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Friedrich Hirzebruch (17 October 1927 - 27 May 2012) was one of the best-known German mathematicians. He was a member of more than 20 national academies of sciences and, since 1988, a foreign member of the Russian Academy of Sciences.

The 2010 mathematics subject classification used in Mathematical Reviews and Zentralblatt für Mathematik includes the section 53C45 Global surface theory (convex surfaces à la A.D. Aleksandrov). Not even Lobachevsky was shown this degree of recognition by the mathematics community. Aleksandrov became the foremost Russian geometer of the 20th century.

The prominent mathematician Vladimir Aleksandrovich Marchenko is the author of more than 130 publications, including 12 monographs. He has obtained fundamental results in harmonic analysis and the theory of almost periodic functions, the spectral theory of differential and finite-difference operators, the theory of inverse problems of spectral analysis and scattering theory, the spectral theory of large random matrices, the theory of diffraction of electromagnetic waves by periodic structures, homogenization theory for boundary-value problems of mathematical physics in domains with complicated microstructure, and the theory of completely integrable non-linear evolution equations.

This article gives a survey of recent research related to the Monge-Kantorovich problem. Principle results are presented on the existence of solutions and their properties both in the Monge optimal transportation problem and the Kantorovich optimal plan problem, along with results on the connections between both problems and the cases when they are equivalent. Diverse applications of these problems in non-linear analysis, probability theory, and differential geometry are discussed.

Bibliography: 196 titles.

This paper surveys the current state of the theory of cobordism, focusing on geometric and universal properties of complex cobordism, the Landweber-Novikov algebra, and the formal group law of geometric cobordisms. The relationships with K-theory, algebraic cycles, formal group laws, compact Lie group actions on manifolds, toric topology, infinite-dimensional Lie algebras, and nilmanifolds are described. The survey contains key results and open problems.

Bibliography: 124 titles.