Oriented cohomology and invariants of homogeneous spaces

线性代数群理论是现代数学中一个建立良好的领域。它最初是李群理论的一个代数版本，这个理论获得了巨大的成功，并得到了广泛的应用，主要是由Chevalley和Borel推动的。在Serre，施普林格，Tits和其他许多人的手中，它发展成为理解几何（齐次空间和环变），数论（以伽罗瓦上同调的形式）和表示理论（群和相关代数）的重要工具。在过去的几十年里，线性代数群理论见证了代数拓扑方法的大量入侵。这些新方法在代数中的一些经典问题上取得了突破，这是早期的纯代数技术所无法达到的。在90年代中期，Voevodsky在二次型的背景下使用同伦和共格理论的技术，导致了米尔诺猜想的解决。这一趋势的另一个显著例子是Eddidin-Graham和Totaro的代数分类空间和等变上同调的发明。将这些结果与Levine-Morel和Panin-Smirnov在20世纪90年代提出的面向代数的上同调的概念相结合，产生了一系列新的代数等变上同调理论（如等变上同调和椭圆上同调），由于其与几何的丰富联系，这些理论在今天得到了积极的研究。

The theory of linear algebraic groups is a well established area of modern mathematics. It started as an algebraic version of the massively successful and widely applied theory of Lie groups, pushed forward most notably by Chevalley and Borel. In the hands of Serre, Springer, Tits and many others, it developed into an important tool for understanding geometry (of homogeneous spaces and toric varieties), number theory (in the form of Galois cohomology) and representation theory (of groups and the associated algebras). In the last decades the theory of linear algebraic groups has witnessed a massive intrusion of the methods of algebraic topology. These new methods have led to breakthroughs on a number of classical problems in algebra, which are beyond the reach of earlier purely algebraic techniques. In the mid of 90's Voevodsky's use of techniques from homotopy and cobordism theory in the context of quadratic forms had resulted in the solution of the Milnor conjecture. Another striking example of this ongoing trend is the invention of an algebraic classifying space and equivariant cohomology by Eddidin-Graham and Totaro. Merging these results with a notion of an algebraic oriented cohomology introduced by Levine-Morel and Panin-Smirnov in 00's have lead to the creation of a family of new algebraic equivariant cohomology theories (e.g. equivariant cobordism and elliptic cohomology) which are actively studied nowadays in view of its rich connections to geometry.