Abstract <p>The paper is devoted to the theory of topological insulators, a new and actively developing direction in condensed matter physics. These are the solid bodies with a wide energy gap stable under small deformations which motivates using the topological methods for their description. One way to understand how the topology comes into play is to employ the spectral flattening method relating the topology of insulators with that of Grassmannians. Another way to construct topological invariants of insulators is to use Berry connection to define the Chern number of insulator.</p>