Moduli spaces and higher representation theory

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Representation theory is the study of symmetries via linear actions. Higher representation theory introduces a new paradigm, wherein spaces are replaced by higher structures (abelian or triangulated categories, or higher categorical structures). Such approaches have been advocated over the last twenty years, in particular by physicists working on quantum gravity, but very little has been achieved so far. It has appeared more and more important to study functors between categories, say to compare a category we are interested in with other categories we understand better. Our claim is that we should study the relations between these functors, and by doing so, we will discover some new symmetries of a fundamental type, analogous to classical symmetries for vector spaces. This would provide concrete (algebraic, numerical) information, while the current study of categories and functors is completed at an abstract level, and concrete data can be obtained only at the expense of a great loss of information. Such a study goes partly in line with usual representation theory: one defines interesting structures (classically one would, for example, consider symmetric groups, simple Lie algebras) and investigates the possible objects they can be symmetries of (classically one tries, for example, to classify simple representations, which are the building bricks for general representations). An important new feature is that, whilst vector spaces are fairly elementary structures, categories (abelian or triangulated) are not. An important consequence would be a better understanding of various categories of algebraic or geometric origin via the study of their higher symmetries. A crucial aspect of the proposal is to provide constructions of categories from other categories. Constructions of moduli spaces should be bypassed and the associated categorical structures should be constructed directly. Developing an algebraic substitute for moduli constructions is the main inspiration for the project. The aim of this project is to develop a new approach to counting invariants in (commutative and non-commutative) geometry, based on the PI's programme of higher representation theory.