Oriented cohomology and invariants of homogeneous spaces

线性代数群理论是现代数学中一个很好的领域。它开始作为一个代数版本的大规模成功和广泛应用的理论李群，推动了最显着的Chevalley和Borel。在手中的塞尔，施普林格，山雀和许多其他人，它发展成为一个重要的工具，了解几何（齐性空间和环面品种），数论（形式的伽罗瓦上同调）和代表性理论（群和相关的代数）。在过去的几十年里，线性代数群的理论见证了代数拓扑方法的大规模入侵。这些新的方法导致了一些经典的代数问题上的突破，这是超越了早期的纯代数技术。在90年代中期Voevodsky的使用技术同伦和配边理论的背景下，二次形式导致了解决米尔诺猜想。这一趋势的另一个突出的例子是Eddidin-Graham和Totaro发明的代数分类空间和等变上同调。将这些结果与Levine-Morel和Panin-Smirnov在2000年引入的代数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（例如等变协边和椭圆上同调），这些理论由于与几何的丰富联系而受到积极研究。

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.