Table of contents

The Fourth European Mathematical Congress will take place in Stockholm from 27 June through 2 July 2004. Further information can be found at the site http://www.math.kth.se/4ecm/.

The chairman of the Scientific Committee is Lennart Carleson, and the chairman of the Organizing Committee is Ari Laptev.

The motto of the congress is: "Mathematics in Science and Technology". Several Nobel prize winners have agreed to address the congress on the role of mathematics in their fields of research. Some satellite conferences are planned to be held around the time of the congress.

At the congress ten awards will be given to the best young mathematicians. Nina Ural'tseva (St. Petersburg) is chairperson of the Prize Committee. The Felix Klein Award will be presented; the formation of the Prize Committee is now complete.

The name of Andrei Nikolaevich Kolmogorov is dear to very many mathematicians who, in one way or another, felt the influence of this remarkable person during the more than sixty years of his research activity. A.N. Kolmogorov is the author of classical results in many branches of mathematics and in contemporary science in general. It is widely acknowledged that he was one of the most outstanding scientists of the 20th century.

The year 2003 was the centenary of the birth of Kolmogorov. A major international conference, "Kolmogorov and Contemporary Mathematics", was dedicated to this event. There were over 1000 participants. The conference programme consisted of talks related to the fields in which Kolmogorov worked.

The conference took place at Moscow State University. Kolmogorov worked at that university throughout his life, as a student and subsequently as a professor in the Mechanics and Mathematics Department. Here he delivered his first talks about many of his remarkable discoveries. Here he founded a unique scientific school of mathematicians who took their first steps in research under his supervision and subsequently became prominent researchers and founders of their own schools.

Kolmogorov was one of the founders of the journal "Uspekhi Matematicheskih Nauk", and its chief editor from 1946 until 1955 and again from 1983 until 1987.

The editorial board has dedicated this and the next issue to A.N. Kolmogorov. This first issue starts with a paper from his files supplied to the editorial board by A.N. Shiryaev. Kolmogorov had prepared it for publication, but for reasons unknown it was not published. The issue is composed mainly of the texts of talks delivered by invited speakers at the conference. This part opens with the paper of A.G. Vitushkin, who was the Kolmogorov Prize laureate for 2003, the award being made to him at the start of the conference.

The paper was apparently written at the end of the 1950s. The text is in its final form, but the list of references was not worked out in detail. Ya.G. Sinai added some comments to the text and included in the list of references some items related to the content of the paper.

Hilbert's thirteenth problem involves the study of solutions of algebraic equations. The object is to obtain a complexity estimate for an algebraic function. As of now, the problem remains open. There are only a few partial algebraic results in this connection, but at the same time the problem has stimulated a series of studies in the theory of functions with their subsequent applications. The most brilliant result in this cycle is Kolmogorov's theorem on superpositions of continuous functions.

Andrei Nikolaevich Kolmogorov firmly believed that in the absence of a rigorous self-contained theory of turbulent fluids and gases one must use hypotheses obtained by processing experimental data. This paper begins with a discussion of the hypothesis of complete self-similarity used in the proof of the widely known (Reynolds-number independent) von Kármán-Prandtl logarithmic law for the distribution of velocity in a turbulent shear flow. It is shown that this hypothesis has not been confirmed experimentally. Instead, a hypothesis of incomplete self-similarity is proposed which leads to a power-law dependence on the Reynolds number. It is shown that this law agrees well with experiments for the most important classes of turbulent shear flows (for flows in pipes and boundary layers).

This paper involves certain types of spatial-temporal models constructed from Lévy bases. The dynamics is described by a field of stochastic processes , on a set of sites , defined as integrals

where denotes a Lévy basis. The integrands are deterministic functions of the form , where has a special form and is a subset of . The first topic is OU (Ornstein-Uhlenbeck) fields , which represent certain extensions of the concept of OU processes (processes of Ornstein-Uhlenbeck type); the focus here is mainly on the potential of for dynamic modelling. Applications to dynamical spatial processes of Cox type are briefly indicated. The second part of the paper discusses modelling of spatial-temporal correlations of SI (stochastic intermittency) fields of the form

This form is useful when explicitly computing expectations of the form

which are used to characterize correlations. The SI fields can be viewed as a dynamical, continuous, and homogeneous generalization of turbulent cascades. In this connection an SI field is constructed with spatial-temporal scaling behaviour that agrees with the energy dissipation observed in turbulent flows. Some parallels of this construction are also briefly sketched.

This paper consists of two parts. The first part traces the history of the boundary problems in probability theory in which Kolmogorov took the most active part. A recently discovered new approach to the solution of boundary problems for random walks satisfying the Cramér condition is presented. This approach is more general, simple, and intuitive than the analytic method proposed by the author in the 1960s. Kolmogorov thought that the construction of such a more general alternative approach was quite desirable because, in his opinion, the purely analytic approach was not quite adequate in essence. The second part of the paper involves the main limit theorems in boundary problems for random walks not satisfying the Cramér condition. A number of recently obtained results are given.

In this paper a new construction of everywhere divergent Fourier-Walsh series is presented. This construction enables one to halve the gap in the Lebesgue-Orlicz classes between the Schipp-Moon lower bound established by using Kolmogorov's construction and the Sjölin upper bound obtained by using Carleson's method. Fourier series which are everywhere divergent after a rearrangement are constructed with respect to the Walsh system (and to more general systems of characters) with the best lower bound for the Weyl factor. Some results related to an upper bound of the majorant for partial sums of series with respect to rearranged multiplicative systems are established. The results thus obtained show certain merits of harmonic analysis on the dyadic group in clarifying and overcoming fundamental difficulties in the solution of the main problems of Fourier analysis.

A constructive proof is given for the classical theorem of Gel'fand and Kolmogorov (1939) characterising the image of the evaluation map from a compact Hausdorff space into the linear space dual to the ring of continuous functions on . Our approach to the proof enabled us to obtain a more general result characterising the image of the evaluation map from the symmetric products into . A similar result holds if and leads to explicit equations for symmetric products of affine algebraic varieties as algebraic subvarieties in the linear space dual to the polynomial ring. This leads to a better understanding of the algebra of multisymmetric polynomials.

The proof of all these results is based on a formula used by Frobenius in 1896 in defining higher characters of finite groups. This formula had no further applications for a long time; however, it has appeared in several independent contexts during the last fifteen years. It was used by A. Wiles and R.L. Taylor in studying representations and by H.-J. Hoehnke and K.W. Johnson and later by J. McKay in studying finite groups. It plays an important role in our work concerning multivalued groups. Several properties of this remarkable formula are described. It is also used to prove a theorem on the structure constants of Frobenius algebras, which have recently attracted attention due to constructions taken from topological field theory and singularity theory. This theorem develops a result of Hoehnke published in 1958. As a corollary, a direct self-contained proof is obtained for the fact that the 1-, 2-, and 3-characters of the regular representation determine a finite group up to isomorphism. This result was first published by Hoehnke and Johnson in 1992.

By using super-Brownian motion, all positive solutions of the non-linear differential equation with in a bounded smooth domain  are characterized by their (fine) traces on the boundary. This solves a problem posed by the author a few years ago. The special case was treated by B. Mselati in 2002.

Some general properties are discussed for normed spaces (or, equivalently, convex bodies) that are finite-dimensional of very high dimension. It turns out that these objects show much less diversity than one could expect a priori. Next, some asymptotic phenomena that govern (in the author's opinion) the asymptotic behaviour of systems of large dimension are analyzed.

Some new aspects are studied for the following general problem: what spectra of frequencies enable one to represent an arbitrary function as a series in the corresponding harmonics. Of particular interest is the `analytic' situation in which only positive frequencies are used.