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

该奖项支持D. Abramovich教授在代数几何领域的研究。Abramovich将继续研究半稳定曲线上2阶向量束模空间的Gieseker-Morrison紧化，目标是识别边界上点对应的向量束，将构造扩展到更高阶，并与其他紧化进行比较。他也将继续他在特征p上的Lang猜想的工作，并将研究椭圆曲线上的积分点和扭转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，它已经发展到解决了几个世纪以来一直存在的问题的地步。最初，它用最简单的方程即多项式来处理平面上的图形。今天，该领域不仅利用代数方法，还利用分析和拓扑学方法；相反，它在这些领域被广泛使用。此外，它已被证明在物理学、理论计算机科学、密码学、编码理论和机器人等多种领域都很有用。

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.