Applications of triangulated categories in representation theory.

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Over the last few decades, triangulated categories have moved from being considered rather technical tools in algebraic geometry to assume central prominence as an object of study in many other algebraic areas of mathematics: notably representation theory of various kinds, commutative algebra and homotopy theory. In all of these fields, the language of triangulated categories has immensely clarified the ideas behind existing central problems, as well as raising important new problems. The initial aim of the project will be to study some recent major theoretical advances and insights involving the triangulated categories that arise in the representation theory of finite dimensional algebras (derived categories and stable module categories), particularly the work of Rouquier on dimensions of triangulated categories, and very recent connections discovered between the unbounded derived category and the long-standing "homological conjectures". These developments have opened up a whole range of questions about generation of triangulated categories, many of which are open even in apparently simple cases. By initially trying to answer questions about these relatively simple examples it is expected that insight will be gained that can be applied to far more complicated and less well-understood examples.