Compactification of Certain Moduli Spaces, and Some Finiteness Problems in Arithmetic Geometry

该奖项支持D·阿布拉莫维奇教授在代数几何领域的研究。阿布拉莫维奇将继续他对半稳定曲线上秩为2的向量丛的模空间的Gieseker-Morison紧化的研究，目的是识别与边界上的点相对应的向量丛，将构造扩展到更高的阶，并与其他紧化进行比较。他还将继续研究特征p中的Lang猜想，并将研究椭圆曲线上的积分点和扭点以及代数堆栈上的积分点。代数几何是现代数学中最古老的部分之一。在过去的十年里，它已经蓬勃发展到解决了几个世纪以来一直存在的问题。最初，它处理由最简单的方程定义的平面上的图形，即多项式。今天，这个领域不仅使用代数的方法，而且还使用分析和拓扑学的方法；相反，它在这些领域被广泛使用。此外，它在物理、理论计算机科学、密码学、编码理论和机器人学等领域都证明了自己的有用。

This award supports the research of Professor D. Abramovich in the field of algebraic geometry. Abramovich will continue his study of the Gieseker-Morrison compactification of the moduli space of vector bundles of rank 2 on semistable curves, with the goals of identifying the vector bundles corresponding to points on the boundary, extending the construction to higher ranks and comparing with other compactifications. He will also continue his work on Lang's conjecture in characteristic p, and will study integral and torsion points on elliptic curves and integral points on algebraic stacks. Algebraic geometry is one of the oldest parts of modern mathematics. In the past ten years, it has blossomed to the point where it has solved problems that have stood for centuries. Originally, it treated figures in the plane defined by the simplest of equations, namely polynomials. Today, the field utilizes methods not only from algebra, but also from analysis and topology; conversely, it is extensively used in those fields. Moreover, it has proved itself useful in fields as diverse as physics, theoretical computer science, cryptography, coding theory and robotics.