Algebraic Varieties, Birational Geometry and the Structure of the Galois Groups

首席研究员打算继续他在代数变异的几何、伽罗瓦群的结构和两族几何的相关问题上的工作。主要研究方向：代数变体的最小优势类的构造，线束上同调的渐近公式，代数和辛流形的基本群和泛覆盖的描述，复辛流形和超卡勒流形的结构。他还建议研究伽罗瓦群的Sylow子群的结构和群上同调的稳定过程的有效版本。这是代数几何领域的研究。代数几何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。

Bogomolov 9801591 The principal investigator intends to continue his work on the geometry of algebraic varieties, the structure of Galois groups, and related problems in birational geometry. He plans to work on constructing minimal dominant classes of algebraic varieties, asymptotic formulas for cohomology of line bundles, the description of fundamental groups and universal coverings of algebraic and symplectic manifolds, and the structure of complex symplectic and hyperkahler manifolds. He also proposes to work on the structure of Sylow subgroups of the Galois groups and on the effective version of the stabilization procedure for group cohomology. This is research in the field of algebraic geometry. Algebraic geometry 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.