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Index defects in the theory of spectral boundary value problems

In this paper, we survey recent results on index defects of elliptic operators on manifolds with boundary. Index defects are similar to the Hirzebruch signature defects in topology, where the defects appear as the correction terms to the signature formula on manifolds with boundary. For some natural classes of elliptic operators, the index defects are found and the corresponding topological indices are computed. The theory is illustrated on two examples: operators satisfying Gilkey's parity condition and operators on twisted Z_n-manifolds. The index defect formula in the latter case is stated in the framework of noncommutative geometry.