Table of contents

Classical and new results on integrable geodesic flows on two-dimensional surfaces are reviewed. The central question is the classification of such flows up to various equivalences, of which the following four kinds are the most interesting ones: 1) isometry; 2) geodesic equivalence; 3) orbital equivalence; 4) Liouville equivalence.

The question of the classification of the phase portraits of optimal synthesis in a neighbourhood of a singular universal manifold is discussed for systems of constant rank that are affine in control. Both phase state and control are assumed to be many-dimensional. The classification is based on the order of the singular extremals and the property of involutiveness (or otherwise) of the velocity indicator. The synthesis of optimal trajectories is shown to be a space fibred over the base W consisting of singular optimal trajectories; its fibres are non-singular optimal trajectories. If the control is many-dimensional, then W is a stratified manifold. In the involutive case the fibres are one-dimensional. In the non-involutive case the fibres are many-dimensional and contain chattering trajectories; the dimension of the fibres and the structure of the field of trajectories in the fibres depend on the order of the singular extremals.

A -algebra servicing the theory of asymptotic representations and its embedding into the Calkin algebra that induces an isomorphism of -groups is constructed. As a consequence, it is shown that all vector bundles over the classifying space that can be obtained by means of asymptotic representations of a discrete group can also be obtained by means of representations of the group into the Calkin algebra. A generalization of the concept of Fredholm representation is also suggested, and it is shown that an asymptotic representation can be regarded as an asymptotic Fredholm representation.

The principles of the construction of adjoint operators in non-linear problems are reviewed. The aim is to draw attention to new approaches in the method of adjoint operators which often enable one to obtain some information about physical processes and unknown parameters of complex systems that may be necessary for the development of the technology of experimental design required in the solution of applied problems.

A generalized graph manifold is a three-dimensional manifold obtained by gluing together elementary blocks, each of which is either a Seifert manifold or contains no essential tori or annuli. By a well-known result on torus decomposition each compact three-dimensional manifold with boundary that is either empty or consists of tori has a canonical representation as a generalized graph manifold. A short simple proof of the existence of a canonical representation is presented and a (partial) algorithm for its construction is described. A simple hyperbolicity test for blocks that are not Seifert manifolds is also presented.

The well-known formula for finding the area of a triangle in terms of its sides is generalized to volumes of polyhedra in the following way. It is proved that for a polyhedron (with triangular faces) with a given combinatorial structure and with a given collection of edge lengths there is a polynomial such that the volume of the polyhedron is a root of it, and the coefficients of the polynomial depend only on  and and not on the concrete configuration of the polyhedron itself. A number of problems in the metric theory of polyhedra are solved as a consequence.

The aim of the paper is to define and study algebraic operations closely related to the group structure on the homotopy groups of topological spaces. These are certain many-place operations on the homotopy groups. The family of these operations induces an algebraic structure on the homotopy groups, which is called an -group structure by analogy with the -structures introduced by Stasheff.

A theory of general boundary-value problems is developed for differential operators with symbols not necessarily satisfying the Atiyah-Bott condition that the corresponding obstruction must vanish. A condition ensuring that these problems possess the Fredholm property is introduced and the corresponding theorems are proved.