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Details of Grant 

EPSRC Reference: EP/T029455/1
Title: Rank functions on triangulated categories, homotopy theory and representations of finite groups
Principal Investigator: Lazarev, Professor A
Other Investigators:
Researcher Co-Investigators:
Project Partners:
Department: Mathematics and Statistics
Organisation: Lancaster University
Scheme: Standard Research
Starts: 01 August 2020 Ends: 31 July 2023 Value (£): 380,778
EPSRC Research Topic Classifications:
Algebra & Geometry
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
EP/T030771/1
Panel History:
Panel DatePanel NameOutcome
01 Jun 2020 EPSRC Mathematical Sciences Prioritisation Panel June 2020 Announced
Summary on Grant Application Form
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.
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Organisation Website: http://www.lancs.ac.uk