This book has a special character. A continuation of the outstanding quality and tradition of Russian mathematical schools at home in new conditions is not at all automatic. The book offers the second volume of papers written by young mathematicians in Russia on their recent achievements. The first volume concentrated on geometry and number theory, while the second volume contains contributions in the field of algebraic geometry, topology and combinatorics. The papers form revised and expanded versions of research articles generally having the character of survey papers. The topics vary a lot; the papers are written by A. E. Guterman (matrices over semi-rings), I. V. Kazachkov (algebraic geometry over Lie algebras), A. V. Malyutin (the Markov destabilization of braids), D. V. Osipov (higher-dimensional local fields and higher adelic theory), T. E. Panov (equivariant topology of torus actions), A.M. Raigorodskii (the Borsuk partition problem), A. B. Skopenkov (embedding and knotting of manifolds) and V. V. Ten (Maxvellian and Boltzmann distributions).