Boris Sternin

Elliptic Theory for Operators Associated with Diffeomorphisms of Smooth Manifolds

In this paper we give a survey of elliptic theory for operators associated with diffeomorphisms of smooth manifolds. Such operators appear naturally in analysis, geometry and mathematical physics. We survey classical results as well as results obtained recently. The paper consists of an introduction and three sections. In the introduction we give a general overview of the area of research. For the reader’s convenience here we tried to keep special terminology to a minimum. In the remaining sections we give detailed formulations of the most important results mentioned in the introduction.

Index of elliptic operators for diffeomorphisms of manifolds

We develop elliptic theory of operators associated with a diffeomorphism of a closed smooth manifold. The aim of the present paper is to obtain an index formula for such operators in terms of topological invariants of the manifold and of the symbol of the operator. The symbol in this situation is an element of a certain crossed product. We express the index as the pairing of the class in K-theory defined by the symbol and the Todd class in periodic cyclic cohomology of the crossed product.

The localization problem in index theory of elliptic operators

Preface.- Introduction.- 0.1 Basics of Elliptic Theory.- 0.2 Surgery and the Superposition Principle.- 0.3 Examples and Applications.- 0.4 Bibliographical Remarks.- Part I: Superposition Principle.- 1 Superposition Principle for the Relative Index.- 1.1 Collar Spaces.- 1.2 Proper Operators and Fredholm Operators.- 1.3 Superposition Principle.- 2 Superposition Principle for K-Homology.- 2.1 Preliminaries.- 2.2 Fredholm Modules and K-Homology.- 2.3 Superposition Principle.- 2.4 Fredholm Modules and Elliptic Operators.- 3 Superposition Principle for KK-Theory.- 3.1 Preliminaries.- 3.2 Hilbert Modules, Kasparov Modules, and KK.- 3.3 Superposition Principle.- Part II: Examples.- 4 Elliptic Operators on Noncompact Manifolds.- 4.1 Gromov-Lawson Theorem.- 4.2 Bunke Theorem.- 4.3 Roes Relative Index Construction.- 5 Applications to Boundary Value Problems.- 5.1 Preliminaries.- 5.2 Agranovich-Dynin Theorem.- 5.3 Agranovich Theorem.- 5.4 Bojarski Theorem and Its Generalizations.- 5.5 Boundary Value Problems with Symmetric Conormal Symbol.- 6 Spectral Flow for Families of Dirac Type Operators.- 6.1 Statement of the Problem.- 6.2 Simple Example.- 6.3 Formula for the Spectral Flow.- 6.4 Computation of the Spectral Flow for a Graphene Sheet.- Bibliography.

Mother body problem

This chapter is devoted to the following problem. Given an external gravitational field, produced by some body, there arises the question of determining the form and mass density of this body. Similar problems appear in geoprospecting, where one measures the gravitational field on the ground surface and wants to determine the source of this gravitational field located inside the earth. Clearly, the solution of this problem is not unique. For example, it has been well known since the time of Newton that a ball with constant mass density produces the same gravitational field as the same mass concentrated at the center of the ball.

Ramified Fourier transform

The Fourier transform plays a fundamental role in the modern theory of differential equations (in real domains!) due primarily to the fact that it converts equations with constant coefficients into algebraic equations.

Singularities of the solution of the Cauchy problem

In this chapter, we use the formula for the solution of the Cauchy problem in terms of the ramified integrals obtained in Chapter 6 and describe the singularities of the solution.

The Cauchy problem for equations with variable coefficients. Leray’s uniformization

In this chapter we deal with the Cauchy problem for equations with variable coefficients. Of course, unlike the case of constant coefficients, it is impossible to obtain the solution exactly and the question of studying qualitative properties of the solution (finding its singularities and asymptotics at singular points) becomes even more important.

Balayage inwards problem

In this chapter we apply the methods developed in previous chapters to one problem in physics, namely, the problem of sweeping the charge. More precisely, the problem of sweeping the charge is reduced to a problem for a differential equation in a real domain and we will show how one can (and should!) study the latter problem using methods of complex theory.

Asymptotics of ramified integrals

In the previous chapter, we saw that the singularities of ramified integrals lie on Landau manifolds. So the question then arises about a more precise description of these singularities. The aim of this chapter is to answer this question. In more detail, we study ramified integrals near generic points of Landau manifolds. Namely, we show that the ramification of the homology class, over which we integrate, is described by Picard–Lefschetz formulas and that the asymptotics of the integral is given by Leray’s formulas. Slightly simplifying the situation, the main result can be formulated as follows: generically, near regular points of the Landau manifold, ramified integrals have singularities of one of the three types: square root singularity, logarithmic singularity, or pole.