Algebraic cycles on homogeneous varieties

上世纪中叶，Solomon Lefschetz引入了数学中的上同调方法，并将其应用于复代数变异的研究。在接下来的十年里，它们主要是由安德烈·韦尔和亚历山大·格罗滕迪克发展起来的，并最终证明了著名的皮埃尔·德列涅的韦尔猜想。从那时起，上同调方法已经成为数学中的一项基本技术，在许多情况下导致了经典代数和几何问题的解决。最近在这方面取得的主要成就包括：定向上同调理论、列文-莫雷尔的代数共调理论以及格罗滕迪克、曼宁和沃沃茨基的动机理论。现代数学的另一个中心对象是作用于变量或流形x的（线性代数）群G的概念。基本的例子是作用于向量空间的线性变换群。近一个世纪以来，G和X的几何形状一直是一个深入研究的主题，阿曼德·博雷尔、克劳德·谢瓦莱、米歇尔·德马祖尔、让-皮埃尔·塞尔、雅克·提茨和其他人都做出了重要贡献。

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.