Operations on equivariant oriented cohomology of homogeneous spaces

线性代数群理论是现代数学中一个久负盛名的领域。它最初是李群理论的代数版本，李群理论非常成功，也得到了广泛的应用，最著名的是由切瓦利和博雷尔推动的。在Serre、Springer、Tits和其他许多人的手中，它发展成为理解(旗簇和各种齐次空间的)几何和(群及其相关代数的)表示理论的重要工具。在过去的几十年里，它见证了代数拓扑学方法的大规模入侵。这些新方法在代数中的几个经典问题上取得了突破，这些问题是早期纯代数技术所无法企及的。拟议的研究计划可以被视为迈向这一理念的下一步。粗略地说，它包括两个方向：第一个方向是研究代数等变理论(所谓的上同调运算)之间的态射；第二个方向是Riemann-Roch型形式及其在齐次空间(例如代数圈、等变Schubert演算)和表示理论(例如矩图上的层、Hecke型代数)的几何中的应用。关于第一个问题，80年代中期，Kostant-Kumar引入了Hecke代数的技巧来“代数化”等变奇异上同调和旗簇的K-理论。通过Bressler-Evans在90年代中期的工作以及作者和合作者最近的工作，这种方法被成功地扩展到任意等变定向理论，因此下一步自然是将这种等变定向理论之间的自同态或更广泛地说，态射(上同调运算)“代数化”。对于第二个方向，SGA6的一般Riemann-Roch形式说，任何操作都会导致一个包含前推和Todd亏格的Riemann-Roch型公式。我们计划研究来自不同运算(如Steenrod、Landweber-Novikov和Adams运算)的Riemann-Roch型定理的不同版本。我们计划利用Schubert簇的类或上同调理论的其他典范基来构造“有趣的”圈。这项研究计划还将为不同的学生和博士后研究人员提供必要的培训，他们将在包容的环境中获得基础数学研究经验。

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.