Mathematical Sciences: Structure of Real Algebraic Varietiesand Morphisms

该项目的主要目标是研究实代数变种的拓扑结构与这些变种的代数几何性质之间的关系。它分为以下几个研究领域：(1)正则映射（即实代数态射）与多项式映射；(2)实代数变种的代数同调与上同调；(3)实代数变种的代数k论与拓扑k论的比较。应用的方法是各种代数和微分拓扑技术以及代数几何技术的组合。与对复数或其他代数闭域上的代数变化的更常见的研究相反，真正的代数几何具有更生动的视觉吸引力和更不发达和令人满意的理论。另一方面，实际几何的应用可能更自然，例如，应用于机器人手臂的运动，其中对虚坐标值的适当解释通常是无法获得相应的位置。

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.