V. E. Shatalov

On the Index of Differential Operators on Manifolds with Conical Singularities

The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah–Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.

The Atiyah–Bott–Lefschetz Theorem for Manifolds with Conical Singularities

We establish an Atiyah–Bott–Lefschetz formula for elliptic operators on manifolds with conical singular points.

On the confluence phenomenon for Fuchsian equations

In the paper we present a method of investigation of asymptotic behavior of solutions to parameter-dependent degenerate differential equations both with regular and irregular points of singularity. We examine the confluence phenomenon for such points, which is the process of the coincidence of different points at a certain value of the parameter. This examination is based on a resurgent representation of the corresponding solutions which also depends on the parameter. In particular, the confluence of the representations in question is considered.

On some global aspects of the theory of partial differential equations on manifolds with singularities

In this paper we investigate the connection between asymptotic expansions of solutions to elliptic equations near different points of singularities of the underlying manifold. We propose the procedure of computation of the asymptotic expansion of the solution at any point of singularity via the asymptotics given at one (fixed) of these points.

An operator algebra on manifolds with cusp type singularities

Equations on manifolds with cusp-type singularities are investigated. The corresponding calculus of pseudodifferential operators is constructed and finiteness theorems (Fredholm property) are established. The resurgent character of solutions is proved for equations with infinitely flat right-hand side.

Asymptotic solutions to Fuchsian equations in several variables

The aim of this paper is to construct asymptotic solutions to multidimensional Fuchsian equations near points of their degeneracy. Such construction is based on the theory of resurgent functions of several complex variables worked out by the authors in [1]. This theory allows us to construct explicit resurgent solutions to Fuchsian equations and also to investigate evolution equations (Cauchy problems) with operators of Fuchsian type in their right-hand parts.

Some Questions of Analysis and Geometry of Complex Manifolds

This section contains definitions of main notions and statements of theory used in the remaining part of the book. Here we only provide the information that is absolutely necessary for understanding the subsequent chapters. A more comprehensive treatment of the problems mentioned and the notions introduced, as well as proofs of the assertions stated, can be found in Sections 1.2–1.5. As for the present section, its aim is in particular to establish the terminology that will be used throughout the book. It is also intended to provide quick and easy reference on the subjects it covers.

Integral Transformations of Ramified Analytic Functions

This chapter is devoted to the construction of a new integral transformation of multivalued complex-analytic functions (in fact, of a series of such transformations). It is known, however, that the proof of invertibility of an integral transformation always uses an integral representation (namely, the composition of the transformation with its inverse)1.

Laplace-Radon Integral Operators

Laplace-Radon integral operators studied in this chapter were designed to supply asymptotic solutions (w.r.t. differentability) to differential equations on complex manifolds. Prior to considering these operators in general let us describe the representation of singular solutions to differential equations we intend to use in our constructions.

Cauchy Problem in Spaces of Ramified Functions

In this section we consider the Cauchy problem