Characteristic p Methods in Commutative Algebra

该奖项支持交换代数和代数几何的研究。首席研究员将继续研究一组长期存在的问题，这些问题围绕着还原为积极特征的过程，特别是关于紧密闭合理论的发展。特别强调的是紧闭、归约为正特征和代数几何中的消失定理（如Kodaira消失定理）之间的关系。密切相关的工作将做关于诺瑟环的一致界的问题，特别是一致Artin-Rees猜想。这是交换代数和代数几何密切相关领域的研究。从本质上讲，这些领域与19世纪对方程及其解的研究非常相似。代数方程，比如多项式方程，和几何之间的关系至少可以追溯到笛卡尔和平面坐标的思想。交换代数通过形成一个称为环的代数对象来研究这种多项式或幂级数方程的解，该代数对象由“一般”解组成。这些泛型解的代数性质可以让我们深入了解方程的几何和代数性质。该领域结合了许多其他领域的技术，包括组合学、拓扑学和分析学。

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.