Derived categories of sheaves over finite partially ordered sets and their homological properties

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Triangulated and derived categories have been successfully used to relate objects of different mathematical origins (e.g. Kontsevich’s Homological mirror symmetry conjecture) as well as objects of the same nature (e.g. Rickard’s Morita theory, Broue’s conjecture). In this project we investigate derived categories arising from combinatorial objects, such as partially ordered sets (posets), quivers with potential and other quivers with relations. Our main goal is to understand how the combinatorial properties of these objects are reflected in representation theoretic and homological properties of the associated derived categories. One of the main questions concerns the existence of an algorithm that given two such objects decides whether their derived categories are equivalent, or not.