Table of contents

This paper gives a survey of classical and modern theorems on polynomials, proved using methods of geometric function theory. Most of the paper is devoted to results of the author and his students, established by applying majorization principles for holomorphic functions, the theory of univalent functions, the theory of capacities, and symmetrization. Auxiliary results and the proofs of some of the theorems are presented.

Bibliography: 124 titles.

A new approach is described to the Schubert calculus on complete flag varieties, using the volume polynomial associated with Gelfand-Zetlin polytopes. This approach makes it possible to compute the intersection products of Schubert cycles by intersecting faces of a polytope.

Bibliography: 23 titles.

This paper studies two-dimensional modal logics of a special type, 'Segerberg squares'. They are defined as the usual squares of modal logics with additional connectives corresponding to the diagonal symmetry and the two projections onto the diagonal. For these logics a finite axiomatization is constructed in many cases, and completeness and the finite model property are proved. A translation of Segerberg squares into classical predicate logic is constructed.

Bibliography: 21 titles.