This proposal centers around two main directions in representation theory, and proposes to develop homotopical methods. The representation theory of finite groups has been driven in a large part by the conjectures of Alperin and Brou'e in the last twenty years. The proposal will bring new topological directions in the area, and new conjectures. The basic idea is to replace actions on vector spaces by actions on topological spaces. Higher representation theory, the study of actions of a given monoidal category on categories, is a much more recent area of research. It has been developed mostly in relation with Kac-Moody algebras. A recent construction is that of tensor structures. Our proposal will bring homotopical methods in the theory. It aims also at bringing higher representation theory in the area of geometers working on counting invariants and topologists working on invariants of four-manifolds, in the continuation of Khovanov.
The project will bring new topological insight in algebra, and in particular in representation theory, the study of symmetries. A major problem is to understand global symmetries from local symmetries. We will approach and recast this within a wider topological setting. The project will also produce methods to build objects in algebra, as a counterpart of the classical constructions of spaces in geometry as families or "moduli spaces". This should lead to an understanding through algebra of some properties of three and four-dimensional geometry.
