Combinatorial aspects of derived representation theory

Calabi-Yau (CY) triangulated categories provide a homological context for ideas that first originated in mathematical physics. In recent years they have acquired independent interest in pure mathematics, particularly in relation to cluster-tilting theory and Bridgeland stability conditions. A triangulated category is Calabi-Yau if it satisfies a particularly nice form of Serre duality: for an integer n, the nth shift functor [n] is naturally isomorphic to the Serre functor S; such a category is called n-CY. When n >1, much is known about the homological and combinatorial structure of such categories, e.g. (higher) cluster-tilting theory and higher homological algebra. The picture when n < 1 is much less well developed despite there being mainstream examples, e.g. stable module categories of symmetric algebras are (-1)-CY and their perfect derived categories are 0-CY. The aim of this project is to develop the structure theory for negative CY categories, for example by classifying (co)torsion pairs in discrete CY triangulated categories, and describing simple-minded systems for stable module categories.