RUI: Fixed Points in Continua

RUI：Continua 中的定点

• 批准号: 9619981
• 负责人: Charles Hagopian
• 金额: $ 2.98万
• 依托单位: University Enterprises, Incorporated
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-09-01 至 2000-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9619981&HistoricalAwards=false
• 关键词: RUI; Fixed; Points; Continua

This research project is based on a program that originated in 1912 with the Brouwer fixed-point theorem. The central problem of the program is to determine whether every plane continuum that does not separate the plane has the fixed-point property . R. H. Bing called this the most interesting problem in plane topology. Aside from providing a beautiful generalization to Brouwer's theorem, a solution to this problem would represent the final step in a historical research effort that has involved many of our brightest mathematicians; P. Alexandroff, R. H. Bing, K. Borsuk, and K. Kuratowski are among the international giants who have worked extensively on this famous problem. Through the years a list of related problems has been developed, each of which has taken on a significance of its own. Recently, the PI has solved two of these problems, establishing the fixed-point property for every simply-connected plane continuum and proving that every deformation of a tree-like continuum has a fixed point. The techniques developed should lead to further advances in this program. Topology is the study of properties that persist when geometric objects are bent, folded, shrunk, stretched, turned, twisted, or in any other way continuously transformed. For example, when points are marked on a rubber band and it is stretched, the order in which the points appear does not change. Other topological facts are more subtle. Among these are the fixed-point theorems, results about points that remain in their original position after an object has been transformed. Consider a cup of coffee that has been stirred (but not whipped) so that the surface of the liquid remains on top. The Brouwer fixed-point theorem tells us that when the liquid stops moving, some point on the surface will be in the same place that it was before we started stirring. There are a wide variety of interesting applications of this theorem in the literature. It has been used to prove a well-known theorem on the existence of roots of complex polynomials, the Fundamental Theorem of Algebra, as well as the existence of equilibrium points in an economy. Fixed points of deformations also appear in a variety of scientific applications. For example, in electromagnetic-wave theory, they are used to show that there are no isotropic antennas and explain why most magnetic plasma containers are tori instead of spheres. This project will deal with generalizations of this theorem.

这个研究项目是基于一个程序，起源于1912年与布劳威尔不动点定理。 该程序的中心问题是确定每个不分离平面的平面连续统是否具有不动点性质。 R. H. Bing称这是平面拓扑学中最有趣的问题。 除了为布劳威尔定理提供一个漂亮的推广外，这个问题的解决方案将代表历史研究工作的最后一步，这涉及到我们许多最聪明的数学家; H. Bing，K. Borsuk和K. Kuratowski是在这个著名问题上进行了广泛研究的国际巨头之一。 这些年来，已经形成了一个相关问题的清单，其中每一个问题都有其自身的重要性。 最近，PI已经解决了其中的两个问题，建立了每个单连通平面连续统的不动点性质，并证明了树状连续统的每个变形都有一个不动点。 所开发的技术应导致该计划的进一步发展。 拓扑学是研究几何对象弯曲、折叠、收缩、拉伸、转动、扭曲或以任何其他方式连续变换时所保持的属性。 例如，当在橡皮筋上标记点并拉伸橡皮筋时，点出现的顺序不会改变。 其他拓扑事实则更为微妙。 其中有不动点定理，即关于物体变换后保持在原来位置的点的结果。 考虑一杯经过搅拌（但不是搅打）的咖啡，这样液体的表面就保持在上面。 布劳威尔不动点定理告诉我们，当液体停止运动时，表面上的某个点将处于我们开始搅拌之前的位置。 这个定理在文献中有许多有趣的应用。 它已被用来证明一个著名的定理的存在性根的复多项式，基本定理的代数，以及存在的平衡点的经济。 变形的固定点也出现在各种科学应用中。 例如，在电磁波理论中，它们被用来表明没有各向同性天线 并解释为什么大多数磁性等离子体容器是圆环而不是球体。 这个项目将处理这个定理的推广。