Derived categories of complete intersections and Hochschild cohomology

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This proposal concerns the structure of the derived categories and associated triangulated categories arising in commutative algebra, and is partly motivated by developments in the representation theory of finite dimensional algebras, and in the cohomology of finite groups. The research focuses both on qualitative aspects of the derived category (for example, support varieties) and on quantitative aspects (for dimension, in the sense of Rouquier). The Hochschild cohomology algebra emerges as a critical player in these investigations, for it acts on the derived category and to capture significant homological information about it, especially over complete intersection rings.In studying a topological space (or other geometric objects) it is often useful to consider how it can be built out of simpler objects, like discs or spheres. One can then attach numerical invariants to the building process (for example, how many steps are required?) to obtain a measure of the complexity of the object. Surprisingly, similar considerations turn out to be extremely fruitful also in the more discrete world of algebraic structures. The questions addressed in the proposal seek to investigate algebraic gadgets called commutative rings, which arise naturally in various branches of mathematics and physics, from this perspective.