The word "symmetry" is of ancient Greek origin. The Greeks interpreted this word as the harmony of different parts of an object, the good proportions between its constituent parts. A more formal contemporary meaning of symmetry is invariance of an object under some kind of transformations. In mathematics the idea of symmetry has taken more abstract forms, leading among other things to the notion of a "quantum group". Quantum groups generalize the idea of symmetry as a collection of transformations, but now such a collection exists as a whole and no transformation makes sense individually. While this may sound abstract, one of the motivations behind quantum groups is to understand symmetries in nature at the subatomic level, where the laws of physics behave in a very bizarre way from our macroscopic point of view. When one studies quantum groups, it turns out that it is useful to think about even more abstract structures, so called "tensor categories", which describe symmetries not of separate objects, but, in some sense, of their core common idea. A fruitful approach to understanding these symmetries is by studying their actions on other abstract structures, "module categories". The goal of the project is to develop analytical tools to study these abstract structures and as application get a better understanding of quantum groups.

Successful but isolated encounters of category theory and operator algebras have appeared for decades. Nevertheless in such encounters categories have been predominantly used to encode certain combinatorial structures which can then be separated from their operator algebraic origins and studied by mostly algebraic methods. It has become clear only recently that tensor categories themselves can be considered as quantizations of groups and as such they possess a rich analytic structure. For example, the subtle differences between various notions of amenability manifest themselves in the Poisson
boundaries of tensor categories, a discovery which has led to a classification of a large class of deformations of compact Lie groups by Neshveyev and Yamashita.
The goal of the project is to develop new methods for studying analytical properties of categories and solve several concrete classification problems by exploring a number of topics, ranging from construction of operator algebras using categorical data to computation of second cohomology of discrete quantum groups, quantum random walks and associated boundary theories, analysis of tensor categories with fusion rules of a compact Lie group, and developing an appropriate notion of quasi-isometry/measure equivalence for quantum groups and tensor categories.