Mathematical Sciences: Studies in Commutative Algebra and Algebraic Geometry

该奖项支持密西根大学安娜堡分校的Melvin Hochster、Igor Dolgachev和Kevin Coombes教授在交换代数和代数几何方面的研究。Hochster博士提出了卓有成效的环紧闭的概念，并将继续他对这个概念的研究，以及局部代数中的几个问题。Dolgachev博士将研究任意特征域上具有有限多个自同构的Enriques曲面的分类，并将研究曲面理论中的其他一些问题。库姆斯博士将研究更多的数论问题，即算术曲面的几何，特别是算术曲面的分支覆盖理论和在算术兴趣领域上定义的曲线。这是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，但在过去的四分之一个世纪里，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，并且反过来在这些领域以及理论计算机科学和机器人技术中得到应用。

This award supports the research in commutative algebra and algebraic geometry of Professors Melvin Hochster, Igor Dolgachev, and Kevin Coombes of the University of Michigan at Ann Arbor. Dr. Hochster has introduced the fruitful notion of tight closure of rings, and will continue his study of this concept, as well as several problems in local algebra. Dr. Dolgachev will be working on the classification of Enriques surfaces with finitely many automorphisms over fields of arbitrary characteristic, and will investigate a number of other questions in the theory of surfaces. Dr. Coombes will be studying more number-theoretic questions, namely the geometry of arithmetic surfaces, especially the theory of branched covers of arithmetic surfaces and of curves defined over fields of arithmetic interest. 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 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 theoretical Computer Science and in Robotics.