Visualizing Complex Projective Spaces and their Applications

【摘要】复变量构成了现代科学几乎所有方面的基础，而复射影空间是由复变量产生的物体和结构的丰富世界的家园。然而，由于高维可视化的巨大困难以及对涉及复杂投影空间的大量相关问题的理解不足，试图描绘这些空间的几何方面的尝试一直很少。现代计算机技术提供了许多非常重要的新工具——符号操作程序、数值优化程序和交互式计算机图形可视化，它们可以为人类的数学智力提供强大的辅助。利用这些工具的好处可以分为两大类：(i)降低门槛，可以这么说，这样更多的人，甚至可能是感兴趣的外行人，可以理解以前无法接触到的数学概念的意义和兴奋；（ii）提高研究成就的潜在水平，以解决使用传统数学研究方法几乎不可能解决的问题。

ABSTRACT0204112Hanson, AndrewIndiana U BloomingtonComplex variables form the basis for virtually every aspect of modern science, and complexprojective spaces are the home to the rich world of objects and structures that arise from com-plexvariables. Yet attempts to depict geometric aspects of these spaces have been scarce due theprodigious difficulty of higher dimensional visualization and poor understanding of the wealth ofrelevant problems that touch on complex projective spaces. Modern computer technology providesa number of very significant new tools symbolic manipulation programs, numerical optimiza-tionprograms, and interactive computer graphics visualizations that can provide powerful as-sistancefor the human mathematical intellect. The benefits of exploiting these tools fall into twomajor categories: (i) lowering the bar, so to speak, so that a larger number of people, possiblyeven interested lay persons, can understand the significance and excitement of previously inacces-siblemathematical concepts; and (ii) raising the potential level of research attainment to allow thesolution of problems that would be nearly impossible using conventional mathematical researchmethods.