Algebraic Calabi-Yau categories

Calabi-Yau categories are fundamental mathematical structures. They encode symmetries appearing in different areas of mathematics, such as the study of derived categories of coherent sheaves on manifolds, representation theory of quivers and finite-dimensional algebras, and Fomin and Zelevinsky's theory of cluster algebras. In any of these contexts, some kind of "tilting theory" occurs. A development of a general tilting theory for Calabi-Yau categories seems to be highly desireable and should lead to new connections between the mentioned areas.