RUI: Fixed Points in Continua

这个研究项目是基于1912年用布劳威尔不动点定理编写的一个程序。该程序的中心问题是确定是否每一个不分离平面的平面连续体都具有不动点性质。R.H.宾称这是平面拓扑学中最有趣的问题。除了为布劳沃定理提供一个漂亮的推广，这个问题的解决方案将代表着历史研究工作的最后一步，这项工作涉及我们许多最聪明的数学家；P.Alexandroff、R.H.Bing、K.Borsuk和K.Kuratowski都是在这个著名问题上进行了广泛工作的国际巨人。多年来，已经形成了一系列相关的问题，每个问题都具有自己的重要性。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体的不动点性质，并证明了树形连续体的每个变形都有一个不动点。开发的技术应该会在这个项目中取得进一步的进展。拓扑学是研究几何对象在弯曲、折叠、收缩、拉伸、旋转、扭曲或以任何其他方式连续变换时保持不变的属性。例如，当在橡皮筋上标记点并将其拉伸时，点的显示顺序不会更改。其他的拓扑学事实更为微妙。其中包括不动点定理，这是关于物体变换后保持在其原始位置的点的结果。考虑一杯已经搅拌(但没有搅拌)的咖啡，这样液体的表面就会保持在顶部。布劳沃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相同位置。这个定理在文献中有很多有趣的应用。它被用来证明一个著名的关于复多项式的根的存在定理，代数基本定理，以及经济中平衡点的存在。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，它们被用来证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环状的而不是球状的。这个项目将处理这个定理的推广。

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.