Table of contents

The centenary of P.S. Novikov's birth provides an inspiring motivation to present, with full proofs and from a modern standpoint, the presumably definitive solutions of some classical problems in descriptive set theory which were formulated by Luzin [Lusin] and, to some extent, even earlier by Hadamard, Borel, and Lebesgue and relate to regularity properties of point sets. The solutions of these problems began in the pioneering works of Aleksandrov [Alexandroff], Suslin [Souslin], and Luzin (1916-17) and evolved in the fundamental studies of Gödel, Novikov, Cohen, and their successors. Main features of this branch of mathematics are that, on the one hand, it is an ordinary mathematical theory studying natural properties of point sets and functions and rather distant from general set theory or intrinsic problems of mathematical logic like consistency or Gödel's theorems, and on the other hand, it has become a subject of applications of the most subtle tools of modern mathematical logic.

Monotone Boolean functions are an important object in discrete mathematics and mathematical cybernetics. Topics related to these functions have been actively studied for several decades. Many results have been obtained, and many papers published. However, until now there has been no sufficiently complete monograph or survey of results of investigations concerning monotone Boolean functions. The object of this survey is to present the main results on monotone Boolean functions obtained during the last 50 years.

This paper is devoted to a detailed description of the notion of boson-fermion correspondence introduced by Coleman and Mandelstam and to applications of this correspondence to integrable and related models. An explicit formulation of this correspondence in terms of massless fermionic fields is given, and properties of the resulting scalar field are studied. It is shown that this field is a well-defined operator-valued distribution on the fermionic Fock space. At the same time, this is a non-Weyl field, and its correlation functions do not exist. Further, realizing a bosonic field as a current of massless (chiral) fermions, we derive a hierarchy of quantum polynomial self-interactions of this field determined by the condition that the corresponding evolution equations of the fermionic fields are linear. It is proved that all the equations of this hierarchy are completely integrable and admit unique global solutions; however, in the classical limit this hierarchy reduces to the dispersionless KdV hierarchy. An application of our construction to the quantization of generic completely integrable interactions is shown by examples of the KdV and mKdV equations for which the quantization procedure of the Gardner-Zakharov-Faddeev bracket is carried out. It is shown that in both cases the corresponding Hamiltonians are sums of two well-defined operators which are bilinear and diagonal with respect to either fermion or boson (current) creation-annihilation operators. As a result, the quantization procedure needs no spatial cut-off and can be carried out on the whole axis of the spatial variable. It is shown that, in the framework of our approach, soliton states exist in the Hilbert space, and the soliton parameters are quantized.