Algebraic cycles on homogeneous varieties

上个世纪中叶，所罗门·莱夫谢茨引入了数学上同调方法，并应用于研究复代数簇。在接下来的十年里，它们基本上是由安德烈·韦尔和亚历山大·格罗森迪克发展起来的，导致了皮埃尔·迪林对著名的韦尔猜想的证明。从那时起，上同调方法已经成为数学中的一种基本技术，在许多情况下，它导致了经典代数和几何问题的解决。在这一方向最近的主要成就中，值得一提的是：定向上同调理论，Levine-Morel的代数上同调理论和Grothendieck，Manin和Voevodsky的动机理论。现代数学的另一个中心目标是作用在簇或流形X上的(线性代数)群G的概念。基本的例子是作用在向量空间上的线性变换群。近一个世纪以来，G和X的几何一直是深入研究的主题，Armand Borel、Claude Chvalley、Michel Demazure、Jean-Pierre Serre、Jaques Tits等人做出了重要贡献。

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.