Rank functions on triangulated categories, homotopy theory and representations of finite groups

This proposed research is in the area of pure mathematics called homotopical algebra created around 50 years ago by a great British mathematician D. Quillen. Homotopical algebra takes its motivation from homotopy theory of topological spaces, the study of those properties of geometric forms and shapes that do not change under continuous deformation. Axiomatization of relevant properties of homotopy theory leads to the concept of a closed model category; this concept has been enormously successful for tackling topological problems, but also impacted a vast array of neighbouring fields: homological algebra, algebraic and differential geometry, representation theory and even mathematical physics.

Recently, homotopical algebra received a new impetus due to the development of a new powerful circle of ideas related to differential graded and infinity categories; these have been developed by J. Lurie and his school in the USA, B. Toen and C.-D. Cisinski in Europe and others. This circle of ideas implicitly underpins the present project. More precisely, the latter should be classed as belonging to applied homotopical algebra in the sense that it uses the abstract categorical constructions of homotopical algebra, particularly derived localization, and applies it to a wide variety of problems in homotopy theory, noncommutative geometry and representation theory, some of which are of much current interest and others has lain dormant for many years for lack of new ideas.

Derived localization is a concept developed in a recent work by proposers; roughly speaking, it is a way to invert elements in non-commutative rings in a homotopy invariant way. Because of this homotopy invariance, this procedure is in many respects similar to the classical procedure of commutative localization (which is one of the cornerstones of algebraic geometry). Derived localization has already found numerous applications in algebraic topology and (derived) algebraic geometry. The goal of this project is to extend applicability of derived localization in new, possibly unexpected, directions.

The proposed project belongs to the area of fundamental research in pure mathematics. It is, therefore, expected to have impact within the academic community only, at least in the foreseeable future.

On the other hand, the proposal is of intra-disciplinary nature; it is situated at the interface of several areas of pure mathematics: algebraic topology, noncommutative ring theory, higher category theory, homotopical algebra and representation theory of finite groups. The main beneficiaries will be the researchers in these fields.

Promoting the idea of derived localization and, more generally, the applicability of homotopical algebra, infinity-categories and differential graded categories to more down-to-earth problems of pure mathematics will be the main impact of this project. Another aspect of the impact that the present project will deliver, is the training, through the involvement of postdoctoral researchers, of early career pure mathematicians in the area of homotopical algebra and its applications. Given that the UK is somewhat under-represented in this field, the proposal is timely and has strategic importance to the healthy development of the pure mathematics community in the UK.