Table of contents

The Nöbeling space , a -dimensional analogue of the Hilbert space, is considered; this is a topologically complete separable (that is, Polish) -dimensional absolute extensor in dimension (that is, ) and a strongly -universal space. The conjecture that the above-listed properties characterize the Nöbeling space in an arbitrary finite dimension  is proved. In the first part of the paper a full axiom system of the Nöbeling spaces is presented and the problem of the improvement of a partition connectivity is solved on its basis. Bibliography: 29 titles.

For a smooth strictly convex set in a reflexive Banach space a criterion is given for this set to be generating. This criterion enabled us to discover new classes of generating sets. Bibliography: 6 titles.

As a generalization of the well-known Racah coefficients (which are usually defined for finite-dimensional representations of semisimple Lie groups), the concept of Racah operators is introduced for locally compact groups with a `nice' dual space. Explicit expressions for these operators are presented for . Bibliography: 8 titles.

The existence of fractional monodromy is proved for the compact Lagrangian fibration on a symplectic 4-manifold that corresponds to two oscillators with arbitrary non-trivial resonant frequencies. Here one means by the monodromy corresponding to a loop in the total space of the fibration the transformation of the fundamental group of a regular fibre, which is diffeomorphic to the 2-torus. In the example under consideration the fibration is defined by a pair of functions in involution, one of which is the Hamiltonian of the system of two oscillators with frequency ratio , where , are arbitrary coprime positive integers distinct from the trivial pair . This is a generalization of the result on the existence of fractional monodromy in the case , published before. Bibliography: 39 titles.

Asymptotically best possible lower bounds for separable approximations are obtained for the function . The method used for the derivation of such bounds is based on the generalization of the maximal volume principle for low-rank approximations. Bibliography: 10 titles.

The aim of the paper is to prove in the case of the Fano threefolds and  Golyshev's conjecture on the modularity of the equations for smooth Fano threefolds with Picard group . More precisely, the counting matrices of prime two-pointed invariants of and  are found with the help of a method allowing one to find the Gromov-Witten invariants of complete intersections in varieties for which these invariants are (partially) known. Bibliography: 33 titles.