Theory of boundary value problems for operators, which violate Atiyah–Bott condition (homotopy classification, index). Gilkeys problem on the fractional part of the eta-invariant was solved. Index formula for elliptic operators in pseudodifferential subspaces. Homotopy classification of elliptic operators on stratified manifolds in terms of K-homology groups. Index formulas on stratified manifolds. Index formulas for elliptic operators associated with group actions (operators with shifts).

Main publications:

A. Yu. Savin, “On the index of elliptic operators associated with a diffeomorphism of a manifold”, Dokl. Math., 82:3 (2010), 884-886

V.E. Nazaikinskii, A.Yu. Savin, B.Yu. Sternin, Elliptic theory and noncommutative geometry. Nonlocal elliptic operators, Birkhäuser, Basel, 2008 , 224 pp.

V.E. Nazaikinskii, A.Yu. Savin, B.-W. Schulze, B.Yu. Sternin, Elliptic theory on singular manifolds, Chapman & Hall/CRC, Boca Raton, FL, 2006 , 356 pp.

A. Savin, “Elliptic operators on manifolds with singularities and $K$-homology”, K-Theory, 34:1 (2005), 71-98

A. Savin, B. Sternin, “The eta invariant and parity conditions”, Adv. Math., 182:2 (2004), 173–203