Compact Torus Action on the Complex Grassmann Manifolds

We survey recent achievements in the theory of the canonical action of the compact torus on the complex Grassmann manifolds. The fundamental problem of this theory is to describe the equivariant topology of Grassmann manifolds and combinatorial structure of their orbit spaces. We introduce new notions that we use to solve this problem and give a series of seminal examples of corresponding constructions. We discuss results from various areas of mathematics that are used to solve the problem under consideration. This survey also contains a detailed description of results that formed the basis of our theory of \((2n,k)\) -manifolds. We formulate several open problems, which emerged as natural concretizations of problems from the classical theory of compact torus actions on smooth manifolds.