The Topology and Arithmetic of Real Algebraic Varieties

这项工作的主要目标是研究连接 真实的代数簇的拓扑和算术之间的关系。 研究者及其合作者的贡献 在真实的代数理论中发现了这种联系 真实的代数簇上的态射和复结构 他 他打算继续朝这个方向进行研究。 另一 进一步研究的主题是光滑的近似 流形的真实的代数簇。 代数簇是几何对象， 由坐标满足的多项式恒等式描述 他们的观点。 “真实的”意味着坐标数是 仅限于普通的真实的数字，避免平方根 减一。 虽然这些都是自然物体， 矛盾的是，治疗它们的理论远没有 令人满意的状态比一个允许虚数， 那些涉及负一的平方根的。 研究者 已作出努力纠正这种情况， 取得了一些可喜的成就。

The principal objective of this work is to study connections between the topology and arithmetic of real algebraic varieties. The contributions of the investigator and his collaborator have uncovered such connections in the theory of real algebraic morphisms and complex structures on real algebraic varieties. He intends to continue his research in this direction. Another topic for further investigation is approximation of smooth manifolds by real algebraic varieties. Algebraic varieties are geometric objects which can be described by polynomial identities satisfied by the coordinates of their points. "Real" means that the coordinate numbers are limited to ordinary real numbers, avoiding the square root of minus one. While these are physically natural objects, paradoxically, the theory for treating them is in a far less satisfactory state than the one permitting imaginary numbers, those involving the square root of minus one. The investigator's efforts have been directed to remedying this situation and have had some gratifying successes.