Mathematical Sciences: Galois Covers and Arithmetic Varieties

该奖项支持宾夕法尼亚大学大卫·哈巴特教授在代数几何和数论方面的研究。Harbater博士建议继续他的研究伽罗瓦盖算术品种和算术交叉理论。他将致力于构建伽罗瓦覆盖的合理线具有特殊类型的伽罗瓦群和分歧，并在他的交叉理论的工作，他计划使用一个水平配对，他最近定义，以研究算术表面和扩大阿拉克洛夫因子群和研究精细结构在无穷大。这是算术代数几何领域的研究，一个结合了代数几何和数论技术的学科。在其最初的公式中，代数几何处理的图形可以在平面上由最简单的方程定义，即多项式。数论开始于整数和一个整数被另一个整除的问题。这两门学科看似相距甚远，实际上从最早的时候就相互影响，但在过去的四分之一个世纪里，相互影响大大增加。算术代数几何领域现在使用的技术，从所有的现代数学，并有相应的影响超出自己的边界。

This award supports the research in algebraic geometry and number theory of Professor David Harbater of the University of Pennsylvania. Dr. Harbater has proposed to continue his study of Galois covers of arithmetic varieties and of arithmetic intersection theory. He will work to construct Galois covers over the rational line having special type of Galois group and ramification, and in his intersection-theoretic work, he plans to use a horizontal pairing that he has recently defined in order to study arithmetic surfaces and to enlarge the Arakelov divisor group and study the fine structure at infinity. This is research in the field of arithmetic algebraic geometry, a subject that combines the techniques of algebraic geometry and number theory. In its original formulation, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Number theory started with the whole numbers and such questions as divisibility of one whole number by another. These two subjects, seemingly so far apart, have in fact influenced each other from the earliest times, but in the past quarter century the mutual influence has increased greatly. The field of arithmetic algebraic geometry now uses techniques from all of modern mathematics, and is having corresponding influence beyond its own borders.