On homotopic classification of elliptic problems with contractions and $K$-groups of corresponding $C^*$-algebras

We consider calculation of a group of stable homotopic classes for pseudodifferential elliptic boundary problems. We study this problem in terms of topological $K$-groups of some spaces in the following cases: for boundary-value problems on manifolds with boundaries, for conjugation problems with conditions on a closed submanifold of codimension one, and for nonlocal problems with contractions.