A. Yu. Savin

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.

On the symbol of nonlocal operators in Sobolev spaces

We consider nonlocal operators generated by pseudodifferential operators and the operator of shift along the trajectories of an arbitrary diffeomorphism of a smooth closed manifold. We introduce the notion of symbol of such operators acting in Sobolev spaces. As examples, we consider specific diffeomorphisms, namely, isometries and dilations.

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.

On the index of nonlocal elliptic operators corresponding to a nonisometric diffeomorphism

We consider nonlocal elliptic operators corresponding to diffeomorphisms of smooth closed manifolds. The index of such operators is calculated. More precisely, it was shown that the index of the operator is equal to that of an elliptic boundary-value problem on the cylinder whose base is the original manifold. As an example, we study nonlocal operators on the two-dimensional Riemannian manifold corresponding to the tangential Euler operator.

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.

Elliptic differential dilation–contraction problems on manifolds with boundary

We give a statement of dilation–contraction boundary value problems on manifolds with boundary in the scale of Sobolev spaces. For such problems, we introduce the notion of symbol and prove the corresponding finiteness theorem.

Index formulas for stratified manifolds

We consider elliptic operators on stratified manifolds with stratification of arbitrary length. Under some (symmetry-like) conditions imposed on the symbols of these operators, we obtain index formulas in which the index of an operator is expressed as the sum of indices of some (explicitly written out) elliptic operators on the strata.