Mathematical Sciences: Algebraic Varieties, Birational Geometry and the Structure of the Galois Groups

这支持了Fedor Bogomolov教授对代数变异几何和两族几何相关问题的研究。他提出在高维中构造代数变异的极小优势类、线束上同调的渐近公式和Lang猜想的一个反例。他还想研究某些函数域的伽罗瓦群的Sylow子群。这是代数几何领域的研究，但它直接关系到方程解中最伟大的进步之一——伽罗瓦理论。代数几何本身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，相反，它在这些领域以及物理学、理论计算机科学和机器人技术中也得到了应用。

This supports the work of Professor Fedor Bogomolov to work on the geometry of algebraic varieties and related problems in birational geometry. He proposes to work on constructing minimal dominant classes of algebraic varieties, asymptotic formulas for cohomology of lines bundles and a counter-example to Lang's conjecture in higher dimensions. He also wants to work on the Sylow subgroups of Galois Groups of certain function fields. This is research in the field of algebraic geometry, yet it directly connects to one of the greatest advances made in the solution of equations--Galois theory. Algebraic geometry itself is 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 physics, theoretical computer science, and robotics.