Oriented cohomology and invariants of homogeneous spaces

齐次空间的有向上同调和不变量

• 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: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=686751
• 关键词: Oriented; cohomology; invariants; homogeneous; spaces

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. **

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。******所提出的项目可以被视为迈向这一哲学的下一步，粗略地说，它由两个方向组成：首先，是一个'。面向等变的上同调的代数程序（在代数闭域上）；第二，处理其在扭转和扭曲旗簇理论中的应用（在任意域上），其目标是匹配旗簇和元素的各种上同调环。具有古典兴趣在其中（例如舒伯特簇）以及一些代数和组合对象中，预期研究的一个重要部分是研究生和博士后学生的参与，例如与定向上同调计算相关的子任务。成为拟议的硕士和博士项目的主题。***因此，我们期望在代数群和齐次空间几何理论方面获得新的成果，这将扩展我们在这些领域的知识数学并将有助于提高加拿大的基础科学能力**。