Operations on equivariant oriented cohomology of homogeneous spaces

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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.