Homology and Cohomology over Commutative Rings

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Avramov is investigating problems arising at the boundary of algebra, algebraic geometry, and algebraic topology. The basic objects of study are homological invariants of commutative noetherian rings that reflect geometric propertied of algebraic varieties. Special attention will be focused on the geometric meaning of the vanishing of Hochschild cohomology, on it finite generation as an algebra, and on the vanishing of Andre-Quillen homology. Asymptotic vanishing of classical homology and cohomology over complete intersections will be studied by geometric methods.This is a project in the areas of mathematics known as Commutative Algebra and Algebraic Geometry. Commutative algebra and algebraic geometry may be thought of as studying solutions of many equatons in many unknowns when, typically, the solution is not unique. The set of solutions can then be viewed geometrically, but it is often best encoded into a family of functions defined on this set. The abstract version of such families of functions are called commutative rings.