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The 19th of January 2012 was the 100th anniversary of the birth of Leonid Vital'evich Kantorovich, an outstanding mathematician and economist of international fame. A child prodigy, who graduated from the university at 18 and became a professor at 20, an academician in the mathematical sciences and a laureate of the Nobel Prize in economics, – these are extraordinary circumstances of his life. They are remarkable in themselves, but also the results he achieved were exceptional and immensely impressive, and the younger generations of researchers, first and foremost mathematicians and economists, must know about them.

A survey is given of results related to the inverse function theorem and to necessary and sufficient first- and second-order conditions for extrema in smooth extremal problems with constraints. The main difference between the results here and the classical ones is that the former are valid and meaningful without a priori normality assumptions.

Bibliography: 48 titles.

In 1942 M.H.A. Newman formulated and proved a simple lemma of great importance for various fields of mathematics, including algebra and the theory of Gröbner-Shirshov bases. Later it was called the Diamond Lemma, since its key construction was illustrated by a diamond-shaped diagram. In 2005 the author suggested a new version of this lemma suitable for topological applications. This paper gives a survey of results on the existence and uniqueness of prime decompositions of various topological objects: three-dimensional manifolds, knots in thickened surfaces, knotted graphs, three-dimensional orbifolds, and knotted theta-curves in three-dimensional manifolds. As it turned out, all these topological objects admit a prime decomposition, although it is not unique in some cases (for example, in the case of orbifolds). For theta-curves and knots of geometric degree 1 in a thickened torus, the algebraic structure of the corresponding semigroups can be completely described. In both cases the semigroups are quotients of free groups by explicit commutation relations.

Bibliography: 33 titles.

A brief survey is given of the classical Langlands programme to construct a correspondence between n-dimensional representations of Galois groups of local and global fields of dimension 1 and irreducible representations of the groups GL(n) connected with these fields and their adelic rings. A generalization of the Langlands programme to fields of dimension 2 is considered and the corresponding version for 1-dimensional representations is described. A conjecture on the direct image of automorphic forms is stated which links the Langlands correspondences in dimensions 2 and 1. In the geometric case of surfaces over a finite field the conjecture is shown to follow from Lafforgue's theorem on the existence of a global Langlands correspondence for curves. The direct image conjecture also implies the classical Hasse-Weil conjecture on the analytic behaviour of the zeta- and L-functions of curves defined over global fields of dimension 1.

Bibliography: 57 titles.

This work presents the state of the art in the theory of potentials for the solutions of systems of linear difference equations, which was proposed by the author in 1969. The role played by difference potentials in the solution of linear difference schemes of general form is for the first time compared in detail to the role played by Cauchy-type integrals in the theory of analytic functions. New vistas are exposed, which are opened up by the theory of difference potentials and arise through combining the universality and algorithmicity of difference schemes with certain properties of Cauchy-type integrals. A brief bibliographical review covers some of the fundamental applications of the theory which have already been implemented.

Bibliography: 61 titles.

On 17 February 2012 Lev Dmitrievich Kudryavtsev passed away. He was a well-known expert in the theory of functions and differential equations, a corresponding member of the Russian Academy of Sciences, a professor and doctor of the physical and mathematical sciences, a laureate of the USSR State Prize and the Prize of the Government of the Russian Federation, and a member of the European Academy of Sciences.

A remarkable mathematician, one of the most prominent specialists in the theory of dynamical systems and bifurcation theory, a laureate of the Lyapunov Prize of the Russian Academy of Sciences and of the Lavren'ev Prize of the National Academy of Sciences of Ukraine, a Humboldt Professor, Head of the Department of Differential Equations of the Research Institute of Applied Mathematics and Cybernetics of Nizhnii Novgorod University, Professor Leonid Pavlovich Shil'nikov passed away on 26 December 2011.