Mathematical Sciences: Structure of Real Algebraic Varietiesand Morphisms

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The project's main goal is to study relationships between the topology of real algebraic varieties and algebro-geometric properties of these varieties. It is divided into the following areas of research: (1) regular mappings (i.e. real algebraic morphisms) and polynomial mappings, (2) algebraic homology and cohomology of real algebraic varieties, (3) comparison between algebraic and topological K-theory of real algebraic varieties. Methods to be applied are the combinations of various techniques of algebraic and differential topology, and algebraic geometry. As opposed to the more common investigations of algebraic varieties over the complex numbers or another algebraically closed field, real algebraic geometry has a more vivid visual appeal and a much less highly developed and satisfactory theory. On the other hand, applications of real geometry may be more natural, for example, applications to the motion of robot arms, where the appropriate interpretation of imaginary coordinate values is ordinarily that the corresponding positions cannot be attained.