Mathematical Sciences: Topics in Commutative Algebra

该奖项支持交换代数中两个主题的研究。第一部分讨论了优环、形式纤维以及优秀局部域R的一般形式纤维的某些素理想与R的一些二元扩张之间的关系；第二部分讨论了Lech关于Hilbert函数在局部Notherian环的忠实平坦态射下的行为的猜想。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。

This award supports research on two topics in commutative algebra. The first is concerned with excellent rings, formal fibres and the relationship between certain prime ideals of the generic formal fibre of an excellent local domain R and some birational extensions of R. The second is related to a conjecture by Lech concerning the behavior of the Hilbert function under faithfully flat morphisms of local noetherian rings. 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.