HUNEKE, 97-31512 This award supports research in commutative algebra and algebraic geometry. The principal investigator will continue to investigate a group of long standing questions centered around the process of reduction to positive characteristic, especially in regard to the developing theory of tight closure. Particular emphasis will be placed on the relationship between tight closure, reduction to positive characteristic, and vanishing theorems in algebraic geometry such as the Kodaira vanishing theorem. Closely related work will be done on questions concerning uniform bounds in Noetherian rings, and in particular on the uniform Artin-Rees conjectures. This is research in the closely related fields of commutative algebra and algebraic geometry. In essence these fields are quite similar to the 19th century study of equations and their solutions. The relationship between algebraic equations, such as polynomial equations, and geometry goes back at least to Descartes and the idea of coordinatizing the plane. Commutative algebra studies the solutions of such polynomial or power series equations by forming an algebraic object, called a ring, which consists of the 'generic' solutions. The algebraic properties of these generic solutions then give insight into the geometric and algebraic nature of the equations. The field combines techniques from a number of other areas including combinatorics, topology, and analysis.
