Algebraic cycles on homogeneous varieties

上个世纪中期，所罗门·莱夫谢茨引入并应用数学上的上同调方法来研究复代数簇。在接下来的十年里，他们基本上是由安德烈·韦尔和亚历山大格罗滕迪克导致证明著名的韦尔猜想皮埃尔德利涅。从那时起，上同调方法已成为一个基本的技术在数学中，在许多情况下导致决议的经典代数和几何问题。在最近的主要成就在这个方向应该提到：面向上同调理论，理论的代数cobordism由莱文，莫雷尔和理论的动机Grothendieck，马宁和Voevodsky。现代数学的另一个中心目标是作用于簇或流形X的（线性代数）群G的概念。最基本的例子是作用在向量空间上的线性变换群。几何的G和X一直是一个主题的深入调查了近世纪，与重要贡献的阿曼德博雷尔，克劳德Chevalley，米歇尔Demazure，让皮埃尔塞尔，雅克山雀，和其他人。

Cohomological methods in mathematics were introduced and applied in the middle of the last century by Solomon Lefschetz to study complex algebraic varieties. During the next decade they were essentially developed by Andre Weil and Alexander Grothendieck leading to the proof of the famous Weil's Conjectures by Pierre Deligne. Since then cohomological methods have become a fundamental technique in mathematics which in many cases led to the resolution of classical algebraic and geometric problems. Among the major recent achievements in this direction one should mention: the oriented cohomology theory, the theory of algebraic cobordism by Levine-Morel and the theory of motives by Grothendieck, Manin and Voevodsky. Another central object of the modern mathematics is the notion of a (linear algebraic) group G acting on a variety or manifold X. The basic example is the group of linear transformations acting on a vector space. The geometry of G and X has been a subject of intensive investigations for almost a century, with important contributions by Armand Borel, Claude Chevalley, Michel Demazure, Jean-Pierre Serre, Jaques Tits, and others.