Mathematical Sciences: Local Algebra and Global Cohomology

该奖项的主要研究者将考虑 交换代数中的局部问题 (1)是 单项素数P（a，B，c）n生成符号代数 (2)有没有一个解决方案的黎曼-罗克问题的高 的非奇异投射簇上的nef因子的幂 尺寸大于或等于3？ (3)假设R和 S是具有相同商域K的正则局部环，V K的估值环是否使得V优于R和S？ 有没有 正则局部环T，使得T可以从R 和S的单偶变换序列，使得V 优于T？ 这是代数几何领域的研究， 现代数学中最古老的部分，但其中有一个 在过去的四分之一个世纪里，革命性的开花。 在其 起源，它处理的数字，可以定义在平面上， 最简单的方程，即多项式。 目前，该领域 不仅使用代数方法，而且使用分析方法， 拓扑学，反过来说， 以及理论计算机科学和机器人技术。

The principal investigator on this award will consider the following local questions in commutative algebra. (1) Is the symbolic algebra of a monomial prime P(a,b,c) finitely generated? (2) Is there a solution to the Riemann-Roch problem for high powers of a nef divisor on a nonsingular projective variety of dimension greater than or equal to three? (3) Suppose that R and S are regular local rings with the same quotient field K, and V is a valuation ring of K such that V dominates R and S? Is there a regular local ring T such that T can be obtained from both R and S by sequences of monodial transformations, and such that V dominates T? This is research in the field of algebraic geometry, one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origins, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays, the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in theoretical computer science and robotics.