras through (G-equivariant) K-theoretic and tracial data. My work lies in greater generality, my goal is to understand the classification for actions of unitary tensor categories on C*-algebras in analogy with Vaughan Jones' theory of subfactors and their classification.

Although every group acts trivially on any C*-algebra, some unitary tensor categories may not act on a given C*-algebra. In particular, the question of classification of unitary tensor categories comes with an extra difficulty; one also needs to worry about the existence of actions.