The past two decades have seen spectacular developments in the study of quantum symmetry of operator algebras, through the theories of quantum groups, subfactors, and rigid C*-tensor categories. Indeed, since the early 2000's the operator algebras community has witnessed explosive growth in many new directions, including (1) the axiomatization and systematic study of locally compact quantum groups, (2) the development of geometric group theoretic tools in the study of discrete quantum groups, subfactors, and related operator algebras, (3) a powerful classification program for the combinatorially defined "easy quantum groups", (4) a rich theory of planar algebras with applications to the classification of small index subfactors, and (5) applications of both subfactors and quantum groups to quantum information science. Through these developments and interactions of ideas behind them, the mathematical framework of rigid C*-tensor categories has emerged as one of the fundamental mathematical structures underpinning them. Rigid C*-tensor categories provide a direct path between communities working on quantum groups and subfactors, allowing ideas developed by one to be used by the other to great effect.

 

Despite this underlying connection between these communities, researchers in quantum groups and subfactors remain, for the most part, divided into two camps, with little interaction occuring between these communities. The aim of this concentration week is to bring together leading experts and young researchers working in both subfactors and operator algebraic quantum groups, with the aim of fostering new communications and collaborations between these communities, through the language of tensor categories.

 

We expect that many important unsolved problems in these fields will benefit from input from both sides. A few examples of such problems that we hope to address during the meeting include: planar algebra approaches to the classifications of easy quantum groups, tensor categorical approaches to quantum graphs, constructions of new examples of property (T) discrete quantum groups, and applications of quantum groups and subfactors in quantum information theory.