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: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=612985
• 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.

线性代数群体的理论是现代数学的良好领域。它最初是大规模成功且广泛应用的谎言理论的代数版本，最著名的是Chevalley和Borel。在Serre，Springer，山雀和许多其他人手中，它发展成为理解几何形状（同质空间和曲折变化）的重要工具，数字理论（以Galois共同体学的形式）和表示理论（组和相关代数）。在过去的几十年中，线性代数群体的理论见证了代数拓扑方法的大量侵入。这些新方法导致了代数中许多经典问题的突破，这些问题超出了早期的纯代数技术的范围。在90年代中期，Voevodsky在二次形式的背景下使用同型和辅助理论的技术，导致了Milnor概念的解决方案。这一持续趋势的另一个显着例子是Eddidin-Graham和Totaro的代数分类空间和同等的共同体学的事件。将这些结果与00's Levine-Morel和Panin-Smirnov介绍的代数为导向的共同体的概念合并，导致创建了一个新的代数等质组学理论家族（例如，等价式的共同体和Elliptic Coomologicy），这是富有的，现在是conteries in toties in veatii deviei of viewi of toviei inveiodies inveiodies， 拟议的项目可以看作是迈向这种哲学的下一步。粗略地说，它由两个方向组成：首先，是等效方向的共同体（在代数封闭的领域）的“代数化计划”；其次，涉及其应用于Torsors和Twisted Flag品种（在任意领域）的应用。它的目标是将旗帜品种的各种共同学和古典兴趣的元素（例如舒伯特品种类）与某些代数和组合对象相匹配。拟议的调查的重要组成部分是研究生和博士后学生的参与。一些子任务，例如与定向同胞中的计算相关的子任务，有望成为提出的MSC和PHD项目的主题。 作为一个结果，我们期望在代数群体的理论和同质空间的几何形状中获得新的结果，这将扩大我们在数学领域的知识，并将有助于提高加拿大的基本科学能力。