Mathematical Sciences: RUI: Geometry of Toric Varieties

这个奖支持考克斯教授在代数几何方面的研究。考克斯教授要解决的一个问题是如何以自然的方式为一个环型环分配一个坐标环。这个环应该是一个多项式环，应该只依赖于环的变化，而不依赖于一个特定的投影嵌入。考克斯教授提出的一种结构似乎具有许多需要的特性。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，但在过去的四分之一世纪里，它已经有了革命性的发展。在它的起源中，它处理的图形可以用最简单的方程，即多项式，在平面上定义。如今，该领域不仅使用代数的方法，还使用分析和拓扑的方法，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

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.