Mathematical Sciences: RUI: Geometry of Toric Varieties

该奖项支持考克斯教授致力于代数几何的研究。考克斯教授将解决的问题之一是以一种自然的方式将坐标环分配给环簇。这个环应该是多项式环，并且应该只依赖于环簇而不是特定的射影嵌入。考克斯教授提出了一种似乎具有许多必要性质的建筑方案。这项研究使用了代数几何的技术，这是现代数学中最古老的部分之一，但在过去的25年里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，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

This award supports the research of Professor Cox to work in algebraic geometry. One of the problems that Professor Cox will address will be the issue of assigning a coordinate ring to a toric variety in a natural way. This ring should be a polynomial ring and should depend only on the toric variety and not on a particular projective embedding. Professor Cox has a suggested construction which seems to have many of the needed properties. This research uses the techniques 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 origin, 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 physics, theoretical computer science, and robotics.