At the core of the project lies the concept of triangulated categories. These structures are omnipresent in contemporary mathematics, and give the proper framework for doing modern homological algebra in very different contexts, such as algebraic geometry, stable homotopy theory, and representation theory.
In order to further develop the theory, and sometimes for practical reasons, certain classes of triangulated categories have received more attention. One particularly well-understood class is that of compactly generated triangulated categories; these satisfy strong representability results, which make for a fairly constructive localization theory. Moreover, such categories do arise in real life; for instance, the derived category of any ring is compactly generated.
Alas, a truly potent dual theory is yet to emerge: In many interesting categories, the only cocompact object is trivial. One idea which is to be pursued in this project, is the weaker notion of '0-cocompactness', as introduced in Oppermann--Psaroudakis--Stai "Change of rings and singularity categories". 0-cocompact objects do appear in nature---indeed, compact generation implies 0-cocompact cogeneration---and in the initial paper, the crux was that 0-cocompact objects cogenerate certain t-structures.
An important task will be to study this concept more systematically. In particular, investigating apparent connections between 0-cocompactness and representability results for functors, (generalized) Serre duality, and Auslander--Reiten theory, merits further attention.
The impact of the idea of 0-cocompactness does not seem to be confined to the above, however. In particular, a natural goal will be to determine if and how a given (algebraic) triangulated category can be compared to the derived categories of the hearts of the t-structures cogenerated by its 0-cocompact objects.
The project is a mobility stipend for the project manager, who is to spend two years in Syracuse, NY, and a final year at NTNU Trondheim.
The most critical challenge is finding suitable research partners. The project team consists of leading experts in category theory, representation theory, and non-commutative geometry: The project manager, Steffen Oppermann (NTNU), Dan Zacharia (Syracuse), and Chrysostomos Psaroudakis (Thessaloniki). The members of the team are highly motivated to embark upon the research, and are chosen carefully, based on the extent to which their expertise is relevant to the primary and secondary project objectives, and ties to the project manager.
The objectives of the project are highly ambitious, and the successful execution of the project will fill obvious voids in contemporary mathematics.