Operations on equivariant oriented cohomology of homogeneous spaces

齐次空间的等变导向上同调的运算

• 批准号: RGPIN-2022-03060
• 负责人: Zaynullin, Kirill
• 金额: $ 2.26万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=759221
• 关键词: Operations; equivariant; oriented; cohomology; homogeneous

The theory of linear algebraic groups is a well-established area of modern mathematics. It started as an algebraic version of the largerly  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 flag varieties and various homogeneous spaces) and representation theory (of groups and the associated algebras). In the last decades, it has witnessed a massive intrusion of the methods of algebraic topology. These new methods have led to breakthroughs on several classical problems in algebra, which are beyond the reach of earlier purely algebraic techniques. The proposed research program can be viewed as the next step toward this philosophy. Roughly speaking, it consists of two directions: the first focuses on the study of morphisms between algebraic equivariant theories (the so-called cohomological operations); the second deals with the Riemann-Roch type formalism and its applications to the geometry of homogeneous spaces (e.g., algebraic cycles, equivariant Schubert calculus) and representation theory (e.g., sheaves on moment graphs, Hecke-type algebras). As for the first, in the mid '80s Kostant-Kumar introduced the techniques of Hecke algebras to `algebraize' equivariant singular cohomology and K-theory of flag varieties. By the works of Bressler-Evans in the mid-'90s and the recent works by the author and collaborators, this approach was successfully extended to an arbitrary equivariant oriented theory. So the next natural step would be to `algebraize' endomorphisms, or more generally, morphisms (cohomological operations) between such equivariant oriented theories. As for the second direction, the general Riemann-Roch formalism of SGA6 says that any operation leads to a Riemann-Roch type formula involving the push-forwards and the Todd genus. We plan to study various versions of the Riemann-Roch type theorems arising from different operations (e.g., Steenrod, Landweber-Novikov and Adams operations). We plan to construct `interesting' cycles using classes of Schubert varieties or other canonical bases for cohomology theories. This research program will also provide necessary training to a diverse group of students and postdoctoral researchers who will gain research experience in fundamental mathematics in an inclusive environment.

线性代数群体的理论是现代数学的公认领域。它最初是较大成功且广泛应用的谎言群体的代数版本，最著名的是Chevalley和Borel。在Serre，Springer，山雀和其他许多人的手中，它发展成为理解几何形状（标志品种和各种同质空间）和表示理论（组和相关代数）的重要工具。在过去的几十年中，它目睹了代数拓扑方法的大量入侵。这些新方法导致了对代数的几个经典问题的突破，这些方法超出了早期的纯代数技术的影响。拟议的研究计划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：第一个侧重于对代数模棱两可理论（所谓的共同体作业）之间的形态学研究；第二个涉及Riemann-Roch型格式及其在同质空间几何形状（例如，代数循环，同等的Schubert calculus）和表示理论（例如，在矩图上，Hecke-type代数）。至于第一个，在80年代中期，Kostant-Kumar引入了Hecke代数的技术，以“代数”等效的奇异共同体和K理论的旗帜变化。通过90年代中期布雷斯勒·埃文斯（Bressler-Evans）的作品以及作者和合作者的最新作品，这种方法已成功地扩展到了任意等效的方向理论。因此，下一个自然的步骤将是“代数”内态性，或更普遍地，这种等效的理论之间的态度（同学操作）。至于第二个方向，SGA6的一般Riemann-Roch格式化表明，任何操作都会导致涉及推动力和TODD属的Riemann-Roch类型公式。我们计划研究由不同操作（例如Steenrod，Landweber-Novikov和Adams操作）引起的各种版本的Riemann-Roch型定理。我们计划使用Schubert品种或其他规范基础来构建“有趣的”周期，以进行共同体学理论。该研究计划还将为众多的学生和博士后研究人员提供必要的培训，这些研究人员将在包容性环境中获得基本数学的研究经验。