Oriented cohomology and invariants of homogeneous spaces

齐次空间的有向上同调和不变量

基本信息

批准号：
RGPIN-2015-04469
负责人：
Zaynullin, Kirill
金额：
$ 2.62万
依托单位：
University of Ottawa
依托单位国家：
加拿大
项目类别：
Discovery Grants Program - Individual
财政年份：
2015
资助国家：
加拿大
起止时间：
2015-01-01 至 2016-12-31
项目状态：
已结题

来源：
https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=589823
关键词：
Oriented cohomology invariants homogeneous spaces

项目摘要

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.

线性代数群理论是现代数学的一个成熟领域，它最初是由谢瓦利和博雷尔推动的广泛成功和广泛应用的李群理论的代数版本。 Springer、Tits 和许多其他人，它发展成为理解几何（齐次空间和环面簇）、数论（以伽罗瓦上同调的形式）和表示论（群和相关的）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规模入侵，这些新方法在代数中的许多经典问题上取得了突破，这是早期纯代数无法企及的。 90 年代中期，Voevodsky 在二次形式的背景下使用同伦和配边理论的技术得出了 Milnor 的解决方案。这一持续趋势的另一个引人注目的例子是 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 00 年代提出的面向代数的上同调概念相结合。导致创建了一系列新的代数等变上同调理论（例如等变共边和椭圆上同调）由于其与几何的丰富联系而受到当今积极的研究。