B. Yu. Sternin

On a minimal element for a family of bodies producing the same external gravitational field

A minimal element, mother body, for a family of bodies producing the same external gravitational field is the body in the family, whose support has Lebesgue measure zero and satisfies some additional requirements. The finite algorithm of constructing mother bodies in is suggested. The local structure of mother bodies near singular points of continued logarithmic potential is investigated in generic positions.

Homotopy classification of elliptic operators on stratified manifolds

We give a homotopy classification of elliptic operators on a stratified manifold. Namely, we establish an isomorphism between the set of elliptic operators modulo stable homotopy and the K-homology group of the manifold. By way of application, we obtain an explicit formula for the obstruction of Atiyah-Bott type to the existence of Fredholm problems in the case of stratified manifolds.

Uniformization of nonlocal elliptic operators and KK-theory

By a pseudodifferential uniformization of a nonlocal elliptic operator we mean the procedure of reducing the operator to a pseudodifferential operator with a controlled modification of the index. In the paper, we suggest an approach to solving the uniformization problem; this approach uses the reduction of the symbol of a nonlocal operator to the symbol of a pseudodifferential operator. The technical apparatus here is Kasparov’s KK-theory.

Elliptic translators on manifolds with point singularities

We consider translators on manifolds with singularities of the type of a transversal intersection of smooth manifolds. We give the definition of ellipticity of translators, prove the finiteness (Fredholm property) theorem, and establish an index formula for the case of point singularities.

Elliptic translators on manifolds with multidimensional singularities

We consider translators on manifolds with many-dimensional singularities. We state the definition of ellipticity for translators, prove a finiteness (Fredholm property) theorem, and establish an index formula.

Elliptic dilation–contraction problems on manifolds with boundary. C *-theory

We study boundary value problems with dilations and contractions on manifolds with boundary. We construct a C*- algebra of such problems generated by zero-order operators. We compute the trajectory symbols of elements of this algebra, obtain an analog of the Shapiro–Lopatinskii condition for such problems, and prove the corresponding finiteness theorem.

Index of Sobolev problems on manifolds with many-dimensional singularities

We consider Sobolev spaces on manifolds with many-dimensional singularities. We prove the Fredholm property of such problems and derive the corresponding index formula. The results are based on the theory of translators on manifolds with singularities.

Noncommutative elliptic theory. Examples

We study differential operators with coefficients in noncommutative algebras. As an algebra of coefficients, we consider crossed products corresponding to the action of a discrete group on a smooth manifold. We give index formulas for the Euler, signature, and Dirac operators twisted by projections over the crossed product. The index of Connes operators on the noncommutative torus is computed.

ATIYAH-BOTT INDEX ON STRATIFIED MANIFOLDS

We define the Atiyah–Bott index on stratified manifolds and propose a formula for it in topological terms. Moreover, we give examples of the calculation of the Atiyah–Bott index for geometric operators on manifolds with edges.

The canonic operator (complex case)

We present the canonic operator method for the complex case. We prove the cocyclicity of a canonic cochain and establish a fundamental theorem on commutation.