Cohomology over Commutative Rings: Structure and Applications

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Avramov will investigate problems arising at the crossroads of commutative algebra, homological algebra, and representation theory. Properties of commutative noetherian rings will be studied through numerical, algebraic, and geometric invariants generated by homological constructions. New methods will be developed for computing such invariants, to be tested on problems that have resisted more conventional approaches. Homological algebra will be utilized to establish further connections between the theories of commutative rings and of finite-dimensional algebras, with the goal of transferring viewpoints and adapting techiques developed in one of the fields for use in the other. Commutative algebra and algebraic geometry may be thought of as studying solutions of several equations 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. Homological algebra brings to the study of rings methods of algebraic topology, developed to study geometric structures. Methods for decomposing complicated objects into primitive building blocks are developed as part of a different branch of algebra, known as representation theory.