Mathematical Methods in the Theory of Ttopological Insulators

In this talk we consider mathematical problems arising in the theory of topological unsulators. These are the solid bodies having the broad energy gap stable under small deformations which motivates the application of topological methods for their study.
A key role in the theory of topological insulators is played by the study of their symmetry groups, going back to Kitaev, who proposed a classification of topological objects in the solid state physics based on the representation theory of Clifford algebras. In our talk we pay a special attention to the topological insulators invariant under time reversion. From the mathematical point of view their invextigation reduces to the study of topological spaces, provided with involution, and quaternion vector bundles over them.