Topological insulators invariant under time reversal: an overview

The talk is devoted to the theory of topological insulators which is an actively developing direction in the solid state physics. The search for new topological objects is reduced to the search of appropriate topological invariants and systems having non-trivial invariants. Such systems are characterized by wide energetic gaps stable with respect to small deformations. The quantum spin Hall insulator may be considered as a non-trivial example of such systems. It is a two-dimensional insulator invariant under time reversal having a non-zero topological $\mathbb Z_2$-invariant introduced by Kane and Mele.
Our talk is devoted to the topological insulators invariant under time reversal transform. In the first part we consider the physical basics of the theory of topological insulators while in the second part we deal with its mathematical aspects.