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00年代引入的代数定向上同调概念相结合，创建了一系列新的代数等变上同调理论（例如等变上边论和椭圆上同调），鉴于其与几何的丰富联系，这些理论如今正受到积极研究。 拟议的项目可以被视为朝着这一理念迈出的下一步。粗略地说，它包括两个方向：第一，是一个'代数化程序'等变定向上同调（在代数闭域）;第二，涉及其应用理论的torsors和扭曲的标志品种（在任意领域）。它的目标是将旗簇的各种上同调环和其中的经典元素（如舒伯特簇的类）与一些代数和组合对象相匹配。拟议的调查的一个重要组成部分是研究生和博士后学生的参与。一些子任务，例如那些与计算定向上同调，预计将成为拟议的硕士和博士项目的主题。 作为一个结果，我们希望获得新的成果，在理论的代数群和几何的齐次空间，这将扩大我们的知识在这些领域的数学，并将有助于推进加拿大的基础科学能力。

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. The proposed project can be viewed as the next step toward this philosophy. Roughly speaking, it consists of two directions: first, is an 'algebraization program' for equivariant oriented cohomology (over an algebraically closed field); second, deals with its applications to the theory of torsors and twisted flag varieties (over an arbitrary field). Its goal is to match various cohomology rings of flag varieties and elements of classical interest in them (such as classes of Schubert varieties) with some algebraic and combinatorial objects. An essential part of the proposed investigations is the involvement of graduate and postdoctoral students. Some of the subtasks, for instance those related with computations in oriented cohomology, are expected to become the subjects of the proposed MSc and PhD-projects. As an outcome we expect to obtain new results in the theory of algebraic groups and geometry of homogeneous spaces that will extend our knowledge in those areas of mathematics and will help advance Canada's fundamental science capabilities.