Representation theory is a study of symmetries of space, such as our 3-dimensional space, or, more generally, a space with any (even infinite number) of dimensions. In this theory, symmetries are represented by linear transformations of this space, or, more explicitly, by matrices. Thus, a representation of a given symmetry structure is basically a collection of matrices which satisfy a certain natural system of nonlinear equations. The equations are determined by the exact type of symmetry structure we are representing - a group, a Lie algebra, or an associative algebra. Representations of a given structure themselves form a quite intricate and rich structure, which encodes relations (or mappings) between different representations. This higher-level structure is called the category of representations. For some type of structures, for example groups, Lie algebras, and quantum groups, representations can be multiplied; in this case the corresponding categories are tensor categories (as multiplication of representations is similar to multiplication of tensors). It turns out that the notion of a tensor category is very interesting in its own right, and that many tensor categories don't arise as categories of representations. This project will study ordinary and tensor categories, some of which arise as representation categories and some of which don't, and connections between them. In particular, the PI will study non-integer rank generalizations of representation categories p