Higher representation theory

It is a common occurrence in mathematics that in order to understand a given thing, we add extra structure to it - seemingly making it more complicated, but in fact making it easier to obtain insight on it. An analogy would be ordering things in the real world. The thought of all possible words in a language seems quite overwhelming, but ordering them lexicographically and writing them in a dictionary, we have an easy way to obtain information about any given one, despite seemingly having complicated things by imposing extra structure (knowing when a word comes before another). In mathematical terms, we have turned words into a category.

Categories are commonly studied through their representations, meaning actions on things that are more easily accessible, not unlike books written in a certain language: Not all of the language will be visible in a given book, but the more books we read, the better we know the language. For certain important categories called algebras, there is a very well-developed theory of such representations.

In recent years, it has been found that in order to study questions about categories, it is often useful to add even more structure to obtain 2-categories, e.g. the "category of all categories". The study of examples of representations of such 2-categories has led to some exciting breakthroughs on long-standing problems in pure mathematics. Our project is to extend the theory of representations of algebras to representations of 2-categories with analogous properties to those of algebras, with the goal of providing a structured framework for the examples that have been observed, hence facilitating future use of these successful concepts.

As with most pure mathematics proposals, the impact of this proposal will first and foremost be academic, namely in related mathematical disciplines. As described under Acandemic Beneficiaries, these areas include algebraic and geometric representation theory, topology, in particular knot theory, quantum field theories, and number theory.
The strong impact of higher representation theory in various branches of mathematics has led to considerable international research efforts and this project will be instrumental in maintaining the health of UK research in this extremely active area.

Since we aim to disseminate our research by uploading it on arxiv.org immediately after completion, as well as publishing it in leading peer-reviewed journals and speaking about it in seminars and at conferences, and since our results will be directly applicable to many situations where 2-representations appear, we expect the impact of our research to happen within a few years in the closer surroundings (algebraic and geometric representation theory), and maybe slightly later further afield in areas like number theory, where there seems to be promise of higher representations providing insight, but where these routes have not been as throroughly investigated.