Mathematical Sciences: Galois Covers and Arithmetic Varieties

该奖项支持宾夕法尼亚大学的David Harbater教授在代数几何和数论方面的研究。哈波特博士建议继续他的伽罗瓦覆盖的算术变种和算术交集理论的研究。他将在具有特殊类型伽罗瓦群和分支的有理线上构造伽罗瓦盖，在他的相交理论工作中，他计划使用他最近定义的水平对来研究算术曲面和扩大Arakelov除数群并研究无穷远处的精细结构。这是算术代数几何领域的研究，是一门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用最简单的方程，即多项式在平面上定义。数论始于整数和整数能否被整数整除之类的问题。这两个看似相距甚远的主题，实际上从最早的时代起就相互影响，但在过去的四分之一世纪中，相互影响大大增加。算术代数几何领域现在使用了所有现代数学的技术，并在其本身的边界之外产生了相应的影响。

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.