Mathematical Sciences: Local Algebra and Global Cohomology

该奖项的主要研究人员将考虑交换代数中的下列局部问题。(1)单项素数P(a，b，c)的符号代数是有限生成的吗？(2)在大于或等于3的非奇异射影簇上，NEF因子的高次方的Riemann-Roch问题是否有解？(3)设R和S是具有相同商域K的正则局部环，V是K的赋值环，使得V支配R和S？是否存在正则局部环T，使得T可以由R和S通过一元变换序列得到，并且使得V支配T？这是对代数几何领域的研究，代数几何是现代数学中最古老的部分之一，但在过去的25年里，它取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及理论计算机科学和机器人学中也找到了应用。

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.